Numerical Mathematics and Advanced Applications ENUMATH 2017

We present an SCRBE PDE App framework for accurate and interactive calculation and visualization of the parametric dependence of the pressure field and associated Quantities of Interest (QoI)—such as impedance and transmission loss—for an extensive family of acoustic duct models. The Static Condensation Reduced Basis Element (SCRBE) partial differential equation (PDE) numerical approach incorporates several principal ingredients: component-to-system model construction, underlying “truth” finite element PDE discretization, (Petrov)-Galerkin projection, static condensation at the component level, parametrized model-order reduction for both the inter-component (port) and intra-component (bubble) degrees of freedom, and offline-online computational decompositions; we emphasize in this paper reduced port spaces and QoI evaluation techniques, especially frequency sweeps, particularly germane to the acoustics context. A PDE App constitutes a Web User Interface (WUI) implementation of the online, or deployed, stage of the SCRBE approximation for a particular parametrized model: User model parameter inputs to the WUI are interpreted by a PDE App Server which then invokes a parallel cloud-based SCRBE Online Computation Server for calculation of the pressure and associated QoI; the Online Computation Server then downloads the spatial field and scalar outputs (as a function of frequency) to the PDE App Server for interrogation and visualization in the WUI by the User. We present several examples of acoustic-duct PDE Apps: the exponential horn, the expansion chamber, and the toroidal bend; in each case we verify accuracy, demonstrate capabilities, and assess computational performance.

We study perfusion by a multiscale model coupling diffusion in the tissue and diffusion along the one-dimensional segments representing the vasculature. We propose a block-diagonal preconditioner for the model equations and demonstrate its robustness by numerical experiments. We compare our model to a macroscale model by Tofts [Modelling in DCE MRI, 2012].

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method, which results in a stable and locally mass-conservative scheme. At each time step one has to solve a non-linear algebraic system, for which one needs adequate iterative solvers. Finding robust ones is particularly challenging here, since the problems considered are double degenerate (i.e. two type of degeneracies are allowed: parabolic-elliptic and parabolic-hyperbolic).Commonly used schemes, like Newton and Picard, are defined either for non-degenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss an iterative linearisation scheme which builds on the L-scheme, and does not employ any regularisation. We prove its rigorous convergence, which is obtained for Hölder type non-linearities. Finally, we present numerical results confirming the theoretical ones, and compare the behaviour of the proposed scheme with schemes based on a regularisation step.

Starting with the discovery of X-rays by Röntgen in 1895, the progress in medical imaging has been extraordinary and immensely beneficial to diagnosis and therapy. Parallel to the increase of imaging accuracy, there is the quest of moving from qualitative to quantitative analysis and patient-tailored therapy. Mathematics, modelling and simulations are increasing their importance as tools in this quest.In this paper we give an overview of relations between mathematical modelling and imaging and focus particularly on the estimation of perfusion in the brain. In the forward model, the brain is treated as a porous medium and a two compartment model (arterial/venous) is used. Motivated by the similarity with techniques in reservoir modelling, we propose an ensemble Kalman filter to perform the parameter estimation and apply the method to a simple example as an illustrative example.

In this paper we give a short note showing convergence rates for periodic approximation of smooth functions by multilevel Gaussian convolution. We will use the Gaussian scaling in the convolution at the finest level as a proxy for degrees of freedom d in the model. We will show that, for functions in the native space of the Gaussian, convergence is of the order d − ln ( d ) ln ( 2 ) $$d^{-\frac {\ln (d)}{\ln (2)}}$$ . This paper provides a baseline for what should be expected in discrete convolution, which will be the subject of a follow up paper.

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has already been proved to be an effective tool for solving interpolation or collocation problems when large data sets are considered. It decomposes the original domain into several subdomains or patches so that only linear systems of relatively small size need to be solved. In research on such partition of unity methods, such subdomains usually consist of spherical patches of a fixed radius. However, for particular data sets, such as track data, ellipsoidal patches seem to be more suitable. Therefore, in this paper, we propose a scheme based on a priori error estimates for selecting the sizes of such variable ellipsoidal subdomains. We jointly solve for both these domain decomposition parameters and the anisotropic RBF shape parameters on each subdomain to achieve superior accuracy in comparison to the standard partition of unity method.

Option contracts under actual market conditions which are more complex than a simple Black-Scholes model are important hedging strategies in the modern financial market. Basket options are attractive products which required the reliable pricing method to take all the beneficial characteristics of a basket option such as correlation effect of underlying assets. The focus of this paper is to present the radial basis function partition of unity method (RBF–PUM) for evaluation of basket options in which underlying assets price follow the Merton jump diffusion model. Numerical experiments are performed for the resulting partial integro-differential equation (PIDE). The resulting valuation method allow for an adaptive space discretization in region near the exercise price to reduce computational cost. The domain truncation effects on the computational error is investigated for the proposed numerical approach. Our numerical examples with two and three underlying assets show that the proposed scheme is accurate, capability of local adaptivity, and efficient in comparison of alternative methods for accurate option prices.

We are interested in approximating vector-valued functions on a compact set Ω ⊂ ℝ d $$\varOmega \subset \mathbb {R}^d$$ . We consider reproducing kernel Hilbert spaces of ℝ m $$\mathbb {R}^m$$ -valued functions which each admit a unique matrix-valued reproducing kernel k. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers n is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the P-Greedy which requires no function evaluations, second and third, the f-Greedy and f∕P-Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.

We present a GPU implementation of a large-scale eigenvalue solver as a part of the ELPA library. We describe the methodology of utilizing the GPU accelerators within an already well optimized MPI-based code. We present numerical results using two different HPC systems equipped with modern GPU accelerators and show the performance benefits of the GPU version.

