M. Braack | Christian-Albrechts-Universität zu Kiel (original) (raw)

Papers by M. Braack

Research paper thumbnail of The effect of numerical methods on the simulation of mid-ocean ridge hydrothermal models

Theoretical and Computational Fluid Dynamics, 2011

This work considers the effect of the numerical method on the simulation of a 2D model of hydroth... more This work considers the effect of the numerical method on the simulation of a 2D model of hydrothermal systems located in the high-permeability axial plane of mid-ocean ridges. The behavior of hot plumes, formed in a porous medium between volcanic lava and the ocean floor, is very irregular due to convective instabilities. Therefore, we discuss and compare two different numerical methods for solving the mathematical model of this system. In concrete, we consider two ways to treat the temperature equation of the model: a semi-Lagrangian formulation of the advective terms in combination with a Galerkin finite element method for the parabolic part of the equations and a stabilized finite element scheme. Both methods are very robust and accurate. However, due to physical instabilities in the system at high Rayleigh number, the effect of the numerical method is significant with regard to the temperature distribution at a certain time instant. The good news is that relevant statistical quantities remain relatively stable and coincide for the two numerical schemes. The agreement is larger in the case of a mathematical model with constant water properties. In the case of a model with nonlinear dependence of the water properties on the temperature and pressure, the agreement in the statistics is clearly less pronounced. Hence, the presented work accentuates the need for a strengthened validation of the compatibility between numerical scheme (accuracy/resolution) and complex (realistic/nonlinear) models. Keywords Black smokers • Numerical simulation • Porous media • Statistical comparisons • Semi-Lagrangian method • Finite element method Communicated by Klein.

Research paper thumbnail of Optimal Control in Fluid Mechanics by Finite Elements with Symmetric Stabilization

SIAM Journal on Control and Optimization, 2009

Research paper thumbnail of Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

Computer Methods in Applied Mechanics and Engineering, 2011

For the Darcy-Brinkman equations, which model porous media flow, we present an equal-order H 1-co... more For the Darcy-Brinkman equations, which model porous media flow, we present an equal-order H 1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L 2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L 2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy-Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.

Research paper thumbnail of INSTITUTE OF PHYSICS PUBLISHING Combust. Theory Modelling 8 (2004) 661–682 COMBUSTION THEORY AND MODELLING

We present an algorithm for parameter identification in combustion problems modelled by partial d... more We present an algorithm for parameter identification in combustion problems modelled by partial differential equations. The method includes local mesh refinement controlled by a posteriori error estimation with respect to the error in the parameters. The algorithm is applied to two types of combustion problems. The first one deals with the identification of Arrhenius parameters, while in the second one diffusion coefficients for a hydrogen flame are calibrated.

Research paper thumbnail of Consistent Finite Elements for Optimal Control Problems in Computational Fluid Dynamics Contents

Research paper thumbnail of Finite Element Discretizations of Optimal Control Flow Problems with Boundary Layers

Lecture Notes in Computational Science and Engineering, 2011

We study the effect of different stabilized finite element methods to distributed control problem... more We study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. On the one hand, the residual based stabilized finite element method SUPG/PSPG leads to different optimality systems depending on the discretization approach: first discretize the state equation and then formulate the corresponding optimality system or derive first the optimality system on the continuous level and then discretize it. On the other hand, for symmetric stabilization as for instance the local projection stabilization (LPS) both approaches lead to the same symmetric optimality system. In particular, we address the question whether a possible commutation error in optimal control problems with boundary layers discretized by stabilized finite element methods may affect the accuracy significantly or not.

Research paper thumbnail of Fast and reliable solution of the Navier–Stokes equations including chemistry

Computing and Visualization in Science, 1999

In this paper, we describe recent developments in the design and implementation of Navier-Stokes ... more In this paper, we describe recent developments in the design and implementation of Navier-Stokes solvers based on finite element discretization. The most important ingredients are residual driven a posteriori mesh refinement, fully coupled defect-correction iteration for linearization, and optimal multigrid preconditioning. These techniques were systematically developed for computing incompressible viscous flows in general domains. Recently they have been extended to compressible low-Mach flows involving chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by several examples, (1) the accurate prediction of drag and lift coefficients, (2) the determination of CARS-signals of species concentration in flow reactors, (3) the computation of laminar flames.

