Numerical Mathematics and Advanced Applications ENUMATH 2019

In the frame of high order finite element approximations of PDEs, we are interested in an explicit and efficient way for constructing finite element functions with assigned gradient, curl or divergence in domains with general topology. Three ingredients, that bear the name of their scientific fathers, are involved: the de Rham’s diagram and theorem, Hodge’s decomposition for vectors, Whitney’s differential forms. Some key images are presented in order to illustrate the mathematical concepts.

We discuss the application of a multiscale scheme to a medical flow problem, the so called Candy Wrapper problem. This problem describes the re-stenosis of a stented blood vessel, which will take several months but which is governed by the rapidly oscillating dynamics of the blood flow. A long term simulation of this three dimensional free-boundary flow problem resolving the fast dynamics is not feasible. Our multiscale approach which has been recently published is based on capturing the fast dynamics by locally isolated periodic-in-time problems which have to be approximated once in each macro step of the long term process. Numerical results show the accuracy and efficiency of this multiscale approach.

A general conservation law that defines a class of physical field theories is constructed. First, the notion of a general field is introduced as a formal sum of differential forms on a Minkowski manifold. By the action principle the conservation law is defined for such a general field. By construction, particular field notions of physics, e.g., magnetic flux, electric field strength, stress, strain etc. become instances of the general field. Hence, the differential equations that constitute physical field theories become also instances of the general conservation law. The general field and the general conservation law together correspond to a large class of relativistic hyperbolic physical field models. The parabolic and elliptic models can thereafter be derived by adding constraints. The approach creates solid foundations for developing software systems for scientific computing; the unifying structure shared by the class of field models makes it possible to implement software systems which are not restricted to certain predefined problems. The versatility of the proposed approach is demonstrated by numerical experiments with moving and deforming domains.

In this work we establish the convergence of an adaptation of the fixed-stress split coupling scheme in fractured heterogeneous poro-elastic media. Here, fractures are modeled as possibly non-planar interfaces, and the flow in the fracture is described by a lubrication type system. The flow in the reservoir matrix and in the fracture are coupled to the geomechanics model through a fixed-stress split iteration, in which mass balance equations (for both flow in the matrix and in the fracture) are augmented with fixed-stress split regularization terms. The convergence proof determines the appropriate localized values of these regularization terms.

The magnetohydrodynamic (MHD) flow of an electrically conducting fluid is considered in a long channel of rectangular cross-section along with the z-axis. The fluid is driven by a pressure gradient along the z-axis. The flow is steady, laminar, fully-developed and is influenced by an external magnetic field applied perpendicular to the channel axis. So, the velocity field V = (0, 0, V ) and the magnetic field B = (0, B 0, B) have only channel-axis components V and B depending only on the plane coordinates x and y on the cross-section of the channel which is a rectangular duct. The finite difference method (FDM) is devised to solve the problem tackling mixed type of boundary conditions such as no-slip and insulated walls and both slipping and variably conducting walls. Thus, the numerical results show the effects of the Hartmann number Ha, the conductivity parameter c and the slipping length α on both of the velocity and the induced magnetic field, especially near the walls. It is observed that the well-known characteristics of the MHD flow are also caught.

Optimal control problems constrained by partial differential equations arise in a multitude of important applications. They lead mostly to the solution of very large scale algebraic systems to be solved, which must be done by iterative methods. The problems should then be formulated so that they can be solved fast and robust, which requires the construction of an efficient preconditioner. After reduction of a variable, a two-by-two block matrix system with square blocks arises for which such a preconditioner, PRESB is presented, involving the solution of two algebraic systems which are a linear combination of the matrix blocks. These systems can be solved by inner iterations, involving some available classical solvers to some relative, not very demanding tolerance.

We propose a new model order reduction (MOR) approach to obtain effective reduction for transport-dominated problems or hyperbolic partial differential equations. The main ingredient is a novel decomposition of the solution into a function that tracks the evolving discontinuity and a residual part that is devoid of shock features. This decomposition ansatz is then combined with Proper Orthogonal Decomposition applied to the residual part only to develop an efficient reduced-order model representation for problems with multiple moving and possibly merging discontinuous features. Numerical case-studies show the potential of the approach in terms of computational accuracy compared with standard MOR techniques.

We consider a Darcy-scale model for mineral precipitation and dissolution in a porous medium. This model is obtained by homogenization techniques starting at the scale of pores. The model is based on a phase-field approach to account for the evolution of the pore geometry and the outcome is a multi-scale strongly coupled non-linear system of equations. In this work we discuss a robust numerical scheme dealing with the scale separation in the model as well as the non-linear character of the equations. We combine mesh refinement with stable linearization techniques to illustrate the behaviour of the multi-scale iterative scheme.

We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer et al., A structure-preserving split finite element discretization of the rotating shallow water equations in split Hamiltonian form (2019). https://hal.inria.fr/hal-02020379 ), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes’ properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.

We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge–Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.

This paper addresses model reduction with data assimilation by elaborating on the Parametrized Background Data-Weak (PBDW) approach (Maday et al. Internat J Numer Methods Engrg 102(5):933–965, 2015) recently introduced to combine numerical models with experimental measurements. This approach is here extended to a time-dependent framework by means of a POD-greedy reduced basis construction.

Different types of vegetation barriers are frequently used for reduction of dust and noise levels. The effectivity of the measures depending on the type of used vegetation (decideous, coniferous) is studied in this article. The mathematical model is based on Reynolds—averaged Navier–Stokes (RANS) equations for turbulent fluid flow in Boussinesq approximation completed by the standard k-?? model. Pollutants, considered as passive scalar, were modelled by additional transport equation. An advanced vegetation model was used. The numerical method is based on finite volume formulation. Two fractions of pollutants, PM10 and PM75, emitted from a four–lane highway were numerically simulated. Forty-nine cases of coniferous and deciduous-type forest differing in density, width and height were studied. The main processes that play a role in modelled cases are described. The differences between the effects of coniferous and deciduous trees on pollutants deposition were studied.

We consider a model for two-phase flow in a porous medium posed in a domain consisting of two adjacent regions. The model includes dynamic capillarity and hysteresis. At the interface between adjacent subdomains, the continuity of the normal fluxes and pressures is assumed. For finding the semi-discrete solutions after temporal discretization by the θ-scheme, we proposed an iterative scheme. It combines a (fixed-point) linearization scheme and a non-overlapping domain decomposition method. This article describes the scheme, its convergence and a numerical study confirming this result. The convergence of the iteration towards the solution of the semi-discrete equations is proved independently of the initial guesses and of the spatial discretization, and under some mild constraints on the time step. Hence, this scheme is robust and can be easily implemented for realistic applications.

The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.

We consider the use of multipreconditioning to solve linear systems when more than one preconditioner is available but the optimal choice is not known. In particular, we consider a selective multipreconditioned GMRES algorithm where we incorporate a weighting that allows us to prefer one preconditioner over another. Our target application lies in the simulation of incompressible two-phase flow. Since it is not always known if a preconditioner will perform well within all regimes found in a simulation, we also consider robustness of the multipreconditioning to a poorly performing preconditioner. Overall, we obtain promising results with the approach.

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

The thermally coupled full magnetohydrodynamic (MHD) flow is numerically investigated in a square cavity subject to an externally applied uniform magnetic field. The governing equations given in terms of stream function, vorticity, temperature, magnetic stream function, and current density, are discretized spatially using both the dual reciprocity boundary element method (DRBEM) and the Chebyshev spectral collocation method (CSCM) while an unconditionally stable backward difference scheme is employed for the time integration. Apart from the novelty of the methodology that allows the use of two different methods, the work aims to accommodate various characteristics related to the application of approaches differ in nature and origin. The qualitative and quantitative comparison of the methods are conducted in several test cases. The numerical simulations indicate that the effect of the physical controlling parameters of the MHD problem on the flow and heat transfer can be monitored equally well by both proposed schemes.

