Numerical Mathematics and Advanced Applications 2011

We present an adaptive mimetic finite difference method for the approximate solution of variational inequalities. The adaptive strategy is based on a heuristic hierarchical type error indicator. Numerical experiments that validate the performance of the adaptive MFD method are also presented.

We present a unified approach to build error estimators based on

H

(div)-reconstructed fluxes on the primal mesh, inspired by the hypercircle method. Here, the transport equation is considered and discretized by discontinuous Galerkin, nonconforming and conforming finite elements. We describe the local computation of fluxes on patches, obtain upper error bounds and show some numerical tests.

This work presents an a posteriori error estimation technique for

Q

1

finite elements on quadrilateral triangulations by residual evaluations with respect to biquadratic test functions. The localization is performed in terms of nodal error indicators instead of cell contributions. The reliability and efficiency of the estimator is shown. Further, we discuss a simplified estimator which is even more attractive from the computational point of view.

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed.

We combine the good properties of recovery-based error estimators with the richness of information typical of an anisotropic a posteriori analysis. This merging yields error estimators which are general purpose yet simple and easy to implement, and automatically incorporate detailed geometric information about the computational mesh. This allows us to devise an effective anisotropic mesh adaptation procedure suited to control the discretization error both in the energy norm and in a goal-oriented framework. The advection-diffusion-reaction problem is considered as a computational paradigm.

We give an overview of results related to computable and guaranteed bounds of modeling errors derived with the help of a posteriori estimates of the functional type and discuss estimates of errors arising in dimension reduction, defeaturing (simplification) of highly structured models, and homogenization.

In this work, we present an adaptive finite element method for the numerical simulation of fluid-structure interaction problems using anisotropic meshes. By formulating the coupled problem in a fully monolithic variational Arbitrary Lagrangian Eulerian framework, sensitivities for guiding goal-oriented error estimation are easily at hand. The errors are locally estimated separately in the different element-coordinate directions. This allows for a directional splitting of elements and the generation of anisotropic meshes. The goal-oriented error estimator is applied to a stationary benchmark problem coupling the incompressible Navier-Stokes equations with a nonlinear hyper-elastic material law.

We propose an adaptive finite element method to approximate the solutions to reaction-diffusion systems on time-dependent domains and surfaces. We derive a computable error estimator that provides an upper bound for the error in the semidiscrete (space) scheme. We reconcile our theoretical results with benchmark computations.

We present a moment-based approach for implementing boundary conditions in a lattice Boltzmann formulation of magnetohydrodynamics. Hydrodynamic quantities are represented using a discrete set of distribution functions that evolve according to a cut-down form of Boltzmann’s equation from continuum kinetic theory. Electromagnetic quantities are represented using a set of vector-valued distribution functions. The nonlinear partial differential equations of magnetohydrodynamics are thus replaced by two constant-coefficient hyperbolic systems in which all nonlinearities are confined to algebraic source terms. Further discretising these systems in space and time leads to efficient and readily parallelisable algorithms. However, the widely used bounce-back boundary conditions place no-slip boundaries approximately half-way between grid points, with the precise position being a function of the viscosity and resistivity. Like most lattice Boltzmann boundary conditions, bounce-back is inspired by a discrete analogue of the diffuse and specular reflecting boundary conditions from continuum kinetic theory. Our alternative approach using moments imposes no-slip boundary conditions precisely at grid points, as demonstrated using simulations of Hartmann flow between two parallel planes.

We study a multi-element Discontinuous Galerkin Time Domain (DGTD) method for solving the system of unsteady Maxwell equations. This method is formulated on a non-conforming and hybrid mesh combining a structured (orthogonal, large size elements) quadrangulation of the regular zones of the computational domain with an unstructured triangulation for the discretization of the irregularly shaped objects. The main objective is to enhance the flexibility and the efficiency of DGTD methods. Within each element, the electromagnetic field components are approximated by a high order nodal polynomial, using a centered flux for the surface integrals and a second order Leap-Frog scheme for the time integration of the associated semi-discrete equations. We formulate the 3D discretization scheme, present the results of mathematical analysis (

L

2

stability and a-priori convergence in 3D). Finally, the 2D numerical performance and convergence is demonstrated.

We consider the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc A 457(2016):2983–2991, 2001). In the case of

soft

material, one of the energy contributions is negligible, and the problem becomes degenerate. The analysis and the numerical scheme recently developed in Ferraz-Leite et al. (Numer Math, Accepted for publication, 2012) are not satisfactory in this case. In the present work, we overcome the degeneracy and extend the numerical scheme by introducing a stabilizing energy term. Convergence of the method is established and a numerical experiment concludes the paper.

A second order finite volume scheme for numerical solution of Maxwell’s equations with discontinuous dielectric permittivity on prismatic meshes is suggested. The scheme is based on the approaches of Godunov, Lax-Wendoff and Van Leer. The key feature of the scheme is gradient calculation and limitation that guarantee approximation even near dielectric permittivity discontinuities. Numerical tests confirm second order of approximation of the proposed scheme.

We study the numerical solution of 3d time-harmonic Maxwell’s equations by a hybridizable discontinuous Galerkin method. A hybrid term representing the tangential component of the numerical trace of the magnetic field is introduced. The global system to solve only involves the hybrid term as unknown. We show that the reduced system has properties similar to wave equation discretizations and the tangential components of the numerical traces for both electric and magnetic fields are single-valued. On the example of a plane wave propagation in vacuum the approximate solutions for both electric and magnetic fields have an optimal convergence order.