The Peaceman–Rachford alternating direction implicit (ADI) method is considered for the time-integration of a class of wave-type equations for linear, isotropic materials on a tensorial domain, e.g., a cuboid in 3D or a rectangle in 2D. This method is known to be unconditionally stable and of conventional order two. So far, it has been applied to specific problems and is mostly combined with finite differences in space, where it can be implemented at the cost of an explicit method.In this paper, we consider the ADI method for a discontinuous Galerkin (dG) space discretization. We characterize a large class of first-order differential equations for which we show that on tensorial meshes, the method can be implemented with optimal (linear) complexity.

Methods based on Discontinuous Finite Element approximation (DG FEM) are basically well-adapted to specifics of wave propagation problems in complex media, due to their numerical accuracy and flexibility. However, they still lack of computational efficiency, by reason of the high number of degrees of freedom required for simulations.The Trefftz-DG solution methodology investigated in this work is based on a formulation which is set only at the boundaries of the mesh. It is a consequence of the choice of test functions that are local solutions of the problem. It owns the important feature of involving a space-time approximation which requires using elements defined in the space-time domain.Herein, we address the Trefftz-DG solution of the Elastodynamic System. We establish its well-posedness which is based on mesh-dependent norms. It is worth noting that we employ basis functions which are space-time polynomial. Some numerical experiments illustrate the proper functioning of the method.

We analyze the performance of a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method (Toselli and Widlund, Domain decomposition methods—algorithms and theory. Springer series in computational mathematics, vol 34, 2005), for the efficient solution of very large linear systems arising from elliptic problems discretized by the Virtual Element Method (VEM) (Beirão da Veiga et al., Math Models Methods Appl Sci 24:1541–1573, 2014). We provide numerical experiments on a model linear elliptic problem with highly heterogeneous diffusion coefficients on arbitrary Voronoi meshes, which we modify by adding nodes and edges deriving from the intersection with an unrelated coarse decomposition. The experiments confirm also in this case that the FETI-DP method is numerically scalable with respect to both the problem size and number of subdomains, and its performance is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements.

We develop a hybrid finite volume—finite element method for solving a coupled system of advection-diffusion equations in a bulk domain and on an embedded surface. The method is applied for modeling of flow and transport in fractured porous medium. Fractures in a porous medium are considered as sharp interfaces between the surrounding bulk subdomains. We take into account interaction between fracture and bulk domain. The method is based on a monotone nonlinear finite volume scheme for equations posed in the bulk and a trace finite element method for equations posed on the surface. The surface of fracture is not fitted by the mesh and can cut through the background mesh in an arbitrary way. The background mesh is an octree grid with cubic cells and we get a polyhedral octree mesh with cut-cells after grid-surface intersection. The numerical properties of the hybrid approach are illustrated in a series of numerical experiments.

We design a Hybrid High-Order method for elliptic problems on curved domains. The method uses a cut cell technique for the representation of the curved boundary and imposes Dirichlet boundary conditions using Nitsche’s method. The physical boundary can cut through the cells in a very general fashion and the method leads to optimal error estimates in the H 1-norm.

We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018).

We consider the recently introduced idea of isoparametric unfitted finite element methods and extend it from simplicial meshes to quadrilateral and hexahedral meshes. The concept of the isoparametric unfitted finite element method is the construction of a mapping from a reference configuration to a higher order accurate configuration where the reference configuration is much more accessible for higher order quadrature. The mapping is based on a level set description of the geometry and the reference configuration is a lowest order level set approximation. On simplices this results in a piecewise planar and continuous approximation of the interface. With a simple geometry decomposition quadrature rules can easily be applied based on a tesselation. On hyperrectangles the reference configuration corresponds to the zero level of a multilinear level set function which is not piecewise planar. In this work we explain how to achieve higher order accurate quadrature with only positive quadrature weights also in this case.

The Neumann expansion of Bessel functions (of integer order) of a function g : ℂ → ℂ $$g:\mathbb {C}\rightarrow \mathbb {C}$$ corresponds to representing g as a linear combination of basis functions φ 0, φ 1, …, i.e., g ( s ) = ∑ ℓ = 0 ∞ w ℓ φ ℓ ( s ) $$g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)$$ , where φ i(s) = J i(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

Interior penalty discontinuous Galerkin discretisations (IPDG) and especially the symmetric variant (SIPG) for time-domain wave propagation problems are broadly accepted and widely used due to their advantageous properties. Linear systems with block structure arise by applying space-time discretisations and reducing the global system to time-slab problems. The design of efficient and robust iterative solvers for linear systems from interior penalty discretisations for hyperbolic wave equations is still a challenging task and relies on understanding the properties of the systems. In this work the numerical properties such as the condition number and the distribution of eigenvalues of different representations of the linear systems coming from space-time discretisations for elastic wave propagation are numerically studied. These properties for interior penalty discretisations depend on the penalisation and on the time interval length.

Multilevel methods, such as Geometric and Algebraic Multigrid, Algebraic Multilevel Iteration, Domain Decomposition-type methods have been shown to be the methods of choice for solving linear systems of equations, arising in many areas of Scientific Computing. The methods, in particular the multigrid methods, have been efficiently implemented in serial and parallel and are available via many scientific libraries.The multigrid methods are primarily used as preconditioners for various Krylov subspace iteration methods. They exhibit convergence that is independent or nearly independent on the number of degrees of freedom and can be tuned to be also robust with respect to other problem parameters. Since these methods utilize hierarchical structures, their parallel implementation might exhibit lesser scalability.In this work we utilize a different framework to construct multigrid methods, based on an analytical function representation of the matrix, that keeps the amount of computation high and local, and reduces the memory requirements. This approach is particularly suitable for modern computer architectures. An implementation of the latter for the three-dimensional discrete Laplace operator is derived and implemented. The same function representation technology is used to construct smoothers of sparse approximate inverse type.