Whitney forms—degree one trimmed polynomials—are a crucial tool for finite element analysis of electromagnetic problem. They not only induce several finite element methods, but they also bear interesting geometrical features. If, on the one hand, features of degree one elements are well understood, when it comes to higher degree elements one is forced to choose between an analytical approach and a geometric one, that is, the duality that holds for the lower degree gets lost. Using tools of finite element exterior calculus, we show a correspondence between the usual basis of a high order Whitney forms space and a subset of the weights, that is, degrees of freedom obtained by integration over subsimplices of the mesh.

Uncertainty Quantification (UQ) is an important field to quantify the propagation of uncertainties, analyze sensitivities or realize statistical inversion of a mathematical model. Sampling-based estimation techniques evaluate the model for many different parameter samples. For computationally intensive models, this might require long runtimes or even be infeasible. This so-called multi-query problem can be speeded up or even be enabled with surrogate models from model order reduction (MOR) techniques. For accurate and physically consistent MOR, structure-preserving reduction is essential.We investigate numerically how so-called symplectic model reduction techniques can improve the UQ results for Hamiltonian systems compared to conventional (non-symplectic) approaches. We conclude that the symplectic methods give better results and more robustness with respect to the size of the reduced model.

In this note we introduce a mixed dimensional Stokes-Darcy coupling where a d dimensional Stokes’ flow is coupled to a Darcy model on the d − 1 dimensional boundary of the domain. The porous layer introduces tangential creeping flow along the boundary and allows for the modelling of boundary flow due to surface roughness. This leads to a new model of flow in fracture networks with reservoirs in an impenetrable bulk matrix. Exploiting this modelling capability, we then formulate a fluid-structure interaction method with contact, where the porous layer allows for mechanically consistent contact and release. Physical seepage in the contact zone due to rough surfaces is modelled by the porous layer. Some numerical examples are reported, both on the Stokes’-Darcy coupling alone and on the fluid-structure interaction with contact in the porous boundary layer.

Parabolic fully nonlinear equations may be found in various applications, for instance in optimal portfolio management strategy. A numerical method for the approximation of a canonical parabolic Monge-Ampère equation is investigated in this work. A second order semi-implicit time-stepping method is presented, coupled to safeguarded Newton iterations A low order finite element method is used for space discretization. Numerical experiments exhibit appropriate convergence orders and a robust behavior.

We are interested in the a posteriori error analysis based on locally reconstructed fluxes for the 2D Signorini problem. We start from a P 1-conforming approximation where the contact condition is treated by means of a Nitsche method. We propose an extension of a general approach previously developed for the Laplace operator, allowing to obtain H(div)-conforming conservative fluxes by a local post-process. The reconstructed flux yields an a posteriori error indicator, which is completed by two additional terms taking into account the non-linear contact condition. We then prove the reliability of the indicator, without any additional assumption.

In this work we present an adaptive matrix-free finite element solver for the Cahn–Hilliard equation modelling phase separation in electrode particles of lithium ion batteries during lithium insertion. We employ an error controlled variable-step, variable-order time integrator and a regularity estimator for the adaptive mesh refinement. In particular, we propose a matrix-free applicable preconditioner. Numerical experiments demonstrate the importance of adaptive methods and show for our preconditioner practically no dependence of the number of GMRES iterations on the mesh size, even for locally refined meshes.

We propose a new definition of the normal fracture diffusion-dispersion coefficient for a reduced model of passive transport in fractured porous media.

Mathematical modeling sheds light on cancer research. In addition to reducing animal-based experiments, mathematical modeling is able to provide predictions and prevalidate hypotheses quantitatively. In this work, two different agent-based frameworks regarding cancer modeling are summarised. In contrast, cell-based models focus on the behavior of every single cell and presents the interaction of cells on a small scale, whereas, cellular automata models are used to simulate the interaction of cells with their microenvironment on a large tissue scale.

In certain applications involving the solution of a Bayesian inverse problem, it may not be possible or desirable to evaluate the full posterior, e.g. due to the high computational cost of doing so. This problem motivates the use of approximate posteriors that arise from approximating the data misfit or forward model. We review some error bounds for random and deterministic approximate posteriors that arise when the approximate data misfits and approximate forward models are random.

In this paper we develop and investigate numerical algorithms for solving the fractional powers of discrete elliptic operators A h α U = F $${\mathcal A}_h^\alpha U = F$$ , 0 < α < 1, for F ∈ V h with V h a finite element or finite difference approximation space. Our goal is to construct efficient time stepping schemes for the implementation of the method based on the solution of a pseudo-parabolic problem. The second and fourth order approximations are constructed by using two- and three-level schemes. In order to increase the accuracy of approximations the geometric graded time grid is constructed which compensates the singular behavior of the solution for t close to 0. This apriori adaptive grid is compared with aposteriori adaptive grids. Results of numerical experiments are presented, they agree well with the theoretical results.

This study presents a subgrid artificial viscosity method for approximating solutions to the Boussinesq equations. The stability is obtained by adding a term via an artificial viscosity and then removing it only on the coarse mesh scale. The method includes both vorticity in the viscous term and a grad-div stabilization. We analyze the method from both analytical and computational point of view and show that it is unconditionally stable and optimally convergent. Several numerical experiments are provided that support the derived theoretical results and demonstrate the efficiency and accuracy of the method.

Starting from the linear equations for magneto-elastic coupling, the unit cell problem and the homogenized problem are derived as limits of a two-scale convergence process in a periodic homogenization setting. Exploiting the periodicity of the cell problem and the properties of its Fourier series representation allows for a reformulation as a Lippmann–Schwinger type equation. Iterative algorithms to solve these equations are presented and validated by an analytically solvable test problem.

The physical modeling of transport in multi-component mixtures results in systems of coupled equations for the mass fractions. This contribution discusses the mathematical structure of such transport systems and presents a novel approximation scheme for the associated mass fluxes. The scheme respects the coupled nature of the equations and allows for a linearized source term. An illustrative example is presented.

We present a method for solving optimal control problems constrained by a partial differential equation, where we simultaneously impose sparsity-promoting L 1-regularization on the control as well as box constraints on both the control and the state. We focus on numerical implementation aspects and on preconditioners used when solving the arising linear systems.

We present a time-simultaneous multigrid scheme for parabolic equations that is motivated by blocking multiple time steps together. The resulting method is closely related to multigrid waveform relaxation and is robust with respect to the spatial and temporal grid size and the number of simultaneously computed time steps. We give an intuitive understanding of the convergence behavior and briefly discuss how the theory for multigrid waveform relaxation can be applied in some special cases. Finally, some numerical results for linear and also nonlinear test cases are shown.

Rational Krylov methods are a powerful alternative for computing the product of a function of a large matrix times a given vector. However, the creation of the underlying rational subspaces requires solving sequences of large linear systems, a delicate task that can require intensive computational resources and should be monitored to avoid the creation of subspace different to those required whenever, e.g., the underlying matrices are ill-conditioned. We propose the use of robust preconditioned iterative techniques to speedup the underlying process. We also discuss briefly how the inexact solution of these linear systems can affect the computed subspace. A preliminary test approximating a fractional power of the Laplacian matrix is included.