Research paper thumbnail of A Posteriori Error Estimation by a Q 1 ∕ Q 2 Hierarchical Basis

Numerical Mathematics and Advanced Applications 2011, 2012

This work presents an a posteriori error estimation technique for Q 1 finite elements on quadrila... more 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.

Research paper thumbnail of Mesh and Model Adaptivity for Flow Problems

Reactive Flows, Diffusion and Transport

This work addresses two aspects. The first one is the practical realization of the dual weighted ... more This work addresses two aspects. The first one is the practical realization of the dual weighted residual (DWR) method of Becker & Rannacher [6] to three dimensional flow problems. The second aspect is its extension to modeling errors. This gives the possibility to use on the one hand locally refined meshes, and on the other hand, a hierarchy of models in order to reduce the numerical costs without sacrifying accuracy. The issue of modeling errors is of importance in many fields, especially in reactive flow computations where sophisticated models are developed but not feasable to be used in multidimensional computations due to the high numerical costs.

Research paper thumbnail of Solving Multidimensional Reactive Flow Problems with Adaptive Finite Elements

Reactive Flows, Diffusion and Transport

Research paper thumbnail of Adaptive finite element methods for flow problems

Foundations of Computational Mathematics, 2001

ABSTRACT We present a general approach to error control and mesh adaptation for computing viscous... more ABSTRACT We present a general approach to error control and mesh adaptation for computing viscous flows by Galerkin finite element method. A posteriori error estimates are derived for quantities of physical interest by duality arguments. In these estimates local cell residuals are multiplied by influence factors which are obtained from the numerical solution of a global dual problem. This provides the basis of a feedback algorithm, by which economical meshes can be constructed which are tailored to the particular needs of computation. The performance of this method is illustrated by several flow examples.

Research paper thumbnail of A posteriori control of modelling and discretization errors for quasi periodic solutions

Journal of Numerical Mathematics, 2014

We propose a duality based a posteriori error estimator for the computation of functionals averag... more We propose a duality based a posteriori error estimator for the computation of functionals averaged in time for nonlinear time dependent problems. Such functionals are typically relevant for (quasi-)periodic solutions in time. Applications arise, e.g. in chemical reaction models. In order to reduce the numerical complexity, we use simultaneously locally refined meshes and adaptive (chemical) models. Hence, considerations of adjoint problems measuring the sensitivity of the functional output are needed. In contrast to the classical dual-weighted residual (DWR) method, we favor a fixed mesh and model strategy in time. Taking advantage of the (quasi-)periodic behaviour, only stationary dual problems have to be solved.

Research paper thumbnail of Parallel Multigrid on Locally Refined Meshes

Reactive Flows, Diffusion and Transport

The use of PC-clusters gives the opportunity to solve three-dimensional flow problems to high acc... more The use of PC-clusters gives the opportunity to solve three-dimensional flow problems to high accuracy at reasonable costs. The use of adaptive mesh refinement can increase the efficiency of such computations even more. However, the combination of multigrid, local mesh refinement and parallelization is not trivial. Therefore, this work addresses the following two topics in order to develope a tool

Research paper thumbnail of On Error Control for Reactive Flow Problems

Scientific Computing in Chemical Engineering II, 1999

In this note, we present the application of weighted-residual techniques for a posteriori error e... more In this note, we present the application of weighted-residual techniques for a posteriori error estimation to reactive flow problems. The main goal is to control arbitrary functionals of the solutions and to obtain economical locally refined meshes. The discretization error is controlled by considering an associated dual problem, where the dual solution represents the sensitivity of the error to the local mesh-size distribution. The Galerkin orthogonality of the finite element discretization leads to an error estimator in which local residuals of the computed solution are multiplied by weights involving second derivatives of the dual solution. This estimator is used as a stopping criterion for the simulation and for local mesh refinement. In an iterative process the mesh is adapted in order to produce the minimal overall error for a prescribed number of grid nodes.