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, when combined with an explicit integration method to numerically solve a time-dependent partial differential equation, this readily leads to unduly large step size restrictions caused by the smallest grid elements. If the local refinement is strongly localized such that the ratio of fine to coarse elements is small, the unduly step size restrictions can be overcome by blending an implicit and an explicit scheme where only solution variables living at fine elements are implicitly treated. The counterpart of this approach is having to solve a linear system per time step. But due to the assumed small fine to coarse elements ratio, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. We propose to present two locally implicit methods for the time-domain Maxwell’s equations. Our purpose is to compare the two with DG spatial discretization so that the most efficient one can be advocated for future use. Finally we will present a preliminary numerical investigation to increase the order of convergence.

This study focuses on the dissolution and growth of small possibly initially non-smooth particles within a diffusive phase. The dissolution or growth of the particle is assumed to be affected by concentration gradients of a single chemical element within the diffusive phase at the particle boundary caused by diffusion and by an interface reaction. The combined formulation results in a mixed-mode formulation. The moving boundary problem is solved using a level-set method and finite-element techniques such as SUPG. The appropriate meshes are derived using a fixed background mesh and the level-set function. We experimentally show that these techniques give mass-conserving solutions in the limit of infinite resolution, give a linear experimental order of convergence, can handle arbitrary particles and give the possibility to incorporate surface tensions using the Gibbs-Thomson effect and the local curvature.

An iterative method for the simultaneous determination of multiple zeros of algebraic polynomials is stated. This method is more efficient compared to all existing simultaneous methods based on fixed point relations. To attain very high computational efficiency, a suitable correction resulting from Li-Liao-Cheng’s two-point fourth-order method of low computational complexity is applied. The presented convergence analysis shows that the convergence rate of the basic method is increased from three to six using this special type of correction and applying only ν additional polynomial evaluations per iteration, where ν is the number of distinct zeros. Computational aspects and some numerical examples are given to demonstrate high computational efficiency and very fast convergence of the proposed method.

A multilevel sparse kernel-based interpolation (MLSKI) method, suitable for moderately high-dimensional function interpolation problems has been recently proposed in (Georgoulis et al. Multilevel sparse kernel-based interpolation, submitted for publication). The method uses both level-wise and direction-wise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions (RBFs) are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. Here, we investigate the use of conditionally positive definite RBFs within the MLSKI setting, thus extending the results from (Georgoulis et al. Multilevel sparse kernel-based interpolation, submitted for publication), where (strictly) positive definite RBFs are used only.

The time discretization of contact-problems in elasticity is a difficult task, since the non-penetration condition at the contact interface can lead to instabilities in displacements, stresses, and energy. For the case of linear elasticity, in (Deuflhard et al., Int J Numer Methods Eng 73(9):1274–1290, 2008), a contact stabilized Newmark scheme has been proposed, which employs a discrete

L

2

-projection at the contact boundary for stabilization and which can shown to be energy dissipative. Here, we combine this contact-stabilization with an approach presented on (Gonzalez, Comput Methods Appl Mech Eng 190(13–14):1763–1783, 2000) for the time discretization of unconstrained problems in nonlinear mechanics. We apply the resulting combined scheme to contact problems with non-linear non-penetration constraints and non-linear material laws and numerically investigate its behavior. Although our combined scheme is not proven to be energy dissipative, it does not show any decrease in energy and the resulting displacements and forces at the contact boundary show a highly stable behaviour.

The aim of this contribution is the numerical simulation of anisotropic surface diffusion of graphs in the context of the epitaxial growth of quantum dots. The numerical scheme is based on the method of lines where the spatial derivatives are approximated by finite differences (Beneš, Appl Math, 48:437–453, 2003). We then solve the resulting ODE system by means of the adaptive Runge-Kutta-Merson method. Finally, we show computational results with various anisotropy settings leading to singular behaviour.

A multiwavelet basis construction for the interval (0, 1) with the special property that the corresponding wavelet discretization of second order constant coefficient differential operators is sparse, is extended to the realline

$$\mathbb{R}$$

and the half-space

$$\mathbb{R}_{+}$$

. The advantage of these new bases is their very convenient usage within adaptive wavelet schemes applied to operator problems on unbounded domains as performance of these schemes is increased while their implementation is facilitated. The construction is explained and underlined by selected numerical experiments.

In this contribution we consider an optimization problem constrained by a system of state equations coupling the nonstationary model for gray-value transport in an image sequence to the physical model of a transport field resulting in the gray-value evolution. Since in this situation the movement over the boundaries is often unknown, we use a Dirichlet-boundary control formulation for the determination of the transport field.

We propose a numerical method for solving linear Volterra integral equations of the second kind with logarithmic kernels which, in addition to a diagonal singularity, may have a weak boundary singularity. The attainable order of global and local convergence of proposed algorithms is discussed and a collection of numerical results is given.

The present work illustrates a difficulty with the level-set method to accurately capture the curvature of interfaces in regions that are of equal distance to two or more interfaces. Such regions are characterized by kinks in the level-set function where the derivative is discontinuous. Thus the standard discretization scheme is not suitable. Three discretization schemes are outlined that are shown to perform better than the standard discretization on two selected test cases.

Model order reduction of a linear time-invariant system consists in approximating its

p

×

m

rational transfer function

H

(

s

) of high degree by another

p

×

m

rational transfer function

$$\widehat{H}(s)$$

of much smaller degree. Minimizing the

$$\mathcal{H}_{2}$$

-norm of the approximation error can be achieved iteratively. The convergence behavior of the algorithm depends on the choice of the initial condition. If a large scale dynamical system is obtained by discretizing a partial differential equation on a fine mesh, the efficiency can be improved by taking advantage of several discretizations on coarser meshes. This idea is illustrated on the advection–diffusion equation.