This paper addresses the numerical solution of the Westervelt equation, which arises as one of the model equations in nonlinear acoustics. The problem is rewritten in a canonical form that allows the systematic discretization by Galerkin approximation in space and time. Exact energy preserving methods of formally arbitrary order are obtained and their efficient realization as well as the relation to other frequently used methods is discussed.

In this work, we further develop multigoal-oriented a posteriori error estimation for the nonlinear, stationary, incompressible Navier-Stokes equations. It is an extension of our previous work on two-side a posteriori error estimates for the DWR method. We now focus on h enrichment and p enrichment for the error estimator. These advancements are demonstrated with the help of a numerical example.

We investigate the feasibility of Bayesian parameter inference for chemical reaction networks described in the low copy number regime. Here stochastic models are often favorable implying that the Bayesian approach becomes natural. Our discussion circles around a concrete oscillating system describing a circadian rhythm, and we ask if its parameters can be inferred from observational data. The main challenge is the lack of analytic likelihood and we circumvent this through the use of a synthetic likelihood based on summarizing statistics. We are particularly interested in the robustness and confidence of the inference procedure and therefore estimates a priori as well as a posteriori the information content available in the data. Our all-synthetic experiments are successful but also point out several challenges when it comes to real data sets.

Block Krylov methods have recently gained a lot of attraction. Due to their increased arithmetic intensity they offer a promising way to improve performance on modern hardware. Recently Frommer et al. (Electron Trans Numer Anal 47:100–126, 2017). presented a block Krylov framework that combines the advantages of block Krylov methods and data parallel methods. We review this framework and apply it on the Block Conjugate Gradients method, to solve linear systems with multiple right hand sides. In this course we consider challenges that occur on modern hardware, like a limited memory bandwidth, the use of SIMD instructions and the communication overhead. We present a performance model to predict the efficiency of different Block CG variants and compare these with experimental numerical results.

We design a hybrid high-order (HHO) method to approximate the Stokes problem on curved domains using unfitted meshes. We prove inf-sup stability and a priori estimates with optimal convergence rates. Moreover, we provide numerical simulations that corroborate the theoretical convergence rates. A cell-agglomeration procedure is used to prevent the appearance of small cut cells.

In this study a new type of non-reflective boundary condition (NRBC) for the Lattice Boltzmann Method (LBM) is proposed; the Non-equilibrium Symmetry Boundary Condition (NSBC). The idea behind this boundary condition is to utilize the characteristics of the non-equilibrium distribution function to assign values to the incoming populations. A simple gradient based extrapolation technique and a far-field criterion are used to predict the macroscopic fluid variables. To demonstrate the non-reflective behaviour of the NSBC, two different tests have been carried out, examining the capability of the boundary to absorb acoustic waves respectively vortices. The results for both tests show that the amount of reflection generated by the NSBC is nearly zero.

The paper deals with the discontinuous Galerkin method (DGM) for the solution of compressible Navier-Stokes equations in the ALE form in time-dependent domains combined with the solution of linear and nonlinear dynamic elasticity. The developed methods are oriented to fluid-structure interaction (FSI), particularly to the simulation of air flow in a time-dependent domain representing vocal tract and vocal folds vibrations. We compare results obtained with the aid of linear and nonlinear elasticity models. The results show that it is more adequate to use the nonlinear elasticity St. Venant-Kirchhoff model in contrast to the linear elasticity model.

We focus on a three-field (displacement-velocity-pressure) stabilized mixed method for poroelasticity based on piecewise trilinear (Q1), lowest order Raviart-Thomas (RT0), and piecewise constant (P0) approximations for displacement, Darcy’s velocity and fluid pore pressure, respectively. Since the selected discrete spaces do not intrinsically satisfy the inf-sup condition in the undrained/incompressible limit, we propose a stabilization strategy based on local pressure jumps. Then, we focus on the efficient solution of the stabilized formulation by a block preconditioned Krylov method. Robustness and efficiency of the proposed approach are demonstrated in two sets of numerical experiments.

Here we present a new approach for the analysis of high-order compact schemes for the clamped plate problem. A similar model is the Navier-Stokes equation in streamfunction formulation. In our book “Navier-Stokes Equations in Planar Domains”, Imperial College Press, 2013, we have suggested fourth-order compact schemes for the Navier-Stokes equations. The same type of schemes may be applied to the clamped plate problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. It is proven that the rate of convergence is actually four, thus the error tends to zero as O(h 4).

We consider a bosonic gas of N bosons. Hartree-Fock approximation allows for a product wave function of single particle solutions Ψ ( x i ) $${\Psi (\vec {x_i})}$$ Ψ ( x 1 , x 2 , ⋯ , x N ) = ∏ i N Ψ ( x i ) $$\displaystyle \Psi (x_1, x_2, \cdots , x_N) = \prod _i^N \Psi (\vec {x}_i) $$ Using a pseudo-potential to account for the condensate self-interaction, the Hamiltonian is found to be H = ∑ i = 1 N − ħ 2 2 m ∂ 2 ∂ x i 2 + V ( x i ) + ∑ i < j 4 π ħ 2 a s m δ ( x i − x j ) , $$\displaystyle H=\sum _{{i=1}}^{N}\left (-{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial {\mathbf {x}}_{i}^{2}}+V({\mathbf {x}}_{i})\right )+\sum _{{i< j}}{4\pi \hbar ^{2}a_{s} \over m}\delta ({\mathbf {x}}_{i}-{\mathbf {x}}_{j}), $$ In this setting m is the mass of the particles, a s is the scattering length of the bosons and ħ = h 2 π $$\hbar = \frac {h}{2\pi }$$ . If all single particle solutions satisfy the governing equation, we arrive at 1 − ħ 2 2 m ∂ 2 ∂ x 2 + V ( x ) + γ | ψ ( x ) | 2 ⏟ H GPE [ ψ ] ( x ) ψ ( x ) = μ ψ ( x ) , $$\displaystyle \begin{aligned} \underbrace {\left (-{\frac {\hbar ^{2}}{2m}}{\partial ^{2} \over \partial {\mathbf {x}}^{2}}+V({\mathbf {x}})+\gamma \vert \psi ({\mathbf {x}})\vert ^{2}\right )}_{H_{\text{GPE}}[\psi ]({\mathbf {x}})} \psi ({\mathbf {x}})=\mu \psi ({\mathbf {x}}),{} \end{aligned} $$ where μ is the chemical potential. Equation (1) is the non-linear Gross-Pitaevskii equation and γ = 4 π ħ 2 a s m $$\gamma ={4\pi \hbar ^{2}a_{s} \over m}$$ . We use a spectral element method to discretise (1). To compute the eigenstates {ψ(x)} of the nonlinear Hamiltonian H GPE, we use two different methods the first is an iterative eigenstate solver and in the second we use a constrained Newton method.

In this paper we present a holistic software approach based on the FEAT3 software for solving multidimensional PDEs with the Finite Element Method that is built for a maximum of performance, scalability, maintainability and extensibility. We introduce basic paradigms how modern computational hardware architectures such as GPUs are exploited in a numerically scalable fashion. We show, how the framework is extended to make even the most recent advances on the hardware market accessible to the framework, exemplified by the ubiquitous trend to customize chips for Machine Learning. We can demonstrate that for a numerically challenging model problem, artificial neural networks can be used while preserving a classical simulation solution pipeline through the incorporation of a neural network preconditioner in the linear solver.

Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.