We present a space- and phenotype-structured model of selection dynamics between cancer cells within a solid tumour. In the framework of this model, we combine formal analyses with numerical simulations to investigate in silico the role played by the spatial distribution of oxygen and therapeutic agents in mediating phenotypic selection of cancer cells. Numerical simulations are performed on the 3D geometry of an in vivo human hepatic tumour, which was imaged using computerised tomography. Our modelling extends our previous work in the area through the inclusion of multiple therapeutic agents, one that is cytostatic, whilst the other is cytotoxic. In agreement with our previous work, the results show that spatial inhomogeneities in oxygen and therapeutic agent concentrations, which emerge spontaneously in solid tumours, can promote the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. A novel conclusion we infer from the simulations and analysis is that, for the same total dose, therapeutic protocols based on a combination of cytotoxic and cytostatic agents can be more effective than therapeutic protocols relying solely on cytotoxic agents in reducing the number of viable cancer cells.

We assess the uncertainty in a hybrid cell-based, continuum-based model for wound contraction. We explore the correlations between the final contraction of a wound and the stiffness of the tissue, forcing applied by fibroblasts, plastic forces, death rate of cells, differentiation rate of cells, amount of random walk, and the chemotactic strength. Furthermore, we compute the likelihood that serious contractions occur. Although the current model is very simple, the principles can be used to unravel the most important biological mechanisms behind wound contraction.

When using the lower order Whitney elements on a simplicial complex, the matrices describing the external derivative, namely, the differential operators gradient, curl and divergence, are the incidence matrices between edges and vertices, faces and edges, tetrahedra and faces. For higher order Whitney elements, if one adopts degrees of freedom based on moments, the entries of these matrices are still equal to 0, 1 or − 1 but they are no more incidence matrices. If one uses instead the “weights of the field on small simplices” as alternative degrees of freedom, the matrices representative of the external derivative are incidence matrices for any polynomial degree.

A parametrized equation of motion in the absolute coordinate formulation is derived for an elastic body with large rigid motion using continuum mechanics. The resulting PDE is then discretized using linear FEM which results in a high dimensional system. Such high dimensional systems are expensive to solve especially in multi-query settings. Therefore, the system is reduced using a reduced order basis and we investigate the error introduced due to the reduction step. Simulations illustrate the efficacy of the procedure for a pendulum example.

In this paper we investigate two non-local geometric geodesic curvature driven flows of closed curves preserving either their enclosed surface area or their total length on a given two-dimensional surface. The method is based on projection of evolved curves on a surface to the underlying plane. For such a projected flow we construct the normal velocity and the external nonlocal force. The evolving family of curves is parametrized by a solution to the fully nonlinear parabolic equation for which we derive a flowing finite volume approximation numerical scheme. Finally, we present various computational examples of evolution of the surface area and length preserving flows of surface curves. We furthermore analyse the experimental order of convergence. It turns out that the numerical scheme is of the second order of convergence.

In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is then derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.

Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampère type differential equation with transport boundary condition, using properties of geometrical optics, conservation of energy, and the theory of optimal mass transport. We present a least-squares method to compute freeform lens surfaces corresponding to a non-quadratic cost function. The numerical algorithm is capable to compute both convex and concave surfaces.

We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can be applied. We show that this approach has great potential by a simple numerical example of an ideal lens.

Based on the Hellinger-Reissner principle, accurate stress approximations can be computed directly in suitable H(div)-like finite element spaces treating conservation of momentum and the symmetry of the stress tensor as constraints. Two stress finite element spaces of polynomial degree 2 which were proposed in this context will be compared and relations between the two will be established. The first approach uses Raviart-Thomas spaces of next-to-lowest degree and is therefore H(div)-conforming but produces only weakly symmetric stresses. The stresses obtained from the second approach satisfy symmetry exactly but are nonconforming with respect to H(div). It is shown how the latter finite element space can be derived by augmenting the componentwise next-to-lowest Raviart-Thomas space with suitable bubbles. However, the convergence order of the resulting stress approximation is reduced from two to one as will be confirmed by numerical results. Finally, the weak stress symmetry property of the first approach is discussed in more detail and a post-processing procedure for the construction of stresses which are element-wise symmetric on average is proposed.

We suggest an algorithm to generate the topology of load-bearing structures with help of a phase field model. The objective function homogenizes equivalent stress within the isotropic elastic material. However, local inhomogeneities in the stress field, e.g., at concentrated loads, do not distract the convergence of the algorithm. Beside a certain threshold in the equivalent stress field, the desired filling level of the design space is the main parameter of our objective function. The phase field parameter describes the density and stiffness of the substance in a closed interval. An Allen-Cahn equation regulates the phase transition, which is not conserving the mass of the system. The model evolves continuous regions of voids or dense material, whereas voids retain an infinitesimal residual stiffness, which is a million times smaller than the stiffness of the dense material. The evolution of structures is discussed by numerical examples.

We analyze a modified Newton method that was first introduced by Turek and coworkers. The basic idea of the acceleration technique is to split the Jacobian A ′(x) into a “good part” A 1 ′ ( x ) $$A^{\prime }_1(x)$$ and into a troublesome part A 2 ′ ( x ) $$A^{\prime }_2(x)$$ . This second part is adaptively damped if the convergence rate is bad and fully taken into account close to the solution, such that the solver is a blend between a Picard iteration and the full Newton scheme. We will provide first steps in the analysis of this technique and discuss the effects that accelerate the convergence.

We consider the approximation of (generalized) functions with continuous piecewise polynomials or with piecewise polynomials that are allowed to be discontinuous. Best error localization then means that the best error in the whole domain is equivalent to an appropriate accumulation of best errors in small domains, e.g., in mesh elements. We review and compare such best error localizations in the three cases of the Sobolev-Hilbert triplet ( H 0 1 , L 2 , H − 1 ) $$(H^1_0,L^2,H^{-1})$$ .

We present an adaptive Discontinuous Galerkin discretization for the solution of porous media flow problems. The considered flows are immiscible and incompressible. The fully adaptive approach implemented allows for refinement and coarsening in both the element size, the polynomial degree and the time step size.

