Markus Bause - Academia.edu (original) (raw)

Papers by Markus Bause

In this talk a higher order finite element approach to the coupled variably saturated groundwater... more In this talk a higher order finite element approach to the coupled variably saturated groundwater flow and bioreactive contaminant transport model is considered. Higher order techniques have proved advantageous in the reliable numerical simulation of biochemically reacting transport processes, due to their less inherent numerical diffusion. For the calcu- lation of the groundwater flow field mixed finite element methods are prefered due to their inherent conservation properties and since they provide a flux approximation as part of the formulation itself. Typically, lowest order mixed Raviart-Thomas elements are used for solving the parabolic-elliptic degenerate Richards equation describing the motion of groundwater, since this model admits solutions of low regularity only. Here, our numerical results obtained by a higher order mixed finite element approach of Brezzi-Douglas-Marini type to elliptic, parabolic and degenerate partial differential equations with solutions of low regula...

arXiv: Numerical Analysis, 2018

Even though substantial progress has been made in the numerical approximation of convection-domin... more Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.

Finite Element Approximation of Fluid Structure Interaction (FSI) Optimization in Arbitrary Lagrangian-Eulerian Coordinates

Volume 7B: Fluids Engineering Systems and Technologies, 2013

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

Lecture Notes in Computational Science and Engineering, 2011

... Here, u D u.x;t/ denotes the unknown where x 2 ˝ Rd , with d 2, and t 2 .0; T / for some T &g... more ... Here, u D u.x;t/ denotes the unknown where x 2 ˝ Rd , with d 2, and t 2 .0; T / for some T > 0. Further, a 2 L 1 .0; TIW 1;1.˝// is the diffusion coefficient, b 2 L 1 .0; TIW 1;1.˝// is the velocity field, r 2 C1.R ... Rd let V p hD Xp h\ H1 0 .˝/ with X p h D fv 2 C.˝/ j vjT ı FT 2 PpT .bT / 8T 2 Thg ...

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

Journal of Computational and Applied Mathematics, 2015

ABSTRACT We develop and study numerically two families of variational time discretization schemes... more ABSTRACT 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.

PAMM, 2004

The extensive application of mathematical and computational methods has become an efficient and p... more The extensive application of mathematical and computational methods has become an efficient and powerful approach to the investigation and solution of many problems and processes in fluid dynamics from qualitative as well as quantitative point of view. In this work a new class of advanced numerical approximation schemes to isothermal compressible viscous flow is presented. The schemes are based on an iteration between an Oseen like problem for the velocity and a hyperbolic transport equation for the density. Such schemes seem attractive for computations because they offer a reduction to simpler problems for which highly refined numerical methods either are known or can be built from existing approximation schemes to similar equations, and because of the guidance that can be drawn from an existence theory based on them. For the generalized Oseen subproblem a Taylor-Hood finite element method is proposed that is stabilized by a reduced SUPG and grad-div technique (cf. [1, 4]) in the convection-dominated case. Results of theoretical investigations and numerical studies are presented.

PAMM, 2008

In this paper a new error estimate for a multi point flux approximation (MPFA) control volume met... more In this paper a new error estimate for a multi point flux approximation (MPFA) control volume method on triangular meshes for approximating flow in porous media is presented. The scheme is mass conservative and has shown to simulate reliably flows with discontinuous permeability tensors and on irregular grids which is the major challenge in hydrological engineering.

Higher ordermixed finite element approximation of subsurface water flow

PAMM, 2007

ABSTRACT This paper focuses on the reliable and efficient numerical approximation of subsurface w... more ABSTRACT This paper focuses on the reliable and efficient numerical approximation of subsurface water flows. The locally mass conservative Brezzi-Douglas-Marini (BDM1) mixed finite element method is considered and compared to a lowest order Raviart–Thomas (RT0) mixed finite element approach and a Multi Point Flux Approximation. Appreciable advantage of the BDM1 element is that it yields a formally second order accurate flux approximation whereas the RT0 and MPFA approach are of first order accuracy only. The problem to be analyzed in this work is whether a superiority of the BDM1 element can also be observed in reservoir simulation where discontinuous and full permeability tensors on non-uniform grids arise and the fluxes lack the regularity that is assumed customarily in optimal order error analyses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport

International Series of Numerical Mathematics, 2007

In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, ... more In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, chemical and biological processes of contaminant transport and degradation in the subsurface. Our latest results for the existence, uniqueness and regularity of solutions to the model equations are summarized; cf. [4, 9]. The basic idea of the proof of regularity is sketched briefly. Moreover, our numerical discretization scheme that has proved its capability of approximating reliably and efficiently solutions of the mathematical model is described shortly, and an error estimate is given; cf. [2, 3]. Finally, to illustrate our approach of modelling and simulating bioreactive transport in the subsurface, the movement and expansion of a m-xylene plume is studied numerically under realistic field-scale assumptions.

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

Numerical Mathematics and Advanced Applications 2009, 2010

ABSTRACT In this work stabilized higher-order finite element approximations of convection-diffusi... more ABSTRACT 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.

Computational Study of Field Scale BTEX Transport and Biodegradation in the Subsurface

Numerical Mathematics and Advanced Applications, 2004

Stabilized finite element schemes for generalized Oseen systems

PAMM, 2005

ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of ... more ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Computing and Visualization in Science, 2004

In this work we present and analyze a reliable and robust approximation scheme for biochemically ... more In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newton's method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes.

Numerical Study of Mixed Finite Element and Multi Point Flux Approximation of Flow in Porous Media

In this paper the numerical performance properties of some locally mass conservative numerical sc... more In this paper the numerical performance properties of some locally mass conservative numerical schemes for simulating flow in a porous medium are studied. In particular, the accuracy of approximations of flows with discontinuous permeability tensors and on irregular grids is analysed. We consider the mixed finite element approach with the lowest order Raviart-Thomas (RT0) and Brezzi-Douglas-Marini (BDM1) element and a multi point flux approximation (MPFA) control volume method. The BDM1 method yields a second order accurate approximation of the flux whereas the RT0 and MPFA approach are of first order accuracy only. MPFA methods offer explicit discrete fluxes which is not possible to get from mixed finite element methods and allows a wider class of applications.

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-reacti... more 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.

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

Journal of Computational and Applied Mathematics

We develop and study numerically two families of variational time discretization schemes for mixe... more 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.

DESCRIPTION My contribution to this work was to extend my software DTM++, which is based on the d... more DESCRIPTION My contribution to this work was to extend my software DTM++, which is based on the deal.II FEM-toolbox, in a way to study the H(div;\Omega) convergence behaviour of higher order time discretisations. The used model is a mass-conservative diffusion equation which is discretised with a mixed finite element method in space. The work is submitted to a journal.

An Iterative Scheme for Steady Compressible Viscous Flow, Modified to Treat Large Potential Forces

Mathematical Fluid Mechanics, 2001

ABSTRACT A relatively simple iterative scheme for steady compressible viscous flow was given in a... more ABSTRACT A relatively simple iterative scheme for steady compressible viscous flow was given in a recent paper of Heywood and Padula [5]. It was intended as a basis for both the existence theory and for numerical methods. Among the assumptions they made in introducing the scheme, one was that the force should be small. On the other hand, by another method, Novotnÿ and Pileckas [7] recently extended the existence theory for such flow to the case of forces that are small perturbations of large potential forces. The purpose of the present paper is to incorporate the ideas of Novotnÿ and Pileckas, for treating such forces, into the simpler and more constructive scheme of Heywood and Padula. This is one step in our larger objective, which is to base the existence theory for compressible viscous flow on constructive methods that may be suggestive of new numerical methods.

In this paper a new error estimate for a multi point flux approximation (MPFA) control volume met... more In this paper a new error estimate for a multi point flux approximation (MPFA) control volume method on triangular meshes for approximating flow in porous media is presented. The scheme is mass conservative and has shown to simulate reliably flows with discontinuous permeability tensors and on irregular grids which is the major challenge in hydrological engineering.