In this paper, we consider the numerical approximation of the quasi-static, linear Biot model in a 3D domain Ω when the right-hand side of the flow equation is concentrated on a 1D line source δ Λ. This model is of interest in the context of medicine, where it can be used to model flow and deformation through vascularized tissue. The model itself is challenging to approximate as the line source induces the pressure and flux solution to be singular. To overcome this, we here combine two methods: (1) a fixed-stress splitting scheme to decouple the flow and mechanics equations and (2) a singularity removal method for the pressure and flux variables. The singularity removal is based on a splitting of the solution into a lower regularity term capturing the solution singularities and a higher regularity term denoted the remainder. With this in hand, the flow equations can now be reformulated so that they are posed with respect to the remainder terms. The reformulated system is then approximated using the fixed-stress splitting scheme. We conclude by showing the results for a test case simulating flow through vascularized tissue. Here, the numerical method is found to converge optimally using lowest-order elements for the spatial discretization.

In this paper, we discuss the numerical solution of the non-isothermal steady-state Bingham flow considering that the viscosity and the yield limit variate with temperature. In the present contribution, we focus on the asymptotic limit case of high thermal conductivity. In this case, the energy equation collapses into an implicit energy equation which involves the viscosity and the yield stress functions, while the temperature becomes a constant solution for this equation. Once we obtain the coupled limit system of this energy equation and the classical Bingham variational inequality of the second kind, we propose a mixed formulation for the resulting limit variational inequality and a finite element discretization for the resulting system of PDEs. Next, we develop a semismooth Newton algorithm for the coupled flow model. Finally, we carry on several numerical experiments for validating our method.

We consider optimal sensor placement for hyper-parameterized linear Bayesian inverse problems, where the hyper-parameter characterizes nonlinear flexibilities in the forward model, and is considered for a range of possible values. This model variability needs to be taken into account for the experimental design to guarantee that the Bayesian inverse solution is uniformly informative. In this work we link the numerical stability of the maximum a posterior point and A-optimal experimental design to an observability coefficient that directly describes the influence of the chosen sensors. We propose an algorithm that iteratively chooses the sensor locations to improve this coefficient and thereby decrease the eigenvalues of the posterior covariance matrix. This algorithm exploits the structure of the solution manifold in the hyper-parameter domain via a reduced basis surrogate solution for computational efficiency. We illustrate our results with a steady-state thermal conduction problem.

Greedy kernel approximation algorithms are successful techniques for sparse and accurate data-based modelling and function approximation. Based on a recent idea of stabilization (Wenzel et al., A novel class of stabilized greedy kernel approximation algorithms: convergence, stability & uniform point distribution. e-prints. arXiv:1911.04352, 2019) of such algorithms in the scalar output case, we here consider the vectorial extension built on VKOGA (Wirtz and Haasdonk, Dolomites Res Notes Approx 6:83–100, 2013. We introduce the so called γ-restricted VKOGA, comment on analytical properties and present numerical evaluation on data from a clinically relevant application, the modelling of the human spine. The experiments show that the new stabilized algorithms result in improved accuracy and stability over the non-stabilized algorithms.

We consider C 1-continuous approximations of the Kirchhoff plate problem in combination with a mesh dependent augmented Lagrangian method on a simply supported Signorini boundary.

We are interested in solving flow in large tridimensional Discrete Fracture Networks (DFN) with the hybrid high-order (HHO) method. The objectives of this paper are: (1) to demonstrate the benefit of using a high-order method for computing macroscopic quantities, like the equivalent permeability of fracture rocks; (2) to present the computational efficiency of our C++ software, NEF++, which implements the solving of flow in fractures based on the HHO method.

Different parallel two-level overlapping Schwarz preconditioners with Generalized Dryja–Smith–Widlund (GDSW) and Reduced dimension GDSW (RGDSW) coarse spaces for elasticity problems are considered. GDSW type coarse spaces can be constructed from the fully assembled system matrix, but they additionally need the index set of the interface of the corresponding nonoverlapping domain decomposition and the null space of the elasticity operator, i.e., the rigid body motions. In this paper, fully algebraic variants, which are constructed solely from the uniquely distributed system matrix, are compared to the classical variants which make use of this additional information; the fully algebraic variants use an approximation of the interface and an incomplete algebraic null space. Nevertheless, the parallel performance of the fully algebraic variants is competitive compared to the classical variants for a stationary homogeneous model problem and a dynamic heterogenous model problem with coefficient jumps in the shear modulus; the largest parallel computations were performed on 4096 MPI (Message Passing Interface) ranks. The parallel implementations are based on the Trilinos package FROSch.

Computational Fluid Dynamics (CFD) simulations are a numerical tool to model and analyze the behavior of fluid flow. However, accurate simulations are generally very costly because they require high grid resolutions. In this paper, an alternative approach for computing flow predictions using Convolutional Neural Networks (CNNs) is described; in particular, a classical CNN as well as the U-Net architecture are used. First, the networks are trained in an expensive offline phase using flow fields computed by CFD simulations. Afterwards, the evaluation of the trained neural networks is very cheap. Here, the focus is on the dependence of the stationary flow in a channel on variations of the shape and the location of an obstacle. CNNs perform very well on validation data, where the averaged error for the best networks is below 3%. In addition to that, they also generalize very well to new data, with an averaged error below 10%.

The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

In this paper we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.

We present a constrained pressure residual (CPR) two-stage preconditioner applied to a discontinuous Galerkin discretization of a two-phase flow in strongly heterogeneous porous media. We consider a fully implicit, locally conservative, higher order discretization on adaptively generated meshes. The implementation is based on the open-source PDE software framework Dune and its PETSc binding.

The discrete wavelet transform is defined for functions on the entire real line. One way to implement the transform on a finite interval is by using special boundary functions. For orthogonal multiwavelets, this has been studied in previous papers. We describe the generalization of some of these results to biorthogonal multiwavelets.

The convergence rate of classic domain decomposition methods in general deteriorates severely for large discontinuities in the coefficient functions of the considered partial differential equation. To retain the robustness for such highly heterogeneous problems, the coarse space can be enriched by additional coarse basis functions. These can be obtained by solving local generalized eigenvalue problems on subdomain edges. In order to reduce the number of eigenvalue problems and thus the computational cost, we use a neural network to predict the geometric location of critical edges, i.e., edges where the eigenvalue problem is indispensable. As input data for the neural network, we use function evaluations of the coefficient function within the two subdomains adjacent to an edge. In the present article, we examine the effect of computing the input data only in a neighborhood of the edge, i.e., on slabs next to the edge. We show numerical results for both the training data as well as for a concrete test problem in form of a microsection subsection for linear elasticity problems. We observe that computing the sampling points only in one half or one quarter of each subdomain still provides robust algorithms.

This paper is devoted to algebraically stabilized finite element methods for the numerical solution of convection–diffusion–reaction equations. First, the algebraic flux correction scheme with the popular Kuzmin limiter is presented. This limiter has several favourable properties but does not guarantee the validity of the discrete maximum principle for non-Delaunay meshes. Therefore, a generalization of the algebraic flux correction scheme and a modification of the limiter are proposed which lead to the discrete maximum principle for arbitrary meshes. Numerical results demonstrate the advantages of the new method.

In this paper we summarize our results on the investigation of the Kuramoto-Sivashinsky model, which describes the motion of flame/smoldering interface. We propose the generalization of the model formulated in terms of mathematical theory of moving parametrized curves, and investigate it from numerical and analytical point of view. In the part dedicated to computational studies, we present the verification of our scheme by measurement of experimental order of convergence. In the analytical part of the paper we summarize biffurcation analysis of the model and study of rotational wave solutions.