A string method for the computation of Hamiltonian trajectories linking two given points is presented, based on the Maupertuis principle; trajectories then correspond to geodesics. For local geodesics, convergence of an algorithm based on Birkhoff’s method has been shown recently in Schwetlick and Zimmer (Submitted). We demonstrate how to extend this approach to global geodesics and thus arbitrary boundary values of the corresponding Hamiltonian problem. Numerical illustrations of the algorithm are given, as well as situations are shown in which the method converges to a degenerate solution.

We propose a new local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations. The discretization contains a crosswind diffusion term which depends on the unknown discrete solution in a nonlinear way. Consequently, the resulting method is nonlinear. Solvability of the nonlinear problem is established and an a priori error estimate in the LPS norm is proved. Numerical results show that the nonlinear crosswind diffusion term leads to a reduction of spurious oscillations.

We propose and study the numerical approximation of an advection-diffusion-reaction model equation by a modified Brezzi–Douglas–Marini mixed finite element method.Nonlinear advection is admitted, arising in complex and coupled flow and transport systems.In contrast to the classical variant of this approach, optimal second-order convergence of the scalar and the vector variable is ensured.No loss of rate of convergence due to the presence of the advection term is observed.

In this paper we consider the efficient numerical approximation of a singularly perturbed parabolic convection-diffusion problem having a convective term which degenerates inside the domain, in the case that the right-hand side of the differential equation is discontinuous on the degeneration line. For small values of the diffusion parameter

$${\varepsilon }^{2}$$

(

$$\varepsilon \in (0,1]$$

), in general, the exact solution has an interior layer in a neighborhood of the degeneration line. We construct a classical finite difference scheme combining the implicit Euler method in time, defined on a uniform mesh, and the first order upwind scheme in space, defined on a piecewise-uniform grid condensing in a neighborhood of the interior layer. Then, the method is an

$$\varepsilon $$

-uniformly convergent scheme of first order in time and almost first order in space. We show the numerical results for a test problem, confirming in practice the theoretical results.

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (Jensen and Smears, On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations, arxiv:1111.5423, 2011); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries.

Stabilized finite element methods for convection-dominated problems contain parameters whose optimal choice is usually not known.This paper presents techniques for computing stabilization parameters in an adaptive way by minimizing a target functional characterizing the quality of the approximate solution.This leads to a constrained nonlinear optimization problem.Numerical results obtained for various target functionals are presented.They demonstrate that a posteriori optimization of parameters can significantly improve the quality of solutions obtained using stabilized methods.

A boundary value problem for singularly perturbed differential-difference equation with turning point is considered. Some a priori estimates are obtained on the solution and its derivatives. In general, to tackle such type of problems one encounters three difficulties: (i) due to presence of the turning point, (ii) due to presence of terms containing shifts and (iii) due to presence of the singular perturbation parameter. Due to presence of the singular perturbation parameter the classical numerical methods fail to give reliable numerical results and do not converge uniformly with respect to the singular perturbation parameter. In this paper a parameter uniform finite difference scheme is constructed to solve the boundary-value problem. A parameter uniform error estimate for the numerical scheme so constructed is established. Numerical experiments are carried out to demonstrate the efficiency of the numerical scheme and support the theoretical estimates.

For a model Dirichlet problem to a singularly perturbed ordinary differential convection-diffusion equation, we discuss a “standard” approach to the construction of difference schemes that use standard grid approximations on uniform grids, the step-size of which is chosen sufficiently small for small values of a perturbation parameter

$$\varepsilon $$

,

$$\varepsilon \in (0,1]$$

. It is shown that such a scheme, under its convergence in the maximum norm theoretically proved, is not

$$\varepsilon $$

-uniformly stable to perturbations in the data of the discrete problem. When perturbations take place and the parameter

$$\varepsilon $$

decreases, the actual accuracy of the computed solutions

may deteriorate up to a full accuracy loss

for sufficiently small values of

$$\varepsilon $$

, namely, under the condition

$$t = \mathcal{O}(\ln {\varepsilon }^{-1})$$

, where

t

is the number of computer word digits.

For a Dirichlet problem for an one-dimensional singularly perturbed parabolic convection-diffusion equation, a difference scheme of the solution decomposition method is constructed. This method involves a special decomposition based on the asymptotic construction technique in which the regular and singular components of the grid solution are solutions of grid subproblems solved on

uniform grids

, moreover, the coefficients of the grid equations do not depend on the singular component of the solution unlike the fitted operator method. The constructed scheme converges in the maximum norm

$$\varepsilon $$

-uniformly (i.e., independent of a perturbation parameter

$$\varepsilon $$

,

$$\varepsilon \in (0,1]$$

) at the rate

$$\mathcal{O}\left ({N}^{-1}\ln N + N_{0}^{-1}\right )$$

the same as a scheme of the condensing grid method on a piecewise-uniform grid (here

N

and

N

0

define the numbers of the nodes in the spatial and time meshes, respectively).

For the stationary advection-diffusion problem the standard continuous Galerkin method is unstable without some additional control on the mesh or method. The interior penalty discontinuous Galerkin method is more stable but at the expense of an increased number of degrees of freedom. The hybrid method proposed in [

5

] combines the computational complexity of the continuous method with the stability of the discontinuous method without a significant increase in degrees of freedom. We discuss the implementation of this method using the finite element library

deal.ii

and present some numerical experiments.