Research paper thumbnail of A Multiscale Method for Turbulent Flow Based on Local Projection Stabilization

Research paper thumbnail of Adaptive Finite Element Methods for PDE-Constrained Optimal Control Problems

Reactive Flows, Diffusion and Transport

Research paper thumbnail of Duality Based A Posteriori Error Estimation for Quasi-Periodic Solutions Using Time Averages

SIAM Journal on Scientific Computing, 2011

We propose an a posteriori error estimation technique for the computation of average functionals ... more We propose an a posteriori error estimation technique for the computation of average functionals of solutions for nonlinear time dependent problems based on duality techniques. The exact solution is assumed to have a periodic or quasi-periodic behavior favoring a fixed mesh strategy in time. We show how to circumvent the need of solving time dependent dual problems. The estimator consists of an averaged residual weighted by sensitivity factors coming from a stationary dual problem and an additional averaging error term coming from nonlinearities of the operator considered. In order to illustrate this technique the resulting adaptive algorithm is applied to several model problems: a linear scalar parabolic problem with known exact solution, the nonsteady Navier-Stokes equations with known exact solution, and finally to the well-known benchmark problem for Navier-Stokes (flow behind a cylinder) in order to verify the modeling assumptions.

Research paper thumbnail of Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method

SIAM Journal on Numerical Analysis, 2006

We propose to apply the recently introduced local projection stabilization to the numerical compu... more We propose to apply the recently introduced local projection stabilization to the numerical computation of the Oseen equation at high Reynolds number. The discretization is done by nested finite element spaces. Using a priori error estimation techniques, we prove the convergence of the method. The a priori estimates are independent of the local Peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth. Moreover, we show how this method may be cast in the framework of variational multiscale methods. We indicate what modeling assumptions must be made to use the method for large eddy simulations.

Research paper thumbnail of Multigrid techniques for finite elements on locally refined meshes

Numerical Linear Algebra with Applications, 2000

ABSTRACT We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast... more ABSTRACT We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast to a standard multigrid algorithm, where the hierarchy of meshes is generated by global refinement, we suppose that the finest mesh results from an adaptive refinement algorithm using bisection and ‘hanging nodes’. We discuss the additional difficulties introduced by these meshes and investigate two different algorithms. The first algorithm uses merely the local refinement regions per level, leading to optimal solver complexity even on strongly locally refined meshes, whereas the second one constructs the lower level meshes by agglomeration of cells. In this note, we are mainly interested in implementation details and practical performance of the two multigrid schemes. Copyright © 2000 John Wiley & Sons, Ltd.

Research paper thumbnail of Solution of a stationary benchmark problem for natural convection with large temperature difference

International Journal of Thermal Sciences, 2002

Computation of compressible flows at low Mach number is often performed by modifications of compr... more Computation of compressible flows at low Mach number is often performed by modifications of compressible flow solvers to the low Mach number limit. In particular, time stepping schemes are used where a Poisson problem is solved for pressure updates. In this paper, we propose a different approach for fast computation of stationary solutions. The discretization is made by bilinear and biquadratic finite elements with an elaborated stabilization. It allows the analytical construction of the Jacobian for the whole coupled system. It is used in Newton iterations for solving the nonlinear problems. No time stepping is needed for a wide class of stationary problems. Numerical computations of a benchmark problem for natural convection show the accuracy of the presented method. Different kinds of discretization are compared quantitatively. Furthermore, the aspect of local mesh refinement is addressed. The local refinement is controled by measuring the discretization error of the sought quantity by a posteriori error estimates. This concept is based on solving an adjoint problem which enters in the error estimator as weighting factors of local residuals.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Research paper thumbnail of The effect of numerical methods on the simulation of mid-ocean ridge hydrothermal models

Theoretical and Computational Fluid Dynamics, 2011

This work considers the effect of the numerical method on the simulation of a 2D model of hydroth... more This work considers the effect of the numerical method on the simulation of a 2D model of hydrothermal systems located in the high-permeability axial plane of mid-ocean ridges. The behavior of hot plumes, formed in a porous medium between volcanic lava and the ocean floor, is very irregular due to convective instabilities. Therefore, we discuss and compare two different numerical methods for solving the mathematical model of this system. In concrete, we consider two ways to treat the temperature equation of the model: a semi-Lagrangian formulation of the advective terms in combination with a Galerkin finite element method for the parabolic part of the equations and a stabilized finite element scheme. Both methods are very robust and accurate. However, due to physical instabilities in the system at high Rayleigh number, the effect of the numerical method is significant with regard to the temperature distribution at a certain time instant. The good news is that relevant statistical quantities remain relatively stable and coincide for the two numerical schemes. The agreement is larger in the case of a mathematical model with constant water properties. In the case of a model with nonlinear dependence of the water properties on the temperature and pressure, the agreement in the statistics is clearly less pronounced. Hence, the presented work accentuates the need for a strengthened validation of the compatibility between numerical scheme (accuracy/resolution) and complex (realistic/nonlinear) models. Keywords Black smokers • Numerical simulation • Porous media • Statistical comparisons • Semi-Lagrangian method • Finite element method Communicated by Klein.