An integrated network consists of a transmission network and at least one distribution network which are connected to each other via a substation. One way to do power flow simulations on these integrated networks is the Master-Slave splitting method. This method splits the integrated network and iterates between the separate transmission (the master) and distribution (the slave) network. In this paper, we extend the method to hybrid networks: a network consisting of a balanced transmission and an unbalanced distribution network. An extra handling is necessary to get the Master-slave splitting to work on hybrid networks. We explain two approaches to use the Master-Slave splitting on a hybrid network and compare these approaches on accuracy, computational time, and convergence, by doing test-simulations. The Master-Slave splitting is interesting when distribution and transmission systems have different characteristics, are in geographically distinct locations, or when system operators are not able or allowed to share data of their network with each other. The extension to hybrid networks makes this method generally applicable and an interesting choice to do power flow simulations on integrated networks.

We review here various results (including own very recent ones) on mesh regularity conditions commonly imposed on simplicial finite element meshes in the interpolation theory and finite element analysis. Several open problems are listed as well.

In numerous applications the mathematical model consists of different processes coupled across a lower dimensional manifold. Due to the multiscale coupling, finite element discretization of such models presents a challenge. Assuming that only singlescale finite element forms can be assembled we present here a simple algorithm for representing multiscale models as linear operators suitable for Krylov methods. Flexibility of the approach is demonstrated by numerical examples with coupling across dimensionality gap 1 and 2. Preconditioners for several of the problems are discussed.

We propose a least squares Galerkin based gradient recovery to approximate Dirichlet problems for strong solutions of linear elliptic problems in nondivergence form and corresponding a priori and a posteriori error bounds. This approach is used to tackle fully nonlinear elliptic problems, e.g., Monge–Ampère, Hamilton–Jacobi–Bellman, using the smooth (vanilla) and the semismooth Newton linearization. We discuss numerical results, including adaptive methods based on the a posteriori error indicators.

The objective of this work is to extend the model reduction technique for coupled 3D-1D elliptic PDEs, previously proposed by the authors, with an a posteriori analysis of the model error, defined as the difference between the solutions of the reference and reduced problem. More precisely, we introduce an estimator for a user-defined functional of the error, computed using a duality approach. This result is particularly useful since it allows to localize the model error on the computational mesh and to investigate the reliability of the model reduction approach.

In an earlier paper, we generalized the CGME (Conjugate Gradient Minimal Error) algorithm to the ℓ 2-regularized weighted least-squares problem. Here, we use this Generalized CGME method to reconstruct images from actual signals measured using a low-field MRI scanner. We analyze the convergence of both GCGME and the classical Generalized Conjugate Gradient Least Squares (GCGLS) method for the simple case when a Laplace operator is used as a regularizer and indicate when GCGME is to be preferred in terms of convergence speed. We also consider a more complicated ℓ 1-penalty in a compressed sensing framework.

Many imaging modalities, such as ultrasound and radar, rely heavily on the ability to accurately model wave propagation. In most applications, the response of an object to an incident wave is recorded and the goal is to characterize the object in terms of its physical parameters (e.g., density or soundspeed). We can cast this as a joint parameter and state estimation problem. In particular, we consider the case where the inner problem of estimating the state is a weakly constrained data-assimilation problem. In this paper, we discuss a numerical method for solving this variational problem.

The main objective of our task is to develop mathematical models, numerical techniques to analyse the thermal effects in electric machines, to implement the developed algorithm in multiprocessor or multi-core environments and to apply them to industrial use cases. In this study, we take into account coupled character of the electromagnetic and thermal features of the physical process. Both thermal and electromagnetic processes are considered transient, solved by means of the FEM method on independent meshes and the time-discretization is realized using time operator splitting. Two examples are presented to assess the accuracy of the developed coupled solvers and the numerical results are compared with the experimental ones, which are obtained from a prototype machine.

In this paper, we propose a new finite element solution approach to the multi-compartmental Darcy equations describing flow and interactions in a porous medium with multiple fluid compartments. We introduce a new numerical formulation and a block-diagonal preconditioner. The robustness with respect to variations in material parameters is demonstrated by theoretical considerations and numerical examples.

Coupling single-carrier networks (SCNs) into multi-carrier energy systems (MESs) has recently become more important. Steady-state load flow analysis of energy systems leads to a system of nonlinear equations, which is usually solved using the Newton-Raphson method (NR). Due to various physical scales within a SCN, and between different SCNs in a MES, scaling might be needed to solve the nonlinear system. In single-carrier electrical networks, per unit scaling is commonly used. However, in the gas and heat networks, various ways of scaling or no scaling are used. This paper presents a per unit system and matrix scaling for load flow models for a MES consisting of gas, electricity, and heat. The effect of scaling on NR is analyzed. A small example MES is used to demonstrate the two scaling methods. This paper shows that the per unit system and matrix scaling are equivalent, assuming infinite precision. In finite precision, the example shows that the NR iterations are slightly different for the two scaling methods. For this example, both scaling methods show the same convergence behavior of NR in finite precision.

A semismooth Newton method (refered as DC–SSN) is proposed for the numerical solution of a class of nonconvex optimal control problems governed by linear elliptic partial differential equations. The nonconvex term in the cost functional arises from a Huber-type local regularization of the L q-quasinorm (q ∈ (0, 1)), therefore it promotes sparsity on the solution. The DC–SSN method solves the optimality system of the regularized problem resulting from the application of difference-of-convex functions programming tools.

We introduce a new strategy for the design of second-order accurate discretizations of non-linear second order operators of Bellman type, which preserves degenerate ellipticity. The approach relies on Selling’s formula, a tool from lattice geometry, and is applied to the Pucci equation, discretized on a two dimensional Cartesian grid. Numerical experiments illustrate the robustness and the accuracy of the method.

In this paper, we demonstrate a multi-scale model for studying blood flow in the vascular structures of an organ. The model may be used for a tracer concentration flow simulation replicating Dynamic Contrast-Enhanced Magnetic Resonance Imaging (DCE–MRI) data. A 1D vascular graph model that represents blood flow through a vascular vessel network is coupled with a single-phase Darcy flow model for the capillary bed which is assumed as a porous media. Numerical experiments show the blood circulation in the system closely related to the structure and parameter of the vascular system, that gives qualitatively realistic tracer concentration flow. This model is a starting point for further investigation in development into clinical applications, using both real data and MRI analysis software.

The barycentric partition of a 3D-cube into tetrahedra is carried out by adding a new node to the body at the centroid point and then, new nodes are progressively added to the centroids of faces and edges. This procedure generates three types of tetrahedra in every single step called, Sommerville tetrahedron number 3 (ST3), isosceles trirectangular tetrahedron and regular right-type tetrahedron. We are interested in studying the number of similarity classes generated when the 8T-LE partition is applied to these tetrahedra.

We consider a cell-based approach in which the balance of momentum is used to predict the impact of cellular forces on the surrounding tissue. To this extent, the elasticity equation and Dirac Delta distributions are combined. In order to avoid the singularity caused by Dirac Delta distribution, alternative approaches are developed and a Gaussian distribution is used as a smoothed approach. Based on the application that the pulling force is pointing inward the cell, the smoothed particle approach is probed as well. In one dimension, it turns out that the aforementioned three approaches are consistent. For two dimensions, we report a computational consistence between the direct and smoothed particle approach.

In implementations of the functional data methods, the effect of the initial choice of an orthonormal basis has not been properly studied. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed functional data and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data. In an attempt to address this issue, we propose a strictly data-driven method of orthonormal basis selection. The method uses B-splines and utilizes recently introduced efficient orthornormal bases called the splinets. The algorithm learns from the data in the machine learning style to efficiently place knots. The optimality criterion is based on the average (per functional data point) mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicate efficiency that could be used to analyze responses to a complex physical system.

In a previous work by the authors a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths was considered and a weak solution of the flow was constructed by means of an implicit variational scheme. Long-time existence of the evolution and convergence to a critical point of the energy were shown. The purpose of this note is to prove uniqueness of the weak solution when p = 2.