An adaptive E-scheme for possibly degenerate, viscous conservation laws is presented. The scheme makes use of both given and numerical diffusion to establish the E-property. In the degenerate case it reduces to local Lax–Friedrichs. Both explicit and time-implicit E-schemes are monotone and TVD. Numerical experiments demonstrate the robustness and improved accuracy of the adaptive scheme.

In this paper we consider a class of “filtered” schemes for some first order time dependent Hamilton-Jacobi equations. A typical feature of a filtered scheme is that at the node x j the scheme is obtained as a mixture of a high-order scheme and a monotone scheme according to a filter function F. The mixture is usually governed by F and by a fixed parameter ε = ε(Δt, Δx) > 0 which goes to 0 as (Δt, Δx) is going to 0 and does not depend on n. Here we improve the standard filtered scheme introducing an adaptive and automatic choice of the parameter ε = ε n(Δt, Δx) at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold ε n. The numerical tests presented confirms the effectiveness of the adaptive scheme.

In this article, we consider linear elastic problems, where Poisson’s ratio is close to 0.5 leading to nearly incompressible material behavior. The use of standard linear or d-linear finite elements involves locking phenomena in the considered problem type. One way to overcome this difficulties is given by selective reduced integration. However, the discrete problem differs from the continuous one using this approach. This fact has especially to be taken into account, when deriving a posteriori error estimates. Here, we present goal-oriented estimates based on the dual weighted residual method using only the primal residual due to the linear problem considered. The major challenge is given by the construction of an appropriate numerical approximation of the error identity. Numerical results substantiate the accuracy of the presented estimator and the efficiency of the adaptive method based on it.

The theory behind Nitsche’s method for approximating the obstacle problem of clamped Kirchhoff plates is reviewed. A priori estimates and residual-based a posteriori error estimators are presented for the related conforming stabilised finite element method and the latter are used for adaptive refinement in a numerical experiment.

We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential Runge–Kutta integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order conditions for stochastic exponential Runge–Kutta integrators. The resulting general order theory covers both Itô and Stratonovich integration.

We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.

Using non-commutative shuffle algebra, we outline how the Magnus expansion allows to define explicit stochastic exponentials for matrix-valued continuous semimartingales and Stratonovich integrals.

We show that three classical examples of schemes for the approximation of linear elliptic problems can be cast in a common framework, called the gradient discretisation method (GDM). An error estimate is then obtained by the extension to this framework of the second Strang lemma, which is completed by a second inequality showing that the conditions which are sufficient for the convergence of the method are also necessary.

We introduce new nonconforming finite element methods for elliptic problems of second order. In contrast to previous work, we consider mixed boundary conditions and the domain does not have to lie on one side of its boundary. Each method is quasi-optimal in a piecewise energy norm, thanks to the discretization of the load functional with a moment-preserving smoothing operator.

Recently, we devised an approach to a posteriori error analysis, which clarifies the role of oscillation and where oscillation is bounded in terms of the current approximation error. Basing upon this approach, we derive plain convergence of adaptive linear finite elements approximating the Poisson problem. The result covers arbritray H −1-data and characterizes convergent marking strategies.

We consider the finite element discretization of PDEs on time-dependent domains. Approximation of boundary conditions is one of the crucial aspects, as well as an appropriate approach to adaptive mesh refinement. We present some numerical test results and the application to the thermomechanical simulation of a milling process, where the domain changes in time due to material removal.

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.

Simulation of physical phenomena in networks of fractures is a challenging task, mainly as a consequence of the geometrical complexity of the resulting computational domains, typically characterized by a large number of interfaces, i.e. the intersections among the fractures. The use of numerical strategies that require a mesh conforming to the interfaces is limited by the difficulty of generating such conforming meshes, as a consequence of the large number of geometrical constraints. Here we show how this issue can be effectively tackled by resorting to the Virtual Element Method on polygonal grids. Advection-diffusion-reaction phenomena are considered, also in advection-dominated flow regimes.

A multiscale Hybrid High-Order method has been introduced recently to approximate elliptic problems with oscillatory coefficients. In this work, with a view toward implementation, we describe the general workflow of the method and we present one possible way for accurately approximating the oscillatory basis functions by means of a monoscale Hybrid High-Order method deployed on a fine-scale mesh in each cell of the coarse-scale mesh.

We consider the family of Virtual Elements introduced in Chinosi (Numer Methods Partial Differ Equ 34(4):1117–1144, 2018) for the Reissner-Mindlin plate problem. The family is based on the MITC approach of the FEM context. We analyze the element of degree 2 and compare it with the corresponding finite element MITC9. Moreover we propose a new approximation of the load in order to achieve the proper order of convergence in L 2.

The subject of the paper is the numerical solution of dynamic elasticity problems. We consider linear model and nonlinear Neo-Hookean model. First the continuous dynamic elasticity problem is formulated and then we pay attention to the derivation of the discrete problem. The space discretization is carried out by the discontinuous Galerkin method (DGM). It is combined with the backward difference formula (BDF) for the time discretization. Further, several numerical experiments are presented showing the behaviour of the developed numerical method in dependence on the coefficient in the penalty form. At the end the developed method is applied to the simulation of vibrations of 2D model of human vocal fold formed by four layers with different materials.

In this work, we consider a non-linear extension of the linear, quasi-static Biot’s model. Precisely, we assume that the volumetric strain and the fluid compressibility are non-linear functions. We propose a fully discrete numerical scheme for this model based on higher order space-time elements. We use mixed finite elements for the flow equation, (continuous) Galerkin finite elements for the mechanics and discontinuous Galerkin for the time discretization. We further use the L-scheme for linearising the system appearing on each time step. The stability of this approach is illustrated by a numerical experiment.