The incompressible miscible displacement problem has attracted interest in recent years as it models economically important activities such as oil recovery and groundwater flow. It is important that numerical simulations can accurately model the types of problems seen in industry. We discuss a posteriori finite element indicators for the incompressible miscible displacement problem and propose an extension to a mixed-discontinuous Galerkin scheme. Furthermore we highlight some physically realistic scenarios not covered by the existing analysis and outline the theory of weighted spaces required to address them.

In this article we consider the a posteriori error estimation and adaptive mesh refinement of

hp

–version discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier–Stokes equations.Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or

O

(2) symmetry.Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems.Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on

hp

–adaptively refined computational meshes are presented.

We develop the a posteriori error analysis, with respect to a mesh–dependent energy norm, of two-grid

hp

–version discontinuous Galerkin finite element methods for quasi-Newtonian flows. The performance of the proposed estimators within an

hp

–adaptive refinement procedure is studied through a numerical experiment.

We derive a goal-oriented a posteriori error estimate for hp-adaptive discontinuous Galerkin discretizations of convection-diffusion eigenvalue problems. We consider one-irregular meshes consisting of parallelograms. The estimate yields very accurate measurements of the errors in the two target functionals considered in this paper. The accuracy of our error estimator is also confirmed by the effectivity index very close to 1 in all numerical tests. We apply our goal-oriented estimator as an error indicator in an anisotropic hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples.

The subject of this paper is the numerical solution of the interaction of compressible flow and an elastic body with a special emphasis on the simulation of vibrations of vocal folds during phonation onset. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method and the compressible Navier-Stokes equations are written in the ALE form. The deformation of the elastic body, caused by the aeroelastic forces, is described by the linear dynamical elasticity equations. Both these systems are coupled by transmission conditions. For the space-discretization of the flow problem the discontinuous Galerkin finite element method (DGFEM) is used. The time-discretization is realized by the backward difference formula (BDF). The structural problem is discretized by the conforming finite element method and the Newmark method. The results of the use of two different couplings and their comparison are presented.

In this paper we present the analysis of the discontinuous Galerkin (DG) finite element method applied to a nonstationary nonlinear convection-diffusion problem. Using the technique of Zhang and Shu (SIAM J Numer Anal 42(2):641–666, 2004), originally for explicit schemes, we prove apriori error estimates uniform with respect to the diffusion coefficient and valid even in the purely convective case. We extend the cited analysis to the method of lines using continuous mathematical induction and a nonlinear Gronwall-type lemma. For an implicit scheme, we prove that there does not exist a Gronwall-type lemma capable of proving the desired estimates using standard arguments. Next, we use a suitable continuation of the implicit solution and use continuous mathematical induction to prove error estimates under a CFL-like condition.

We derive a two-sided error bound for the nonstationary heat equation with mixed Dirichlet/Neumann boundary conditions. The space semi-discretization is carried out with the aid of the interior penalty discontinuous Galerkin methods and the backward Euler method is employed for the time discretization. The approach is based on the Helmholtz decomposition and the averaging interpolation operator. The behavior of derived estimates is demonstrated on a numerical example.

We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the

discretize-then-optimize (DO)

and the

optimize-then-discretize (OD)

approach lead to the same discrete systems and in both approaches the observed

L

2

convergence for states and controls is

$$O({h}^{k+1})$$

, where

k

is the degree of polynomials used. The situation is different for NIPG and IIPG, where the the DO and the OD approach lead to different discrete systems. For example, when standard penalization is used, the

L

2

error in the controls is only

O

(

h

) independent of

k

. However, if superpenalization is used, the lack of adjoint consistency is reduced and the observed convergence for NIPG and IIPG is essentially equal to that of the SIPG method in the DO and OD approach.

We address an immersed boundary method applied to the study of cardiovascular drug eluting stents deployed in coronary bifurcations. The problem involves the interaction of arterial deformations, hemodynamics and controlled drug release. Resorting to an immersed boundary method facilitates the handling of complex stent pattern and simplifies the definition of the mathematical model for drug release.

The purpose of the paper is to approximate an elliptic problem coupling two different formulations. The domain is split into two non-overlapping sub-domains. On the first one, the problem is approximated using classical Galerkin method where the primal solution

p

is searched in

H

1

approximation spaces. On the other one, the mixed formulation is applied, which is based on Hdiv and

L

2

approximation spaces for the dual ∇

p

and primal

p

solutions, respectively. On the interface, the continuity of

p

and ∇

p

is imposed strongly, using transmission conditions. The resulting coupled formulation is a saddle point problem, which is solved for high order hierarchical approximation spaces. Numerical simulations for a test problem show consistent rates of convergence when compared with the corresponding classical and mixed formulations in the whole domain.

This paper is about the stability w.r.t. the

H

1

-semi-norm of the nodal interpolation operator acting between non-nested finite element spaces. (An earlier, slightly less general version of the main result has been proved in the author’s thesis (Dickopf, Multilevel methods based on non-nested meshes. Ph.D. thesis, University of Bonn, 2010.

http://hss.ulb.uni-bonn.de/2010/2365

, Chap. 5.1). Lively and fruitful discussions first during the ENUMATH conference in September and then during the Söllerhaus Workshop on Domain Decomposition Methods in October 2011 have encouraged the author to rework the analysis of the nodal interpolation over intervals and present it in this extended and considerably revised form.) We show that, for arbitrary spaces of piecewise linear functions of one variable, the

H

1

-stability constant is bounded by one without any assumptions on the mesh sizes or on the relations between the meshes. We also give counterexamples for the nodal interpolation in higher order finite element spaces.

A novel discretization strategy, dubbed m-adaptation, is developed for solving the acoustic wave equation (in the time domain) on rectangular meshes. The developed method is based on a scheme that is second-order accurate in space and time but has the sixth-order numerical anisotropy on square meshes and the fourth order dispersion on rectangular meshes.

The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge–Ampère type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature however the method can be generalised to any Monge–Ampère operator.

We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the Möbius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed in Berninger et al. (SIAM J Numer Anal 49(6):2576–2597, 2011) is a discretization by geodesic finite elements in the manifold

$$\mathbb{R}$$

with a special metric.

As an alternative to the sharp interface formulation, the phase field approach is a widely used technique for modeling phase transitions. The governing system of reaction-diffusion equations captures the instability of the underlying physical problem and is capable of modeling the evolution of complicated crystal shapes during solidification of an undercooled melt. For its numerical solution, we propose our novel anti-diffusive multipoint flux approximation (MPFA) finite volume scheme on a Cartesian mesh. The scheme is verified against the analytical solution of the modified sharp interface model. Experimental order of convergence (EOC) is measured for the temperature field in the usual norms. In addition, EOC is also obtained for the phase interface through approximating the volume of the symmetric difference of the solid phase subdomains. In the anisotropic cases including unusual higher order symmetries, computational studies with various settings also confirm convergence of our MPFA scheme which is faster than in the case of the reference finite volume scheme with 2nd order flux approximation.

We apply an evolving surface finite element method (ESFEM) to a mathematical model for diffusion induced grain boundary motion. The model involves the coupling of a diffusion equation on a moving surface to an equation for the motion of the surface. We formulate a finite element approximation of the model which involves triangulated surfaces whose vertices move in time. We present numerical simulations.

The article deals with the 2D numerical simulation of the stratified incompressible flows behind the moving thin horizontal strip in the towing tank and over the sinusoidal hill. The mathematical model is based on the Boussinesq approximation of the Navier–Stokes equations. The resulting set of PDE’s is then solved by two different numerical methods. Different boundary conditions are tested.

In this paper we discuss simulation techniques for a rising bubble in viscoelastic fluids via numerical methods based on high order FEM. A level set approach based on the work in (Sethian, Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and material science, 2nd edn. Cambridge University Press, 1999) is used for interface tracking between the bubble and the surrounding fluid. The two matters obey the Newtonian and the Oldroyd-B constitutive law in the case of a viscoelastic fluid while the flow model is given by the Navier-Stokes equations. The total system of equations is discretized in space by the LBB-stable finite element

Q

2

P

1

, and in time by the family of θ-scheme integrators. The solver is based on Newton-multigrid techniques (Damanik et al., J Comput Phys 228:3869–3881, 2009; J Non-Newton Fluid Mech 165:1105–1113, 2010) for nonlinear fluids. First, we validate the multiphase flow results with respect to the benchmark results in (Hysing et al., Int J Numer Methods Fluids 60(11):1259–1288, 2009), then we perform numerical simulations of a bubble rising in a viscoelastic fluid and show cusp formation at the trailing edge.

In this work we focus on a model reduction approach for the treatment of fractures in a porous medium, represented as interfaces embedded in a

n

-dimensional domain, in the form of a (

n

− 1)-dimensional manifold, to describe fluid flow and transport in both domains. We employ a method that allows for non-matching grids, thus very advantageous if the position of the fractures is uncertain and multiple simulations are required. To this purpose we adopt an extended finite element approach, XFEM, to represent discontinuities of the variables at the interfaces, which can arbitrarily cut the elements of the grid. The method is applied to the solution of the Darcy and advection-diffusion problems in porous media.

In this paper, we extend our work for the heat equation in (Hussain et al., J Numer Math 19(1):41–61, 2011) and for the Stokes equations in (Hussain et al., Open Numer Methods J 4:35–45, 2012) to the nonstationary Navier-Stokes equations in two dimensions. We examine

continuous

Galerkin-Petrov (cGP) time discretization schemes for nonstationary incompressible flow. In particular, we implement and analyze numerically the higher order cGP(2)-method. For the space discretization, we use the LBB-stable finite element pair

$$Q_{2}/P_{1}^{\mathit{disc}}$$

. The discretized systems of nonlinear equations are treated by using the fixed-point as well as the Newton method and the associated linear subproblems are solved by using a monolithic multigrid solver with GMRES method as smoother. We perform nonstationary simulations for a benchmarking configuration to analyze the temporal accuracy and efficiency of the presented time discretization scheme.

The new approach, called the density-enthalpy method, has certain advantages over the tradition methods for solving multi-phase flow problems. The system is modeled by the mass and energy conservation, Darcy’s law, and other thermodynamic relations. It is solved by using the standard Galerkin algorithm for spatial discretization, backward Euler for time integration, and Newton-Raphson iteration for linearization. In this paper, the main objective is to study the effect of gravity on Darcy flow.

This work deals with the numerical simulation of viscous and viscoelastic fluids flow. The governing system of equations is based on the system of balance laws for mass and momentum for incompressible laminar fluids. Two models for the stress tensor are tested. For viscous fluids flow Newtonian model is used. By the combination of Newtonian and simple viscoelastic (Maxwell) models the behaviour of the mixture of viscous and viscoelastic fluids can be described. This model is called Oldroyd-B model. Both presented models (Newtonian and Oldroyd-B) can be generalized for the numerical modelling of the generalized Newtonian and Oldroyd-B fluids flow. In this case the viscosity is no more constant but is defined as a shear rate dependent viscosity function

$$\mu (\dot{\gamma })$$

. One of the most frequently used shear-thinning models is the generalized cross model. Numerical solution of the described models is based on cell-centered finite volume method using explicit Runge–Kutta time integration. Steady state solution is achieved for

$$t \rightarrow \infty $$

. In this case the artificial compressibility method can be applied. The numerical results of generalized Newtonian and generalized Oldroyd-B fluids flow obtained by this method are presented and compared.