We consider the efficient numerical approximation of acoustic wave propagation in time domain by a finite element method with mass lumping. In the presence of internal damping, the problem can be reduced to a second order formulation in time for the velocity field alone. For the spatial approximation we consider H(div)-conforming finite elements of second order. In order to allow for an efficient time integration, we propose a mass-lumping strategy based on approximation of the L 2-scalar product by inexact numerical integration which leads to a block-diagonal mass matrix. A careful error analysis allows to show that second order accuracy is not reduced by the quadrature errors which is illustrated also by numerical tests.

In this work we present a data driven method, used to improve mode-based model order reduction of transport fields with sharp fronts. We assume that the original flow field q(x, t) = f(ϕ(x, t)) can be reconstructed by a front shape function f and a level set function ϕ. The level set function is used to generate a local coordinate, which parametrizes the distance to the front. In this way, we are able to embed the local 1D description of the front for complex 2D front dynamics with merging or splitting fronts, while seeking a low rank description of ϕ. Here, the freedom of choosing ϕ far away from the front can be used to find a low rank description of ϕ which accelerates the convergence of q − f ( ϕ n ) $$\left \Vert q- f(\phi _n) \right \Vert $$ , when truncating ϕ after the nth mode. We demonstrate the ability of this new ansatz for a 2D propagating flame with a moving front.

High concentration of atmospheric dust is a well known risk factor to human health. Vegetative barriers are one of the most popular ways how to substantially reduce the high pollution concentration. Correct modelling of air flow inside the Atmospheric Boundary Layer (ABL) is essential to accurately predict concentration of the passive scalar (dust). The question whether the CFD toolbox OpenFOAM is capable of modelling of this type of problems is tested in the contribution. The results obtained from OpenFOAM were compared simultaneously with the experimental data and the CFD results of the program Atifes, developed at CTU for ABL simulations. It is shown that the recommended setting of OpenFOAM’s atmospheric library has several limitations. Special attention is paid to the different wall functions used in both solvers and the differences are discussed.

Regression models are often employed in prospectivity modeling for the targeting of resources. Logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. In this paper, a model selection procedure based on logistic regression enhanced with nonlinearities is proposed. The method is designed to help the researcher in the model building process and can also be used as preprocessing step for other machine learning algorithms such as neural networks.

Designing freeform optical surfaces that control the redistribution of light from a particular source distribution to a target irradiance poses challenging problems in the field of illumination optics. There exists a wide variety of strategies in academia and industry, and there is an interesting link with optimal transport theory. Many freeform optical design problems can be formulated as a generalized Monge-Ampère equation. In this paper, we consider the design of a single freeform lens that converts the light from an ideal point source into a far-field target. We derive the generalized Monge-Ampère equation and numerically solve it using a generalized least-squares algorithm. The algorithm first computes the optical map and subsequently constructs the optical surface. We show that the numerical algorithm is capable of computing a lens surface that produces a projection of a painting on a screen in the far field.

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

Glaucoma is a multifactorial neurodegenerative disease that involves the optic nerve head and the death of the retinal ganglion cells. The main challenge in medicine is to understand the origin of this degeneration. In this paper we present a virtual clinical study employing the Ocular Mathematical Virtual Simulator (OMVS), a mathematical model, which is able to disentangle the hidden mechanisms and to investigate the causes of this ocular neurodegeneration. In particular, we focus our attention on the influence that intraocular pressure, intracranial pressure and arterial blood pressure set on the ocular hemodynamics.

Peer methods are a comprehensive class of time integrators offering numerous degrees of freedom in their coefficient matrices that can be used to ensure advantageous properties, e.g. A-stability or super-convergence. In this paper, we show that implicit-explicit (IMEX) Peer methods are well-balanced and asymptotic preserving by construction without additional constraints on the coefficients. These properties are relevant when solving (the space discretisation of) hyperbolic systems of balance laws, for example. Numerical examples confirm the theoretical results and illustrate the potential of IMEX-Peer methods.

We develop a method for constructing convergent approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. Our results apply to explicit finite difference schemes and Trotter-Kato splitting formulas, and error estimates are found for schemes approximating solutions of stochastic Hamilton-Jacobi equations.

The methods of real data interpolation can be generalized with fractal interpolation. These fractal interpolation functions can be constructed with the so-called iterated function systems. Local iterated function systems are an important generalization of the classical iterated function systems. In order to obtain new approximation methods this methods can be combined with classical interpolation methods. In this paper we focus on the study of the stochastic local fractal interpolation function in the case of a random data set.

Cervical cancer is caused by the human Papillomavirus (HPV) that attacks the cervix. Cervical cancer globally ranks third as the most frequent cancer among women. In this research, a model of HPV infection in cervical cancer consists of five sub categories of cells, namely susceptible cells, infected cells, pre-cancer cells, cancer cells, and viruses. The study was conducted by forming a model of HPV infection with the addition of treatment controls on pre-cancerous cells. The aim is to minimize the number of pre-cancerous cells while minimizing cost. The HPV infection model with control was solved using Pontryagin’s maximum principle in order to obtain optimal control. Numerical simulations are performed on the differential equations for the cell densities using the fourth order Runge-Kutta method. The simulation results indicate that a smart administration of treatment can be tailored such that the number of pre-cancer cells is minimized at minimal cost. This configuration with a minimal number of pre-cancer cells is favourable since it inhibits the development of cancer cells.

We survey the Nitsche’s master-slave finite element method for elastic contact problems analysed in Gustafsson et al. (SIAM J Sci Comput 42:B425–B446, 2020). The main steps of the error analysis are recalled and numerical benchmark computations are presented.

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanics subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arising from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.

This contribution focuses on CFD modeling of the dynamics of the bubbling fluidized bed under conditions specific for oxyfuel combustion. A custom OpenFOAM solver is developed based on the Multiphase Particle-In-Cell framework for handling the fluid-particle and inter-particle interactions. Features of this Euler-Lagrange approach are discussed, and some of the solver design details are given. Some simulation results of a laboratory-scale combustion device or its combustion chamber are demonstrated, showing the capabilities of the solver. The current limitations and plans for further development are also included.

For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of first and second order in one and two space dimensions. In one dimension, for piecewise constant data in space and explicit Euler in time, the stabilized scheme can even be shown to be monotone. In this contribution, we will examine the conditions for monotonicity in more detail.

This article presents a new approach to solve the equations of flow in heterogeneous porous media by using random walks on regular lattices. The hydraulic head is represented by computational particles which are spread globally from the lattice sites according to random walk rules, with jump probabilities determined by the hydraulic conductivity. The latter is modeled as a realization of a random function generated as a superposition of periodic random modes. One- and two-dimensional numerical solutions are validated by comparisons with analytical manufactured solutions. Further, an ensemble of divergence-free velocity fields computed with the new approach is used to conduct Monte Carlo simulations of diffusion in random fields. The transport equation is solved by a global random walk algorithm which moves computational particles representing the concentration of the solute on the same lattice as that used to solve the flow equations. The integrated flow and transport solution is validated by a good agreement between the statistical estimations of the first two spatial moments of the solute plume and the predictions of the stochastic theory of transport in groundwater.

This paper focus on a finite element approximation of aeroelastic problems. The turbulent flow interacting with flexibly supported airfoil is considered. The flow is described by unsteady Reynolds averaged Navier–Stokes equations, where the main attention is paid to the simulation of turbulent flow with the transition. The motion of the computational domain is addressed and the coupled aeroelastic problem is discretized. Numerical results are shown.

We develop a new second-order accurate operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows problems. The scheme is tested on a benchmark test case with an analytical solution; some of its main features, such as unconditional stability and second-order accuracy, are verified.

The steady magnetohydrodynamic (MHD) flow and heat transfer between parallel plates is considered in which the electrically conducting fluid has temperature dependent properties such as viscosity, thermal and electrical conductivity. The fluid is driven by a constant pressure gradient, and a uniform external transverse magnetic field is applied perpendicular to the plates. The effects of viscous and Joule dissipations are considered in the energy equation, and the fluid is assumed to be slipping in the vicinity of the plates. The effects of the magnetic field, the hydrodynamic slip, and convective thermal boundary conditions on the flow and heat transfer are investigated as well as the temperature dependent parameters. The Chebyshev spectral collocation method which is easy to implement is presented for the approximation of the solutions to the governing equations. The velocity and the temperature of the fluid are obtained with a cheap computational expense.

The Trilinos library LOCA ( http://www.cs.sandia.gov/LOCA/ ) allows computing branches of steady states of large-scale dynamical systems like (discretized) nonlinear PDEs. The core algorithms typically are (pseudo-)arclength continuation, Newton–Krylov methods and (sparse) eigenvalue solvers. While LOCA includes some basic techniques for computing bifurcation points and switching branches, the exploration of a complete bifurcation diagram still takes a lot of programming effort and manual interference.On the other hand, recent developments in algorithms for fully automatic exploration are condensed in PyNCT ( https://pypi.org/project/PyNCT/ ). The scope of this algorithmically versatile software is, however, limited to relatively small (e.g. 2D) problems because it relies on linear algebra from Python libraries like NumPy. Furthermore, PyNCT currently does not support problems with a non-Hermitian Jacobian matrix, which rules out interesting applications in chemistry and fluid dynamics.In this paper we aim to combine the best of both worlds: a high-level implementation of algorithms in PyNCT with parallel models and linear algebra implemented in Trilinos. PyNCT is extended to non-symmetric systems and its complete backend is replaced by the PHIST library ( https://bitbucket.org/essex/phist ), which allows us to use the same underlying HPC libraries as LOCA does.We then apply the new code to a reaction-diffusion model to demonstrate its potential of enabling fully automatic bifurcation analysis on parallel computers.

We present the derivation of the generalized Monge–Ampère equation for two optical systems, viz. a freeform lens with parallel incident and refracted light rays, which transforms a source emittance into a desired target illuminance, and a freeform reflector converting the intensity of a point source into a far-field distribution. The derivations are based on Hamilton’s characteristic functions. We outline a least-squares solution method and apply it to a test problem from laser beam shaping.

Isogeometric Analysis (IgA) can be considered as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. The development of efficient solvers for discretizations arising in IgA is a challenging task, as most (standard) iterative solvers have a detoriating performance for increasing values of the approximation order p of the basis functions. Recently, p-multigrid methods have been developed as an alternative solution strategy. With p-multigrid methods, a multigrid hierarchy is constructed based on the approximation order p instead of the mesh width h (i.e. h-multigrid). The coarse grid correction is then obtained at level p = 1, where B-spline basis functions coincide with standard Lagrangian P 1 basis functions, enabling the use of well known solution strategies developed for the Finite Element Method to solve the residual equation. Different projection schemes can be adopted to go from the high-order level to level p = 1. In this paper, we compare a direct projection to level p = 1 with a projection between each level 1 ≤ k ≤ p in terms of iteration numbers and CPU times. Numerical results, including a spectral analysis, show that a direct projection leads to the most efficient method for both single patch and multipatch geometries.

To benefit from current trends in HPC hardware, such as increasing availability of low precision hardware, we present the concept of prehandling as a direct way of preconditioning and the hierarchical finite element method which is exceptionally well-suited to apply prehandling to Poisson-like problems, at least in 1D and 2D. Such problems are known to cause ill-conditioned stiffness matrices and therefore high computational errors due to round-off. We show by means of numerical results that by prehandling via the hierarchical finite element method the condition number can be significantly reduced (while advantageous properties are preserved) which enables us to obtain sufficiently accurate solutions to Poisson-like problems even if lower computing precision (i.e. single or half precision format) is used.

In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. The second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. The finite element formulation that is based on the variational multiscale concept preserves the linearity of the eigenproblem, and accommodates arbitrary interpolations. Several eigenvalues and eigenfunctions are computed, and then the time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.

Parametric curve models are convenient to describe and quantitatively characterize the contour of objects in bioimages. Unfortunately, designing algorithms to fit smoothly such models onto image data classically requires significant domain expertise. Here, we propose a convolutional neural network-based approach to predict a continuous parametric representation of the outline of biological objects. We successfully apply our method on the Kaggle 2018 Data Science Bowl dataset composed of a varied collection of images of cell nuclei. This work is a first step towards user-friendly bioimage analysis tools that extract continuously-defined representations of objects.

This contribution discusses the performance of time stepping schemes within a data assimilation framework, applied to the method of lines solutions of the non-isothermal compressible gas flow equations. We consider important classes of schemes, namely an embedded explicit Runge–Kutta (ERK) scheme, a diagonally implicit Runge–Kutta (DIRK) scheme, a fully implicit Runge–Kutta (IRK) scheme and a Rosenbrock–Krylov (ROK) scheme. For the numerical illustration, we estimated the flow transients in a subsea pipeline system. Errors from numerical discretization, missing and variability of physical parameters and inaccuracy of initial and boundary conditions are assumed non-Gaussian. Efficiency, robustness and estimation accuracy were evaluated. Results showed that the DIRK scheme is a good compromise between efficiency and robustness. Spurious oscillations were filtered out by the sequential Monte–Carlo algorithm.

Variational formulations of time-dependent PDEs in space and time yield (d + 1)-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other hand, this approach enables adaptivity in space and time as well as model reduction w.r.t. both type of variables. In this paper, we show that matrix oriented techniques can significantly reduce the computational timings for solving the arising linear systems outperforming both time-stepping schemes and other solvers.

We consider a variational formulation of Linear Time-Invariant (LTI)-systems and derive a model reduction in dimension and time inspired by space-time variational reduced basis (RB) methods for parabolic problems. A residual-type RB error estimator is derived whose effectivity is investigated numerically.

We describe the simulation of traffic flows on networks. On individual roads we use standard macroscopic traffic models. The discontinuous Galerkin method in space and a multistep method in time is used for the numerical solution. We introduce limiters to keep the density in an admissible interval as well as prevent spurious oscillations in the numerical solution. To simulate traffic on networks, one should construct suitable numerical fluxes at junctions.

This contribution deals with the numerical simulation of a fluid-structure interaction problem. The elastic body is modelled with the aid of a linear elasticity model. The fluid flow is described by the incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian formulation. The coupling conditions are specified and the coupled problem is formulated. The fluid-structure interaction problem is discretized by the finite element method solver applied both to the elastic part as well as to the fluid flow approximation. For the fluid flow approximation the residual based stabilization is used. Special attention is paid to the penalization boundary condition used at the inlet. It allows to relax an exact value of the inlet velocity on the boundary during channel closing phase nearly to complete channel closure. Numerical results for flow-induced vibrations near the stability boundary are presented and the critical velocity of flutter instability is determined.

We study some discrete boundary value problems which are treated as digital approximation for starting boundary value problem for elliptic pseudo-differential equation. Starting from existence and uniqueness theorem we give a comparison between discrete and continuous solutions for certain boundary value problems.