Stabilized finite element schemes for generalized Oseen systems

ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of ... more ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this talk a higher order finite element approach to the coupled variably saturated groundwater... more In this talk a higher order finite element approach to the coupled variably saturated groundwater flow and bioreactive contaminant transport model is considered. Higher order techniques have proved advantageous in the reliable numerical simulation of biochemically reacting transport processes, due to their less inherent numerical diffusion. For the calcu- lation of the groundwater flow field mixed finite element methods are prefered due to their inherent conservation properties and since they provide a flux approximation as part of the formulation itself. Typically, lowest order mixed Raviart-Thomas elements are used for solving the parabolic-elliptic degenerate Richards equation describing the motion of groundwater, since this model admits solutions of low regularity only. Here, our numerical results obtained by a higher order mixed finite element approach of Brezzi-Douglas-Marini type to elliptic, parabolic and degenerate partial differential equations with solutions of low regula...

arXiv: Numerical Analysis, 2018

Even though substantial progress has been made in the numerical approximation of convection-domin... more Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.

Finite Element Approximation of Fluid Structure Interaction (FSI) Optimization in Arbitrary Lagrangian-Eulerian Coordinates

Volume 7B: Fluids Engineering Systems and Technologies, 2013

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

Lecture Notes in Computational Science and Engineering, 2011

... Here, u D u.x;t/ denotes the unknown where x 2 ˝ Rd , with d 2, and t 2 .0; T / for some T &g... more ... Here, u D u.x;t/ denotes the unknown where x 2 ˝ Rd , with d 2, and t 2 .0; T / for some T > 0. Further, a 2 L 1 .0; TIW 1;1.˝// is the diffusion coefficient, b 2 L 1 .0; TIW 1;1.˝// is the velocity field, r 2 C1.R ... Rd let V p hD Xp h\ H1 0 .˝/ with X p h D fv 2 C.˝/ j vjT ı FT 2 PpT .bT / 8T 2 Thg ...

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

Journal of Computational and Applied Mathematics, 2015

ABSTRACT We develop and study numerically two families of variational time discretization schemes... more ABSTRACT 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.

PAMM, 2004

The extensive application of mathematical and computational methods has become an efficient and p... more The extensive application of mathematical and computational methods has become an efficient and powerful approach to the investigation and solution of many problems and processes in fluid dynamics from qualitative as well as quantitative point of view. In this work a new class of advanced numerical approximation schemes to isothermal compressible viscous flow is presented. The schemes are based on an iteration between an Oseen like problem for the velocity and a hyperbolic transport equation for the density. Such schemes seem attractive for computations because they offer a reduction to simpler problems for which highly refined numerical methods either are known or can be built from existing approximation schemes to similar equations, and because of the guidance that can be drawn from an existence theory based on them. For the generalized Oseen subproblem a Taylor-Hood finite element method is proposed that is stabilized by a reduced SUPG and grad-div technique (cf. [1, 4]) in the convection-dominated case. Results of theoretical investigations and numerical studies are presented.

PAMM, 2008

In this paper a new error estimate for a multi point flux approximation (MPFA) control volume met... more In this paper a new error estimate for a multi point flux approximation (MPFA) control volume method on triangular meshes for approximating flow in porous media is presented. The scheme is mass conservative and has shown to simulate reliably flows with discontinuous permeability tensors and on irregular grids which is the major challenge in hydrological engineering.

Higher ordermixed finite element approximation of subsurface water flow

PAMM, 2007

ABSTRACT This paper focuses on the reliable and efficient numerical approximation of subsurface w... more ABSTRACT This paper focuses on the reliable and efficient numerical approximation of subsurface water flows. The locally mass conservative Brezzi-Douglas-Marini (BDM1) mixed finite element method is considered and compared to a lowest order Raviart–Thomas (RT0) mixed finite element approach and a Multi Point Flux Approximation. Appreciable advantage of the BDM1 element is that it yields a formally second order accurate flux approximation whereas the RT0 and MPFA approach are of first order accuracy only. The problem to be analyzed in this work is whether a superiority of the BDM1 element can also be observed in reservoir simulation where discontinuous and full permeability tensors on non-uniform grids arise and the fluxes lack the regularity that is assumed customarily in optimal order error analyses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport

International Series of Numerical Mathematics, 2007

In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, ... more In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, chemical and biological processes of contaminant transport and degradation in the subsurface. Our latest results for the existence, uniqueness and regularity of solutions to the model equations are summarized; cf. [4, 9]. The basic idea of the proof of regularity is sketched briefly. Moreover, our numerical discretization scheme that has proved its capability of approximating reliably and efficiently solutions of the mathematical model is described shortly, and an error estimate is given; cf. [2, 3]. Finally, to illustrate our approach of modelling and simulating bioreactive transport in the subsurface, the movement and expansion of a m-xylene plume is studied numerically under realistic field-scale assumptions.

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

Numerical Mathematics and Advanced Applications 2009, 2010

ABSTRACT In this work stabilized higher-order finite element approximations of convection-diffusi... more ABSTRACT 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.

Computational Study of Field Scale BTEX Transport and Biodegradation in the Subsurface

Numerical Mathematics and Advanced Applications, 2004

Stabilized finite element schemes for generalized Oseen systems

PAMM, 2005

ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of ... more ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Computing and Visualization in Science, 2004

In this work we present and analyze a reliable and robust approximation scheme for biochemically ... more In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newton's method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes.

Numerical Study of Mixed Finite Element and Multi Point Flux Approximation of Flow in Porous Media

In this paper the numerical performance properties of some locally mass conservative numerical sc... more In this paper the numerical performance properties of some locally mass conservative numerical schemes for simulating flow in a porous medium are studied. In particular, the accuracy of approximations of flows with discontinuous permeability tensors and on irregular grids is analysed. We consider the mixed finite element approach with the lowest order Raviart-Thomas (RT0) and Brezzi-Douglas-Marini (BDM1) element and a multi point flux approximation (MPFA) control volume method. The BDM1 method yields a second order accurate approximation of the flux whereas the RT0 and MPFA approach are of first order accuracy only. MPFA methods offer explicit discrete fluxes which is not possible to get from mixed finite element methods and allows a wider class of applications.

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-reacti... more 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.

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

Journal of Computational and Applied Mathematics

We develop and study numerically two families of variational time discretization schemes for mixe... more 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.

DESCRIPTION My contribution to this work was to extend my software DTM++, which is based on the d... more DESCRIPTION My contribution to this work was to extend my software DTM++, which is based on the deal.II FEM-toolbox, in a way to study the H(div;\Omega) convergence behaviour of higher order time discretisations. The used model is a mass-conservative diffusion equation which is discretised with a mixed finite element method in space. The work is submitted to a journal.

An Iterative Scheme for Steady Compressible Viscous Flow, Modified to Treat Large Potential Forces

Mathematical Fluid Mechanics, 2001

ABSTRACT A relatively simple iterative scheme for steady compressible viscous flow was given in a... more ABSTRACT A relatively simple iterative scheme for steady compressible viscous flow was given in a recent paper of Heywood and Padula [5]. It was intended as a basis for both the existence theory and for numerical methods. Among the assumptions they made in introducing the scheme, one was that the force should be small. On the other hand, by another method, Novotnÿ and Pileckas [7] recently extended the existence theory for such flow to the case of forces that are small perturbations of large potential forces. The purpose of the present paper is to incorporate the ideas of Novotnÿ and Pileckas, for treating such forces, into the simpler and more constructive scheme of Heywood and Padula. This is one step in our larger objective, which is to base the existence theory for compressible viscous flow on constructive methods that may be suggestive of new numerical methods.

In this paper a new error estimate for a multi point flux approximation (MPFA) control volume met... more In this paper a new error estimate for a multi point flux approximation (MPFA) control volume method on triangular meshes for approximating flow in porous media is presented. The scheme is mass conservative and has shown to simulate reliably flows with discontinuous permeability tensors and on irregular grids which is the major challenge in hydrological engineering.

Stabilized finite element schemes for generalized Oseen systems

ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of ... more ABSTRACT In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)