We analyze an iterative coupling of mixed and discontinuous Galerkin methods for numerical modelling of coupled flow and mechanical deformation in porous media. The iteration is based on an optimized fixed-stress split along with a discontinuous variational time discretization. For the spatial discretization of the subproblem of flow mixed finite element techniques are applied. The spatial discretization of the subproblem of mechanical deformation uses discontinuous Galerkin methods. They have shown their ability to eliminate locking that sometimes arises in numerical algorithms for poroelasticity and causes nonphysical pressure oscillations.

In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the ∥ ⋅ ∥ L 2 $$\Vert \cdot \Vert _{L^2}$$ -norm and in special ∥⋅∥DG-norm.

Models that involve coupled dynamics in a mixed-dimensional geometry are of increasing interest in several applications. Here, we describe the development of a simulation model for flow in fractured porous media, where the fractures and their intersections form a hierarchy of interacting subdomains. We discuss the implementation of a simulation framework, with an emphasis on reuse of existing discretization tools for mono-dimensional problems. The key ingredients are the representation of the mixed-dimensional geometry as a graph, which allows for convenient discretization and data storage, and a non-intrusive coupling of dimensions via boundary conditions and source terms. This approach is applicable for a wide class of mixed-dimensional problems. We show simulation results for a flow problem in a three-dimensional fracture geometry, applying both finite volume and virtual finite element discretizations.

This paper proposes a new algorithm for fast matrix-free evaluation of linear operators based on hybridizable discontinuous Galerkin discretizations with sum factorization, exemplified for the convection-diffusion equation on quadrilateral and hexahedral elements. The matrix-free scheme is based on a formulation of the method in terms of the primal variable and the trace. The proposed method is shown to be up to an order of magnitude faster than the traditionally considered matrix-based formulation in terms of the trace only, despite using more degrees of freedom. The impact of the choice of basis on the evaluation cost is discussed, showing that Lagrange polynomials with nodes co-located with the quadrature points are particularly efficient.

Based on the benchmark results in Hysing et al (Int J Numer Methods Fluids 60(11):1259–1288, 2009) for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in Adelsberger et al (Proceedings of the 11th world congress on computational mechanics (WCCM XI), Barcelona, 2014). Additionally, we include also an axisymmetric configuration which allows 2.5D simulations and which provides further possibilities for validation and evaluation of numerical multiphase flow components and software tools in 3D.

This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331–355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.

In this paper we study the inverse problem arising in the model describing the transport of two-phase flow in porous media. We consider some physical assumptions so that the mathematical model (direct problem) is an initial boundary value problem for a parabolic degenerate equation. In the inverse problem we want to determine the coefficients (flux and diffusion functions) of the equation from a set of experimental data for the recovery response. We formulate the inverse problem as a minimization of a suitable cost function and we derive its numerical gradient by means of the sensitivity equation method. We start with the discrete formulation and, assuming that the direct problem is discretized by a finite volume scheme, we obtain the discrete sensitivity equation. Then, with the numerical solutions of the direct problem and the discrete sensitivity equation we calculate the numerical gradient. The conjugate gradient method allows us to find numerical values of the flux and diffusion parameters. Additionally, in order to demonstrate the effectiveness of our method, we present a numerical example for the parameter identification problem.

In this work, we present a new implicit scheme for two-phase flow in porous media. The proposed scheme is based on the iterative IMPES (IMplicit Pressure Explicit Saturation) method and, therefore, preserves its efficiency in treatment of nonlinearities, while relaxing the time step condition common for explicit methods. At the same time, it does not involve costly computation of Jacobian matrix required for generic Newtons type methods.Implicit treatment of capillary pressure term ensures the stability and convergence properties of the new scheme. This choice of stabilization is supported by mathematical analysis of the method which also includes the rigorous proof of convergence.Our numerical results indicate that the scheme has superior performance compared with standard IMPES and fully implicit methods on benchmark problems.

This work deals with testing of the Mixed-Hybrid Finite Element Method (MHFEM) for solving two phase flow problems in porous media. We briefly describe the numerical method, it’s implementation, and benchmark problems. First, the method is verified using test problems in homogeneous porous media in 2D and 3D. Results show that the method is convergent and the experimental order of convergence is slightly less than one. However, for the problem in heterogeneous porous media, the method produces oscillations at the interface between different porous media and we demonstrate that these oscillations are not caused by the mesh resolution. To overcome these oscillations, we use the mass lumping technique which eliminates the oscillations at the interface. Tests on the problems in homogeneous porous media show that although the mass lumping technique slightly decreases the accuracy of the method, the experimental order of convergence remains the same.

As demand for water increases across the globe, the availability of freshwater in many regions is likely to decrease due to a changing climate, an increase in human population and changes in land use and energy generation. Many of the world’s freshwater sources are being drained faster than they are being replenished. To solve this problem, new techniques are developed to improve and optimise renewable groundwater sources, which are an increasingly important water supply source globally. One of this emerging techniques is rainwater storage in the subsurface. In this paper, different methods for rainwater infiltration are presented. Furthermore, Monte Carlo simulations are performed to quantify the impact of variation in the soil characteristics and the infiltration parameters on the infiltration rate. Numerical results show that injection pulses may increase the amount of water that can be injected into an aquifer.

Fluid-induced slip of fractures is characterized by strong multiphysics couplings. Three physical processes are considered: Flow, rock deformation and fracture deformation. The fractures are represented as lower-dimensional objects embedded in a three-dimensional domain. Fluid is modeled as slightly compressible, and flow in both fractures and matrix is accounted for. The deformation of rock is inherently different from the deformation of fractures; thus, two different models are needed to describe the mechanical deformation of the rock. The medium surrounding the fractures is modeled as a linear elastic material, while the slip of fractures is modeled as a contact problem, governed by a static-dynamic friction model. We present an iterative scheme for solving the non-linear set of equations that arise from the models, and suggest how the step parameter in this scheme should depend on the shear modulus and mesh size.

In the setting of energy efficient building operation, an optimal boundary control problem governed by a linear parabolic advection-diffusion equation is considered together with bilateral control and state constraints. To keep the temperature in a prescribed range with the less possible heating cost, an economic model predictive control (MPC) strategy is applied. To speed-up the MPC method, a reduced-order approximation based on proper orthogonal decomposition (POD) is utilized. A-posteriori error analysis ensures the quality of the POD models. A numerical test illustrates the efficiency of the proposed strategy.

Motivated by thermal ablation treatments for prostate cancer, the current work investigates the optimal delivery of heat in tissue. The problem is formulated as an optimal control problem constrained by a parametrized partial differential equation (PDE) which models the heat diffusion in living tissue. Geometry and material parameters as well as a parameter entering through the boundary condition are considered. Since there is a need for real-time solution of the treatment planning problem, we introduce a reduced order approximation of the optimal control problem using the reduced basis method. Numerical results are presented that highlight the accuracy and computational efficiency of our reduced model.

In projection-based model reduction (MOR), orthogonal coordinate systems of comparably low dimension are used to produce ansatz subspaces for the efficient emulation of large-scale numerical simulation models. Constructing such coordinate systems is costly as it requires sample solutions at specific operating conditions of the full system that is to be emulated. Moreover, when the operating conditions change, the subspace construction has to be redone from scratch.Parametric model reduction (pMOR) is concerned with developing methods that allow for parametric adaptations without additional full system evaluations. In this work, we approach the pMOR problem via the quasi-linear interpolation of orthogonal coordinate systems. This corresponds to the geodesic interpolation of data on the Stiefel manifold. As an extension, it enables to interpolate the matrix factors of the (possibly truncated) singular value decomposition. Sample applications to a problem in mathematical finance are presented.

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14,259 degrees of freedom. The steady-state snapshot solutions define a reduced order space, which allows to accurately evaluate the steady-state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization. It is shown, how a multilevel static condensation (Karniadakis and Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, Oxford, 2005) in the pressure and velocity boundary degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

The present work considers the optimal control of a convective Cahn-Hilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization.

Two novel approaches are presented for dealing with three dimensional flow simulations in porous media with fractures: one method is based on the minimization of a cost functional to enforce matching conditions at the interfaces, thus allowing for non conforming grids at the interfaces; the other, instead, takes advantage of the new Virtual Elements to easily build conforming polygonal grids at the fracture-porous matrix interfaces. Both methods are designed to minimize the effort in mesh generation process for problems characterized by complex geometries. The methods are described in their simplest fashion in order to keep the notation as compact and simple as possible.

In recent years, the numerical treatment of boundary value problems with the help of polygonal and polyhedral discretization techniques has received a lot of attention within several disciplines. Due to the general element shapes an enormous flexibility is gained and can be exploited, for instance, in adaptive mesh refinement strategies. The Virtual Element Method (VEM) is one of the new promising approaches applicable on general meshes. Although polygonal element shapes may be highly adapted, the analysis relies on isotropic elements which must not be very stretched. But, such anisotropic element shapes have a high potential in the discretization of interior and boundary layers. Recent results on anisotropic polygonal meshes are reviewed and the Virtual Element Method is applied on layer adapted meshes containing isotropic and anisotropic polygonal elements.

In this note we analyse the classical longest-edge n-section algorithm applied to the simplicial partition in R d, and prove that an infinite sequence of simplices violating the Zlámal minimum angle condition, often required in finite element analysis and computer graphics, is unavoidably produced if n ≥ 4. This result implies the fact that the number of different simplicial shapes produced by this version of n-section algorithms is always infinite for any n ≥ 4.

We explore the applicability of a new adaptive stabilized dual-mixed finite element method to a singularly-perturbed convection-diffusion equation with mixed boundary conditions. We establish the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas/Brezzi-Douglas-Marini and continuous piecewise polynomials. We consider a simple a posteriori error indicator and provide some numerical experiments that illustrate the performance of the method.

When discussing shapes of simplices, e.g. in connection with mesh generation and finite element methods, it is important to have a suitable space parametrizing the shapes. In particular, degenerations of different types can appear as boundary components in various ways. For triangles, we will present two natural parametrizing sets that highlight two different types of degenerations, and then combine the properties into a new parametrizing space that allows a good basis for understanding both types of degenerations. The combined space is constructed by the process of blowing up, which in this simple case is introduction of polar coordinates. For tetrahedra, there are many different types of degenerations. In this short paper we will only give one example of what can be achieved by blowing up a natural model, namely to pull apart the two tetrahedral degenerating types known as slivers and caps.

We study the relation between the set of n + 1 vertices of an n-simplex S having ?? n − 1 $$\mathbb {S}^{n-1}$$ as circumsphere, and the set of n + 1 unit outward normals to the facets of S. These normals can in turn be interpreted as the vertices of another simplex Ŝ $$\hat {S}$$ that has ?? n − 1 $$\mathbb {S}^{n-1}$$ as circumsphere. We consider the iterative application of the map that takes the simplex S to Ŝ $$\hat {S}$$ , study its convergence properties, and in particular investigate its fixed points. We will also prove some statements about well-centered simplices in the above context.

In this paper the Synge maximum angle condition for planar triangulations is generalized for higher-dimensional simplicial partitions. In addition, optimal interpolation properties are presented for linear simplicial elements which can degenerate in certain ways.

Biot’s equations of poroelasticity are numerically solved by an Element-based Finite Volume Method (EbFVM). A stabilization technique is advanced to avoid spurious pressure modes in the vicinity of undrained conditions. Classical benchmark problems and more realistic 3D test cases are addressed. The results show that the proposed stabilization is able to eliminate the pressure instabilities preserving the solution accuracy.

We study the numerical solution of the quasi-static linear Biot equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.

The focus of this paper is the numerical solution of a mathematical model for the growth of a permeable biofilm in a microchannel. The model includes water flux inside the biofilm, different biofilm components, and shear stress on the biofilm-water interface. To solve the resulting highly coupled system of model equations, we propose a splitting algorithm. The Arbitrary Lagrangian Eulerian (ALE) method is used to track the biofilm-water interface. Numerical simulations are performed using physical parameters from the existing literature. Our computations show the effect of biofilm permeability on the nutrient transport and on its growth.

The shapes of phospholipid bilayer biomembranes are modeled by the celebrated Canham-Evans-Helfrich model as constrained Willmore minimizers. Several numerical treatments of the model have been proposed in the literature, one of which was used extensively by biophysicists over two decades ago to study real lipid bilayer membranes. While the key ingredients of this algorithm are implemented in Brakke’s well-known surface evolver software, some of its glory details were never explained by either the geometers who invented it or the biophysicists who used it. As such, most of the computational results claimed in the biophysics literature are difficult to reproduce. In this note, we give an exposition of this method, connect it with some related ideas in the literature, and propose a modification of the original method based on replacing mesh smoothing with harmonic energy regularization. We present a theoretical finding and related computational observations explaining why such a smoothing or regularization step is indispensable for the success of the algorithm. A software package called WMINCON is available for reproducing the experiments in this and related articles.

This work deals with the numerical modelling of the different processes related to the phytoremediation methods for remedying heavy metal-contaminated environments. Phytoremediation is a cost-effective plant-based approach of remediation that takes advantage of the ability of plants to concentrate elements and compounds from the environment and to metabolize them in their tissues (toxic heavy metals and organic pollutants are the major targets of phytoremediation). Within the framework of water pollution, biosorption has received considerable attention in recent years because of its advantages: biosorption uses cheap but efficient materials as biosorbents, such as naturally abundant algae. In order to analyse this environmental problem, we propose a two-dimensional mathematical model combining shallow water hydrodynamics with the system of coupled equations modelling the concentrations of heavy metals, algae and nutrients in large waterbodies. Within this novel framework, we present a numerical algorithm for solving the system, and several preliminary computational examples for a simple realistic case.

This work combines numerical modelling, optimization techniques, and optimal control theory of partial differential equations in order to analyze the mitigation of the urban heat island (UHI) effect, which is a very usual environmental phenomenon where the metropolitan areas present a significantly warmer temperature than their surrounding areas, mainly due to the consequences of human activities. At the present time, UHI is considered as one of the major environmental problems in the twenty-first century (undesired result of urbanization and industrialization). Mitigation of the UHI effect can be achieved by using green roofs/walls and lighter-coloured surfaces in urban areas, or (as will be addressed in this study) by setting new green zones inside the city. In order to study the problem, we introduce a well-posed mathematical formulation of the environmental problem (related to the optimal location of green zones in metropolitan areas), we give a numerical algorithm for its resolution, and finally we discuss several numerical results for several realistic 3D examples.

The aim of this paper is to provide some mathematical results for the discrete problem associated to contact with Coulomb friction, in linear elasticity, when finite elements and Nitsche method are considered. We consider both static and dynamic situations. We establish existence and uniqueness results under appropriate assumptions on physical (friction coefficient) and numerical parameters. These results are complemented by a numerical assessment of convergence.

The magnetohydrodynamic (MHD) flow between two parallel slipping and conducting infinite plates containing symmetrically placed electrodes is solved by using the dual reciprocity boundary element method (DRBEM). The flow is driven by the current traveling between the plates and the external magnetic field applied perpendicular to the plates. The coupled MHD equations are solved for the velocity of the fluid and the induced magnetic field as a whole without introducing an iteration. The effects of both the slip ratio and the length of the electrodes are discussed on the flow and magnetic field behavior for increasing values of Hartmann number (Ha). It is found that, an increase in the Hartmann number produces Hartmann layers of thickness 1∕Ha near the conducting parts and shear layers of order of thickness 1 ∕ H a $$1/\sqrt {Ha}$$ in front of the end points of electrodes. When the slip ratio increases Hartmann layers are weakened and the increase in the length of the electrodes retards this weakening effect of the slip on the Hartmann layers. The DRBEM discretizes only a finite portion of the plates and provides the solution inside the infinite region which is mostly concentrated in front of the electrodes. The aim of the study is to numerically simulate the MHD flow under the influence of the slipping velocity on the partly conducting plates which can not be treated theoretically.

Two different methods for the simulation of particulate matter (PM) transport, dispersion and sedimentation on the vegetation are presented. A common sectional model based on a transport equation for each PM size fraction is compared to the innovative model known as moment method. It is based on solving of three transport equations for the moments of the whole PM distribution. Both methods are tested in 2D on a tree patch and in 3D on a hedgerow.The background flow field in the Atmospheric Boundary Layer (ABL) used for both methods is computed by solver based on RANS equations for viscous incompressible flow with stratification due to gravity. The two equations k − ?? turbulence model is used. Three effects of the vegetation are considered: slowdown or deflection of the flow, influence on the turbulence levels inside or near the vegetation and filtering of the particles present in the flow.

For Kirchhoff plate bending problems on domains whose boundaries are curvilinear polygons a discretization method based on the consecutive solution of three second-order problems is presented.In Rafetseder and Zulehner (SIAM J Numer Anal 56(3):1961–1986, 2018) a new mixed variational formulation of this problem is introduced using a nonstandard Sobolev space (and an associated regular decomposition) for the bending moments. In case of a polygonal domain the coupling condition for the two components in the decomposition can be interpreted as standard boundary conditions, which allows for an equivalent reformulation as a system of three (consecutively to solve) second-order elliptic problems.The extension of this approach to curvilinear polygonal domains poses severe difficulties. Therefore, we propose in this paper an alternative approach based on Lagrange multipliers.

In the present paper, a multiobjective optimal control problem governed by a heat equation with time-dependent convection term and bilateral control constraints is considered. For computing Pareto optimal points and approximating the Pareto front, the reference point method is applied. As this method transforms the multiobjective optimal control problem into a series of scalar optimization problems, the method of proper orthogonal decomposition (POD) is introduced as an approach for model-order reduction. New strategies for efficiently updating the POD basis in the optimization process are proposed and tested numerically.

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method’s features.

We have recently developed a third-order limiter function for the reconstruction of cell interface values on equidistant grids (J Sci Comput, 68(2):624–652, 2016). This work now extends the reconstruction technique to non- uniform grids in one space dimension, making it applicable for more elaborate test cases in the context of finite volume schemes.Numerical examples show that the new limiter function maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions.

Various choices of limiters in the framework of algebraic flux correction (AFC) schemes applied to the numerical solution of scalar steady-state convection–diffusion–reaction equations are discussed. A new limiter of upwind type is proposed for which the AFC scheme satisfies the discrete maximum principle and linearity preservation property on arbitrary meshes.

Optimized Schwarz methods use better transmission conditions than the classical Dirichlet conditions that were used by Schwarz. These transmission conditions are optimized for the physical problem that needs to be solved to lead to fast convergence. The optimization is typically performed in the geometrically simplified setting of two unbounded subdomains using Fourier transforms. Recent studies for both homogeneous and heterogeneous domain decomposition methods indicate that the geometry of the physical domain has actually an influence on this optimization process. We study here this influence for an advection diffusion equation in a bounded domain using separation of variables. We provide theoretical results for the min-max problems characterizing the optimized transmission conditions. Our numerical experiments show significant improvements of the new transmission conditions which take the geometry into account, especially for strong tangential advection.

One-level iterative domain decomposition methods share only information between neighboring subdomains, and are thus not scalable in general. For scalability, a coarse space is thus needed. This coarse space can however do more than just make the method scalable: there exists an optimal coarse space in the sense that we have convergence after exactly one coarse correction, and thus the method becomes a direct solver. We introduce and analyze here a new such optimal coarse space for the FETI method for the positive definite Helmholtz equation in one and two space dimensions for strip domain decompositions. We then show how one can approximate the optimal coarse space using optimization techniques. Computational results illustrating the performance and effectiveness of this new coarse space and its approximations are also presented.

In this paper we use a reference trajectory computed by a model predictive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy.

Multiple shooting methods are time domain decomposition methods suitable for solving boundary value problems (BVP). They are based on a subdivision of the time interval and the integration of appropriate initial value problems on this subdivision. In certain critical cases, systematic adaptive techniques to design a proper time domain decomposition are essential. We extend an adaptive shooting points distribution developed in the 1980s for linear boundary value problems based on ordinary differential equations (ODE) to the nonlinear case.

We present an overview of the results of the authors’ paper (Kučera and Shu, IMA J Numer Anal, to appear) on the time growth of the error of the discontinuous Galerkin (DG) method and set them in appropriate context. The application of Gronwall’s lemma gives estimates which grow exponentially in time even for problems where such behavior does not occur. In the case of a nonstationary advection-diffusion equation we can circumvent this problem by considering a general space-time exponential scaling argument. Thus we obtain error estimates for DG which grow exponentially not in time, but in the time particles carried by the flow field spend in the spatial domain. If this is uniformly bounded, one obtains an error estimate of the form C(h p+1∕2), where C is independent of time. We discuss the time growth of the exact solution and the exponential scaling argument and give an overview of results from Kučera and Shu (IMA J Numer Anal, to appear) and the tools necessary for the analysis.

Nonsmoothness of the boundary of polygonal domains limits the regularity of the solutions of elliptic problems. This leads to the presence of the so-called pollution effect in the finite element approximation, which results in a reduced convergence order of the scheme measured in the L 2 and L ∞-norms, compared to the best-approximation order. We show that the energy-correction method, which is known to eliminate the pollution effect in the L 2-norm, yields the same convergence order of the finite element error as the best approximation also in the L ∞-norm. We confirm the theoretical results with numerical experiments.

We consider some correlations between theories of discrete and continuous pseudo-differential equations. The discrete theory is very useful to construct good finite approximations for continuous solutions, and solvability theory for discrete pseudo-differential equations is very similar to the theory of continuous ones. We show certain elements of such a theory, and for simplest cases give comparison estimates.

The non-symmetric coupling for parabolic-elliptic interface problems on Lipschitz domains was recently analysed in Egger et al. (On the non-symmetric coupling method for parabolic-elliptic interface problems, preprint, 2017, arXiv:1711.08487). In Egger et al. (2017, Section 5) a classical FEM-BEM discretisation analysis was provided, but only with polygonal boundaries. In this short paper we will look at the case where the boundary is smooth. We introduce a polygonal approximation of the domain and compute the FEM-BEM coupling on this approximation. Note that the original quasi-optimality cannot be achieved. However, we are able to show a first order convergence result for lowest order FEM-BEM.

We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

We present a novel flux approximation scheme for the viscous Burgers equation. The numerical flux is computed from a local two-point boundary value problem for the stationary equation and requires the iterative solution of a nonlinear equation depending on the local boundary values and the viscosity. In the inviscid limit the scheme reduces to the Godunov numerical flux.

Flow past a circular cylinder embodies many interesting features of fluid dynamics as a challenging fluid phenomenon. In this preliminary study, flow past a cylinder is simulated numerically using a Galerkin procedure based on solenoidal bases. The advantages of using solenoidal bases are twofold: first, the incompressibility condition is exactly satisfied due to the expansion of the flow field in terms of the solenoidal bases and second, the pressure term is eliminated in the process of Galerkin projection onto solenoidal dual bases. The formulation is carried out using a mapped nodal Fourier expansion in the angular variable while a modal polynomial expansion is used in the radial variable. A variational approach to recover the pressure variable is also presented. Some numerical tests are performed.

We introduce a mimetic Cartesian cut-cell method for incompressible viscous flow that conserves mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. In particular we discuss how the no-slip boundary conditions should be applied in a conservative way on objects immersed in the Cartesian mesh. We use the method to compute the flow around a cylinder. We find a good comparison between our results and benchmark results for both a steady and an unsteady test case.