We propose a mathematical model for the description of plaque progression in carotid arteries. This is based on the coupling of a fluid-structure interaction problem, arising between blood and vessel wall, and differential problems for the cellular evolution. A numerical model is also proposed. This is based on the splitting of the coupled problem based on a suitable strategy to manage the multiscale-in-time nature of the problem. We present some preliminary numerical results both in ideal and real scenarios.

Wrinkling or pattern formation of thin (floating) membranes is a phenomenon governed by buckling instabilities of the membrane. For (post-) buckling analysis, arc-length or continuation methods are often used with a priori applied perturbations in order to avoid passing bifurcation points when traversing the equilibrium paths. The shape and magnitude of the perturbations, however, should not affect the post-buckling response and hence should be chosen with care. In this paper, our primary focus is to develop a robust arc-length method that is able to traverse equilibrium paths and post-bifurcation branches without the need for a priori applied perturbations. We do this by combining existing methods for continuation, solution methods for complex roots in the constraint equation, as well as methods for bifurcation point indication and branch switching. The method has been benchmarked on the post-buckling behaviour of a column, using geometrically non-linear isogeometric Kirchhoff-Love shell element formulations. Excellent results have been obtained in comparison to the reference results, from both bifurcation point and equilibrium path perspective.

We consider a morphoelastic framework that models permanent deformations. The text treats a stability assessment in one dimension and a preservation of symmetry in multiple dimensions. Next, we treat the influence of uncertainty in some of the field variables onto the predicted behaviour of tissue.

We investigate the adaptation of the spectral clustering algorithm to the privacy preserving domain. Spectral clustering is a data mining technique that divides points according to a measure of connectivity in a data graph. When the matrix data are privacy sensitive, cryptographic techniques can be applied to protect the data. A pivotal part of spectral clustering is the partial eigendecomposition of the graph Laplacian. The Lanczos algorithm is used to approximate the eigenvectors of the Laplacian. Many cryptographic techniques are designed to work with positive integers, whereas the numerical algorithms are generally applied in the real domain. To overcome this problem, the Lanczos algorithm is adapted to be performed with fixed-point arithmetic. Square roots are eliminated and floating-point computations are transformed to fixed-point computations. The effects of these adaptations on the accuracy and stability of the algorithm are investigated using standard datasets. The performance of the original and the adapted algorithm is similar when few eigenvectors are needed. For a large number of eigenvectors loss of orthogonality affects the results.

This paper verifies a mathematical model that is developed for the open source CFD-toolbox OpenFOAM, which couples turbulent combustion with conjugate heat transfer. This feature already exists in well-known commercial codes. It permits the prediction of the flame’s characteristics, its emissions, and the consequent heat transfer between fluids and solids via radiation, convection, and conduction. The verification is based on a simplified 2D axisymmetric cylindrical reactor. In the first step, the combustion part of the solver is compared against experimental data for an open turbulent flame. This shows good agreement when using the full GRI 3.0 reaction mechanism. Afterwards, the flame is confined by a cylindrical wall and simultaneously conjugate heat transfer is activated and analysed. It is shown that the combustion and conjugate heat transfer are successfully coupled.

We discuss new local regularity estimates related to the p-Poisson equation −div(A(∇u)) = −divF for p > 2. In the planar case d = 2 we are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the forcing term F to the flux A ( ∇ u ) = ∇ u p − 2 ∇ u $$A(\nabla u)=\left | \nabla u \right |{ }^{p-2}\nabla u$$ . In case of higher dimensions or systems we have a smallness restriction on the corresponding smoothness parameter. Apart from that, our results hold for all reasonable parameter constellations related to weak solutions u ∈ W 1, p( Ω) including quasi-Banach cases with applications to adaptive finite element analysis.

Parameter-dependent discretizations of linear fluid-structure interaction problems can be approached with low-rank methods. When discretizing with respect to a set of parameters, the resulting equations can be translated to a matrix equation since all operators involved are linear. If nonlinear fluid-structure interaction problems are considered, a direct translation to a matrix equation is not possible. We present a method that splits the parameter set into disjoint subsets and, on each subset, computes an approximation of the problem related to the upper median parameter by means of the Newton iteration. This approximation is then used as initial guess for one Newton step on a subset of problems.

In this work, the novel “Tensor Diffusion” approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized “Tensor Stokes” problem, avoiding the need of considering a separate stress tensor in the solution process. Besides fully developed channel flows, the “Tensor Diffusion” approach is evaluated as well in the context of general two-dimensional flow configurations, which are simulated by a suitable four-field formulation of the viscoelastic model respecting the “Tensor Diffusion”.

We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called L-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.

This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a hydro-static pressure as additional unknown besides the displacement field and the phase-field variable. To fulfill the fracture irreversibility constraint, we consider a formulation as a variational inequality in the phase-field variable. For adaptive mesh refinement, we use a recently developed residual-type a posteriori error estimator for the phase-field variational inequality which is efficient and reliable, and robust with respect to the phase-field regularization parameter. The proposed model and the adaptive error-based refinement strategy are demonstrated by means of numerical tests derived from the L-shaped panel test, originally developed for concrete. Here, the Poisson’s ratio is changed from the standard setting towards the incompressible limit ν → 0.5.

When using kernel interpolation techniques for constructing a surrogate model from given data, the choice of interpolation points is crucial for the quality of the surrogate. When dealing with vector-valued target functions which are approximated by matrix-valued kernel models, the selection problem is further complicated as not only the choice of points but also the directions in which the data is projected must be determined.We thus propose variants of Matrix P-greedy algorithms that enable us to iteratively select suitable sets of point-direction pairs with which the approximation space is enriched. We show that the selected pairs result in quasi-optimal convergence rates. Experimentally, we investigate the approximation quality of the different variants.

This article is devoted to presenting efficient solvers for time-periodic parabolic optimization problems. The solvers are based on deriving two-sided bounds for the cost functional. Here, we especially employ the time-periodic nature of the problem discussed in order to obtain fully computable and guaranteed upper and lower bounds for the cost functional. We present the multiharmonic finite element method as a proper approach for deriving a discretized solution of the time-periodic problem. The multiharmonic finite element functions can be used as initial guess for the arbitrary functions in the upper and lower bounds, which then can be minimized and maximized, respectively, in order to obtain an approximate solution of any desired accuracy. Finally, new numerical results are presented in order to show the efficiency of the method discussed also in practice.

We consider a finite element approximation for a system consisting of the evolution of a curve evolving by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The curve evolves inside a given domain Ω ⊂ ℝ 2 $$\Omega \subset \mathbb {R}^2$$ and meets ∂ Ω orthogonally. We present numerical experiments and show the experimental order of convergence of the approximation.

For the second–order wave equation, we compare the Newmark Galerkin method with a stabilised space–time finite element method for tensor–product space–time discretisations with piecewise multilinear, continuous ansatz and test functions leading to an unconditionally stable Galerkin–Petrov scheme, which satisfies a space–time error estimate. We show that both methods require to solve a linear system with the same system matrix. In particular, the stabilised space–time finite element method can be solved sequentially in time as the Newmark Galerkin method. However, the treatment of the right–hand side of the wave equation is different, where the Newmark Galerkin method requires more regularity.

We develop a mathematical model for the interaction of the mechanics of a three-dimensional permeable reservoir or aquifer with the flow through wells. We apply a model reduction technique that represents the wells as one-dimensional channels with arbitrary configuration in the space and we introduce proper coupling conditions to account for the interaction of the wells with the bulk region. The resulting problem consists of coupled partial differential equations defined on manifolds with heterogeneous dimensionality. To highlight the potential of this modeling approach in the description of realistic scenarios, we combine it with a suitable discretization method and we discuss the results of preliminary simulations on an idealized test case containing two wells.