Research paper thumbnail of Optimal Control in Fluid Mechanics by Finite Elements with Symmetric Stabilization

SIAM Journal on Control and Optimization, 2009

Research paper thumbnail of Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

Computer Methods in Applied Mechanics and Engineering, 2011

For the Darcy-Brinkman equations, which model porous media flow, we present an equal-order H 1-co... more For the Darcy-Brinkman equations, which model porous media flow, we present an equal-order H 1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L 2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L 2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy-Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.

Research paper thumbnail of INSTITUTE OF PHYSICS PUBLISHING Combust. Theory Modelling 8 (2004) 661–682 COMBUSTION THEORY AND MODELLING

We present an algorithm for parameter identification in combustion problems modelled by partial d... more We present an algorithm for parameter identification in combustion problems modelled by partial differential equations. The method includes local mesh refinement controlled by a posteriori error estimation with respect to the error in the parameters. The algorithm is applied to two types of combustion problems. The first one deals with the identification of Arrhenius parameters, while in the second one diffusion coefficients for a hydrogen flame are calibrated.

Research paper thumbnail of Consistent Finite Elements for Optimal Control Problems in Computational Fluid Dynamics Contents

Research paper thumbnail of Finite Element Discretizations of Optimal Control Flow Problems with Boundary Layers

Lecture Notes in Computational Science and Engineering, 2011

We study the effect of different stabilized finite element methods to distributed control problem... more We study the effect of different stabilized finite element methods to distributed control problems governed by singular perturbed Oseen equations. On the one hand, the residual based stabilized finite element method SUPG/PSPG leads to different optimality systems depending on the discretization approach: first discretize the state equation and then formulate the corresponding optimality system or derive first the optimality system on the continuous level and then discretize it. On the other hand, for symmetric stabilization as for instance the local projection stabilization (LPS) both approaches lead to the same symmetric optimality system. In particular, we address the question whether a possible commutation error in optimal control problems with boundary layers discretized by stabilized finite element methods may affect the accuracy significantly or not.

Research paper thumbnail of Fast and reliable solution of the Navier–Stokes equations including chemistry

Computing and Visualization in Science, 1999

In this paper, we describe recent developments in the design and implementation of Navier-Stokes ... more In this paper, we describe recent developments in the design and implementation of Navier-Stokes solvers based on finite element discretization. The most important ingredients are residual driven a posteriori mesh refinement, fully coupled defect-correction iteration for linearization, and optimal multigrid preconditioning. These techniques were systematically developed for computing incompressible viscous flows in general domains. Recently they have been extended to compressible low-Mach flows involving chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by several examples, (1) the accurate prediction of drag and lift coefficients, (2) the determination of CARS-signals of species concentration in flow reactors, (3) the computation of laminar flames.

Research paper thumbnail of A Posteriori Error Estimation by a Q 1 ∕ Q 2 Hierarchical Basis

Numerical Mathematics and Advanced Applications 2011, 2012

This work presents an a posteriori error estimation technique for Q 1 finite elements on quadrila... more 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.

Research paper thumbnail of Mesh and Model Adaptivity for Flow Problems

Reactive Flows, Diffusion and Transport

This work addresses two aspects. The first one is the practical realization of the dual weighted ... more This work addresses two aspects. The first one is the practical realization of the dual weighted residual (DWR) method of Becker & Rannacher [6] to three dimensional flow problems. The second aspect is its extension to modeling errors. This gives the possibility to use on the one hand locally refined meshes, and on the other hand, a hierarchy of models in order to reduce the numerical costs without sacrifying accuracy. The issue of modeling errors is of importance in many fields, especially in reactive flow computations where sophisticated models are developed but not feasable to be used in multidimensional computations due to the high numerical costs.

Research paper thumbnail of Solving Multidimensional Reactive Flow Problems with Adaptive Finite Elements

Reactive Flows, Diffusion and Transport

Research paper thumbnail of Adaptive finite element methods for flow problems

Foundations of Computational Mathematics, 2001

ABSTRACT We present a general approach to error control and mesh adaptation for computing viscous... more ABSTRACT We present a general approach to error control and mesh adaptation for computing viscous flows by Galerkin finite element method. A posteriori error estimates are derived for quantities of physical interest by duality arguments. In these estimates local cell residuals are multiplied by influence factors which are obtained from the numerical solution of a global dual problem. This provides the basis of a feedback algorithm, by which economical meshes can be constructed which are tailored to the particular needs of computation. The performance of this method is illustrated by several flow examples.

Research paper thumbnail of A posteriori control of modelling and discretization errors for quasi periodic solutions

Journal of Numerical Mathematics, 2014

We propose a duality based a posteriori error estimator for the computation of functionals averag... more We propose a duality based a posteriori error estimator for the computation of functionals averaged in time for nonlinear time dependent problems. Such functionals are typically relevant for (quasi-)periodic solutions in time. Applications arise, e.g. in chemical reaction models. In order to reduce the numerical complexity, we use simultaneously locally refined meshes and adaptive (chemical) models. Hence, considerations of adjoint problems measuring the sensitivity of the functional output are needed. In contrast to the classical dual-weighted residual (DWR) method, we favor a fixed mesh and model strategy in time. Taking advantage of the (quasi-)periodic behaviour, only stationary dual problems have to be solved.

Research paper thumbnail of Parallel Multigrid on Locally Refined Meshes

Reactive Flows, Diffusion and Transport

The use of PC-clusters gives the opportunity to solve three-dimensional flow problems to high acc... more The use of PC-clusters gives the opportunity to solve three-dimensional flow problems to high accuracy at reasonable costs. The use of adaptive mesh refinement can increase the efficiency of such computations even more. However, the combination of multigrid, local mesh refinement and parallelization is not trivial. Therefore, this work addresses the following two topics in order to develope a tool

Research paper thumbnail of On Error Control for Reactive Flow Problems

Scientific Computing in Chemical Engineering II, 1999

In this note, we present the application of weighted-residual techniques for a posteriori error e... more In this note, we present the application of weighted-residual techniques for a posteriori error estimation to reactive flow problems. The main goal is to control arbitrary functionals of the solutions and to obtain economical locally refined meshes. The discretization error is controlled by considering an associated dual problem, where the dual solution represents the sensitivity of the error to the local mesh-size distribution. The Galerkin orthogonality of the finite element discretization leads to an error estimator in which local residuals of the computed solution are multiplied by weights involving second derivatives of the dual solution. This estimator is used as a stopping criterion for the simulation and for local mesh refinement. In an iterative process the mesh is adapted in order to produce the minimal overall error for a prescribed number of grid nodes.

Research paper thumbnail of A Multiscale Method for Turbulent Flow Based on Local Projection Stabilization

Research paper thumbnail of Adaptive Finite Element Methods for PDE-Constrained Optimal Control Problems

Reactive Flows, Diffusion and Transport

Research paper thumbnail of Duality Based A Posteriori Error Estimation for Quasi-Periodic Solutions Using Time Averages

SIAM Journal on Scientific Computing, 2011

We propose an a posteriori error estimation technique for the computation of average functionals ... more We propose an a posteriori error estimation technique for the computation of average functionals of solutions for nonlinear time dependent problems based on duality techniques. The exact solution is assumed to have a periodic or quasi-periodic behavior favoring a fixed mesh strategy in time. We show how to circumvent the need of solving time dependent dual problems. The estimator consists of an averaged residual weighted by sensitivity factors coming from a stationary dual problem and an additional averaging error term coming from nonlinearities of the operator considered. In order to illustrate this technique the resulting adaptive algorithm is applied to several model problems: a linear scalar parabolic problem with known exact solution, the nonsteady Navier-Stokes equations with known exact solution, and finally to the well-known benchmark problem for Navier-Stokes (flow behind a cylinder) in order to verify the modeling assumptions.

Research paper thumbnail of Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method

SIAM Journal on Numerical Analysis, 2006

We propose to apply the recently introduced local projection stabilization to the numerical compu... more We propose to apply the recently introduced local projection stabilization to the numerical computation of the Oseen equation at high Reynolds number. The discretization is done by nested finite element spaces. