Markus Bause

Uniform Error Analysis for Lagrange--Galerkin Approximations of Convection-Dominated Problems

In this paper we present a rigorous error analysis for the Lagrange--Galerkin method applied to convection-dominated diffusion problems. We prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit. This is in contrast to other results in this field. For the time discretization, uniform convergence with respect to the diffusion parameter of order O(k//tn) is shown for initial values in L2 and O(k) for initial values in H2. For the spatial discretization with linear finite elements, we verify uniform convergence of order O(h2+min{h,h2/k) for data in H2. By interpolation of Banach spaces, suboptimal convergence rates are derived under less restrictive assumptions. The analysis is heavily based on a priori estimates, uniform in the diffusion parameter, for the solution of the continuous and the semidiscrete problem. They are derived in a Lagrangian framework by transforming the Eulerian coordinates completely into subcharacteristic coordinates. Finally, we illustrate the error estimates by some numerical results.

Higher order finite element approximation of systems of convection-diffusion-reaction equations with small diffusion

In this work we study numerically the performance properties of a class of approximation schemes for systems of convection-diffusion-reaction models with small diffusion. A coupling of the equations by first order and zero order terms is admitted. Higher order conforming finite element methods are applied to minimize the effects of numerical diffusion and artificial mixing of species. To reduce spurious oscillations close to sharp layers or interfaces, streamline upwind Petrov-Galerkin stabilization and shock capturing as an additional stabilization in the crosswind direction are used. In applications of practical interest the reliability and accuracy of the approach is demonstrated.

Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Abstract We analyze an optimized artificial fixed-stress iterative scheme for a space–time finite element approximation of the Biot system modeling fluid flow in deformable porous media. The iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step. The optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations. The separated subproblems of fluid flow, written as a mixed first order in space system, and mechanical deformation are discretized by space–time finite element methods of arbitrary order. Continuous and discontinuous Galerkin discretizations of the time variable are encountered. The convergence of the iterative schemes is proved for the continuous and fully discrete case. The choice of the optimization parameter is identified in the proofs of convergence of the iterations. The analyses are illustrated and confirmed by numerical experiments.

Variational Space---Time Methods for the Wave Equation

In this work we present some new variational space–time discretisations for the scalar-valued acoustic wave equation as a prototype model for the vector-valued elastic wave equation. The second-order hyperbolic equation is rewritten as a first-order in time system of equations for the displacement and velocity field. For the discretisation in time we apply continuous Galerkin–Petrov and discontinuous Galerkin methods, and for the discretisation in space we apply the symmetric interior penalty discontinuous Galerkin method. The resulting algebraic system of equations exhibits a block structure. First, it is simplified by some calculations to a linear system for one of the variables and a vector update for the other variable. Using the block diagonal structure of the mass matrix from the discontinuous Galerkin discretisation in space, the reduced system can be condensed further such that the overall linear system can be solved efficiently. The convergence behaviour of the presented schemes is studied carefully by numerical experiments. Moreover, the performance and stability properties of the schemes are illustrated by a more sophisticated problem with complex wave propagation phenomena in heterogeneous media.

Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space

Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin–Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach–Nečas–Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

Variational time discretization for mixed finite element approximations of nonstationary diffusion problems

We develop and study numerically two families of variational time discretization schemes for mixed finite element approximations applied to nonstationary diffusion problems. Continuous and discontinuous approximations of the time variable are encountered. The solution of the arising algebraic block system of equations by a Schur complement technique is described and an efficient preconditioner for the iterative solution process is constructed. The expected higher order rates of convergence are demonstrated in numerical experiments. Moreover, superconvergence properties are verified. Further, the efficiency and stability of the approaches are illustrated for a more sophisticated three-dimensional application of practical interest with discontinuous and anisotropic material properties.

Stabilized Finite Element Methods with Shock-Capturing for Nonlinear Convection-Diffusion-Reaction Models

In this work stabilized higher-order finite element approximations of convection-diffusion-reactions models with nonlinear reaction mechanisms are studied. Streamline upwind Petrov–Galerkin (SUPG) stabilization together with anisotropic shock-capturing as an additional stabilization in crosswind-direction is used. The parameter design of the scheme is described precisely and error estimates are provided. Theoretical results are illustrated by numerical computations. The work extends former investigations for linear problems to more realistic nonlinear models.

Iterative Coupling of Variational Space-Time Methods for Biot’s System of Poroelasticity

In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time finite element methods based on a discontinuous Galerkin approximation of the time variable are used. The spatial approximation of the flow problem is done by mixed finite element methods. The stability of the approach is illustrated by numerical experiments. The presented variational space-time framework is of higher order accuracy such that problems with high fluctuations become feasible. Moreover, it offers promising potential for the simulation of the fully dynamic Biot–Allard system coupling an elastic wave equation for solid’s deformation with single-phase flow for fluid infiltration.

Performance of Stabilized Higher-Order Methods for Nonstationary Convection-Diffusion-Reaction Equations

We study the performance properties of a class of stabilized higher-order finite element approximations of convection-diffusion-reaction models with nonlinear reaction mechanisms. Streamline upwind Petrov-Galerkin (SUPG) stabilization together with anisotropic shock-capturing as an additional stabilization in crosswind-direction is used. We show that these techniques reduce spurious oscillations in crosswind-direction and increase the accuracy of simulations.

Adaptive Multigrid Methods for Extended Fluid-Structure Interaction (eXFSI) Problem: Part I — Mathematical Modelling

This contribution is the first part of three papers on Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art of recent developments in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research will focus on the newly developed mathematical model of a new FSI problem which is called eXtended Fluid-Structure Interaction (eXFSI) problem in ALE framework. This model is used to design an on-live Structural Health Monitoring (SHM) system in order to determine the wave propagation in moving domains and optimum locations for SHM sensors. eXFSI is strongly coupled problem of typical FSI with a wave propagation problem on the fluid-structure interface, where wave propagation problems automatically adopted the boundary conditions from of the typical FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as an one step-θ scheme; from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with Newton’s method whereas the spatial discretization is done with a Galerkin finite element scheme. To control computational costs we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaptation during the computation. The implementation is accomplished via the software library package DOpElib and deal.II for the computation of different eXFSI configurations.Copyright