The work deals with numerical 3D simulations of incompressible turbulent flow in channel junction with one inlet and two outlets. The complex flow in the junction includes separation, impingement and secondary flow. The mathematical model is based on unsteady Reynolds averaged Navier-Stokes equations (URANS) with an explicit algebraic Reynolds stress turbulence model (EARSM). The solution method uses dual time artificial compressibility scheme with upwind finite volume discretization. Some methods of ensuring prescribed flow-rate distribution are discussed and tested. The results are compared with PIV measurement.

The paper presents the mathematical model of a two dimensional steady viscous incompressible flow through a rotating radial blade machine. The flow is described and studied in the rotating frame. The paper provides the classical and weak formulation of the corresponding boundary value problem. The boundary condition on the outflow is the so called “natural” boundary condition, with the additional nonlinear term proposed by Bruneau and Fabrie (Math Model Numer Anal 30(7):815–840, 1996), and a new term arising from the rotation of the machine. The existence of a weak solution is proved.

The paper deals with the numerical modeling of compressible single-phase flow of a mixture composed of several components in a porous medium. The mathematical model is formulated by Darcy’s law, components continuity equations, constitutive relations, and initial and boundary conditions. The problem is solved numerically using a combination of the mixed-hybrid finite element method for the total flux discretization and the finite volume method for the discretization of the transport equations. The time discretization is carried out by Euler’s method. The resulting large system of nonlinear algebraic equations is solved by the Newton-Raphson method. The dimensions of the obtained system of linear algebraic equations are significantly reduced so that they do not depend on the number of mixture components. The convergence of the numerical scheme is verified in the single-component case by comparing the numerical solution with an analytical solution.

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness is caused by a prescribed periodic motion of the channel wall. Instudy three different governing systems of equations are considered –

Full system, Adiabatic system, Iso-energetic system

. Unsteady flow fields for inlet Mach number

$$M_{\infty } = 0.012$$

and frequency 100Hz are presented.

This paper is dealing with numerical simulation of generalized Newtonian and generalized Oldroyd-B fluids. The Newtonian model of a fluid cannot capture all the phenomena in many fluids with complex microstructure, such as polymers, suspensions and granular materials. The motion of polymeric fluids is described by the conservation of mass and momentum. One shall assume that the fluid is incompressible and temperature variations are negligible. When one considers viscoelastic behavior of polymeric fluids, the extra stress tensor depends not only on the current motion of the fluid, but also on the history of the motion. In this case the extra stress tensor is decomposed into its Newtonian part and its elastic part. Components of the elastic part of the extra stress tensor are computed using the Oldroyd-B constitutive equation. Numerical solution of the arising system of equations is solved using the artificial compressibility method, finite volume method and Runge-Kutta method. Numerical methods are tested in the geometry of constricted channel.

For a family of variational multiscale methods we perform an a-priori error analysis for inf-sup stable finite element pairs in low-turbulent incompressible flow problems. This is done for underlying layer-adapted meshes with strong Dirichlet boundary conditions and for isotropic meshes with weak Dirichlet boundary conditions. For both approaches we provide first numerical results in a three-dimensional channel at

Re

τ

= 180.

This paper focuses on the mathematical and numerical modelling of interaction of the two-dimensional incompressible fluid flow and a flexibly supported airfoil section with a control section. A simplified problem is considered: The flow is modelled by the system of Navier-Stokes equations and and the structure motion is described with the aid of nonlinear ordinary differential equations. The time-dependent computational domain is taken into account by the Arbitrary Lagrangian-Eulerian method. Higher order time discretization is considered within the stabilized finite element method. The application of the described method is shown.

We consider a simple model for visco–elasticity, that is commonly applied to simulate dermal wound healing. First the problem is formulated, then, convergence to a steady–state equilibrium solution is demonstrated. Subsequently, we construct analytic solutions based on Green’s Functions for one-dimensional sample problems. These solutions enable us to look at the convergence behavior towards equilibrium solutions. We also give some conditions for monotonic convergence.

In the present discussion we present a combination of the volume of fluid method with the front tracking method for the advection of the interface between two immiscible phases in two dimensions. The mass of each phase is conserved up to roundoff accuracy. The interface is thereby represented very accurately. A drawback of the present method is that topological changes are not handled ‘automatically’ as for the volume of fluid method.

It is well-known that the Crank-Nicolson scheme for pure fluid problems suffers from stability for computations over long-term time intervals. In the presence of fluid-structure interaction in which the fluid equations are reformulated with the help of arbitrary Lagrangian-Eulerian (ALE) mapping, the ALE convection also causes stability problems. In this study, we derive a stability estimate of a monolithically coupled time-discretized fluid-structure interaction problem. Moreover, a numerical comparison of all relevant second order time-stepping schemes, such as secant and tangent Crank-Nicolson, shifted Crank-Nicolson, and Fractional-Step-Theta, is demonstrated. The numerical experiments are based on a benchmark configuration for fluid-structure interactions.

We consider a mathematical model for Biogrout, which is a novel soil reinforcement technique based on Microbially Induced Carbonated Precipitation. We focus on an adaptation of the flow equation such that mass is conserved instead of volume. The adaptation is validated by a mass balance. Some numerical simulations are presented and used for the discussion on the various adjustments of the flow equation.

We present performance results of a mixed-precision strategy developed to improve a recently developed massively parallel GPU-accelerated tool for fast and scalable simulation of unsteady fully nonlinear free surface water waves over uneven depths (Engsig-Karup et al., Int J Num Meth, 2011). The underlying wave model is based on a potential flow formulation, which requires efficient solution of a Laplace problem at large-scales. We report recent results on a new mixed-precision strategy for efficient iterative high-order accurate and scalable solution of the Laplace problem using a multigrid-preconditioned defect correction method. The improved strategy improves the performance by exploiting architectural features of modern GPUs for mixed precision computations and is tested in a recently developed generic library for fast prototyping of PDE solvers. The new wave tool is applicable to solve and analyze large-scale wave problems in coastal and offshore engineering.

We are focusing on an iterative solver for the three-dimensional Helmholtz equation on multi-GPU using CUDA (Compute Unified Device Architecture). The Helmholtz equation discretized by a second order finite difference scheme is solved with Bi-CGSTAB preconditioned by a shifted Laplace multigrid method. Two multi-GPU approaches are considered: data parallelism and split of the algorithm. Their implementations on multi-GPU architecture are compared to a multi-threaded CPU and single GPU implementation. The results show that the data parallel implementation is suffering from communication between GPUs and CPU, but is still a number of times faster compared to many-cores. The split of the algorithm across GPUs limits communication and delivers speedups comparable to a single GPU implementation.

We describe a CUDA-based parallel preconditioning method for non-normal matrices. In particular, we are interested in solving the non-isothermal Reynolds-averaged Navier-Stokes equations. These are at the bottom of indoor air-flow simulations which are necessary for predicting the energy consumption of a building configuration. Within each timestep one has to solve linearized auxiliary problems of Oseen and advection-diffusion-reaction type. Solving the linear algebraic subproblems is accelerated by CUDA by nearly an order of magnitude. Particularly suited is the sparse approximate inverse approach which yields promising results.

Efficient stencil operations are essential in explicit schemes for evolutionary PDEs. In particular, for conservation and balance laws, the solution will in many cases have non-constant values only in a portion of the grid. We present novel methods that through simple observation of the stencil and the distribution of conserved quantities, reduce both the memory footprint and the computational burden by only computing in cells in which the solution changes. To this end, we utilize sparse updating of grid cells, in which data values are not stored before they actually contribute in the simulation. This is motivated by the need to perform simulations over very large domains to model real-world dam breaks and various flooding scenarios. The methods are applied to a high-resolution shallow water simulator, but are also applicable to other stencil-based explicit solvers.

In application of the Balancing Domain Decomposition by Constraints (BDDC) to a case with many substructures, solving the coarse problem exactly becomes the bottleneck which spoils scalability of the solver. However, it is straightforward for BDDC to substitute the exact solution of the coarse problem by another step of BDDC method with subdomains playing the role of elements. In this way, the algorithm of three-level BDDC method is obtained. If this approach is applied recursively, multilevel BDDC method is derived. We present a detailed description of a recently developed parallel implementation of this algorithm. The implementation is applied to an engineering problem of linear elasticity and a benchmark problem of Stokes flow in a cavity. Results by the multilevel approach are compared to those by the standard (two-level) BDDC method.

A Bayesian approach for uncertainty quantification of oil reservoir parameters in forecasting the production is straightforward in principle. However, the complexity of flow simulators and the nature of the inverse problem at hand present an ongoing practical challenges to addressing uncertainty in all subsurface parameters. In this paper, we focus on two important subsurface parameters, permeability and porosity, and discuss quantifying uncertainty in those parameters.

In this paper, we discuss some numerical schemes for an upscaled (core scale) model describing the transport, precipitation and dissolution of solutes in a porous medium. We consider two weak formulations, conformal and mixed. We discuss the time discretization in both formulations and prove the convergence of the resulting schemes. A numerical study is presented for the mixed formulation.

In this paper we present the adaptive variational multiscale method for solving the Poisson equation in mixed form. We use the method introduced in Larson and Målqvist (Comput Method Appl Mech Eng 196:2313–2324, 2007), and further analyzed and applied to mixed problems in Larson and Målqvist (Comput Method Appl Mech Eng 19:1017–1042, 2009), which is a general tool for solving linear partial differential equations with multiscale features in the coefficients. We extend the numerics in Larson and Målqvist (Comput Method Appl Mech Eng 19:1017–1042, 2009) from rectangular meshes to triangular meshes which allow for computation on more complicated domains. A new a posteriori error estimate is also included, which is used in an adaptive algorithm. We present a numerical example that shows the efficiency of incorporating a posteriori based adaptivity into the method.

Hydrodynamic problems often feature geometrical configurations that allow a suitable dimensional model reduction. One-dimensional models may be sometimes accurate enough for describing a dynamic of interest. In other cases, localized relevant phenomena require more precise models. To improve the computational efficiency, geometrical multiscale models have been proposed, where reduced (1D) and complete (2D–3D) models are coupled in a unique numerical solver. In this paper we consider an adaptive geometrical multiscale modeling: the regions of the computational domain requiring more or less accurate models are automatically and dynamically selected via a heuristic criterion. To the best of our knowledge, this is a first example of automatic geometrical multiscale model reduction.

Quantitative bounds are presented for the superlinear convergence of the MINRES method of Paige and Saunders (SIAM J Numer Anal 12:617–629, 1975) for the solution of sparse linear systems

Ax

=

b

, with

A

symmetric and indefinite. It is shown that the superlinear convergence is observed as soon as the harmonic Ritz values approximate well the eigenvalues of

A

that are either closest to zero or farthest from zero. This generalizes a well-known corresponding result obtained by van der Sluis and van der Vorst with respect to the Conjugate Gradients method, for

A

symmetric and positive definite.

This contribution focuses on partitioned solution approaches in fluid-structure interaction problems. Depending on certain physical parameters of fluid and structure, the fixed-point iteration that is mostly used to strongly couple the different solvers in each time step is susceptible to deceleration. We present a method that is able to overcome this effect by a specific preconditioning of the fixed-point iteration. Thus, the full convergence order of the underlying time-discretisation schemes is preserved. As computational example, a benchmark problem from hemodynamics is considered where this effect has a particularly strong influence. It turns out that, though a single step of the preconditioned iteration is more expensive, the overall gain in efficiency can be significant.

This note discusses convergence behaviors of multilevel Krylov methods for some simple problems, mainly focusing on the possible choice of transfer operators. This study is part of the search for an optimal multilevel Krylov method.

We consider preconditioning methods for the systems of linear algebraic equations arising from Symmetric Interior Penalty discontinuous Galerkin (SIPG) discretization of linear elasticity problems in primal (displacement) formulation. The presented approach is a generalization of the subspace correction method studied in Ayuso et al. (A Subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations, arXiv:1110.5743v2, 2011) for linear elasticity problems with discontinuous material properties. The application of the preconditioner reduces to the solution of a problem arising from discretization of the equations of linear elasticity by nonconforming Crouzeix-Raviart finite elements plus the solution of a well-conditioned problem on the complementary space.

This work is devoted to the construction and analysis of robust solution techniques for the distributed optimal control problem for the Stokes equations with inequality constraints on the control. There the first order system of necessary and sufficient optimality conditions is nonlinear. A primal-dual active set method is applied in order to linearize the system. In every step a linear saddle point system has to be solved. For this system, we analyze a block-diagonal preconditioner that is robust with respect to the discretization parameter as well as the active set.

The computation of eigenvalues is one of the core topics of numerical mathematics. We will discuss an eigenvalue algorithm for the computation of inner eigenvalues of a large, symmetric, and positive definite matrix

M

based on the preconditioned inverse iteration

$$\begin{array}{rcl} x_{i+1} = x_{i} - {B}^{-1}\left (Mx_{ i} - \mu (x_{i})x_{i}\right ),& & \\ \end{array}$$

and the folded spectrum method (replace

M

by

$${(M - \sigma I)}^{2}$$

). We assume that

M

is given in the tensor train matrix format and use the TT-toolbox from I.V. Oseledets (see

http://spring.inm.ras.ru/osel/

) for the numerical computations. We will present first numerical results and discuss the numerical difficulties.

We present how entropy estimates and logarithmic Sobolev inequalities on the one hand, and the notion of quasi-stationary distribution on the other hand, are useful tools to analyze metastable overdamped Langevin dynamics, in particular to quantify the degree of metastability. We discuss the interest of these approaches to estimate the efficiency of some classical algorithms used to speed up the sampling, and to evaluate the error introduced by some coarse-graining procedures. This paper is a summary of a plenary talk given by the author at the ENUMATH 2011 conference.

This paper is concerned with studying the effects of uncertain data in the context of error indicators, which are often used in mesh adaptive numerical methods. We consider the diffusion equation and assume that the coefficients of the diffusion matrix are known not exactly, but within some margins (intervals). Our goal is to study the relationship between the magnitude of uncertainty and reliability of different error indication methods. Our results show that even small values of uncertainty may seriously affect the performance of all error indicators.

We present a reduced basis method for the simulation of American option pricing. To tackle this model numerically, we formulate the problem in terms of a time dependent variational inequality. Characteristic ingredients are a POD-greedy and an angle-greedy procedure for the construction of the primal and dual reduced spaces. Numerical examples are provided, illustrating the approximation quality and convergence of our approach.

In physics and engineering problems, model input is never exact. The effect of small uncertainties on the solution is thus an important question. In this study, a direct statistical-visual approach to approximate the solution set is investigated in the context of axially moving materials. The multidimensional probability distribution for the input uncertainties is assumed known. It is considered as a deterministic object, which is then mapped through the model. The resulting probability density of the model output is visualized. The proposed system consists of three non-trivial parts, which are briefly discussed: a multidimensional sampler, a density estimator, and a high dynamic range (HDR) plotter. Dynamic range compression is achieved via tone mapping techniques from HDR photography. This allows a contrast-preserving representation of the HDR data on a regular computer display or on paper. The model itself is treated as a black box; hence, the same approach can be used for investigating e.g. floating-point rounding errors or approximation error instead of uncertain data.

We discuss the singularity of the Fisher information arising from statistical inference for continuous-time stochastic processes of practical interest, such as asset price dynamics in finance and individual animal movement in biology, under high frequency discrete sampling schemes. Singularity seems to be caused by the scale parameter and the selfsimilarity index, while there exists a different type of singularity resulting from some redundancy of parameters in the short time framework. We derive the speed of convergence of the Fisher information to singularity for some instances and show that the convergence to singularity may be delayed through a wise expansion of the total observation window.

We present three different approaches to model, in a computationally cheap way, problems characterized by strong horizontal dynamics, even though in the presence of transverse heterogeneities. The three approaches are based on the hierarchical model reduction setting introduced in Ern et al. (Hierarchical model reduction for advection-diffusion-reaction problems. In: Kunisch K, Of G, Steinbach O (eds) Numerical mathematics and advanced applications. Springer (2008), pp 703–710) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010).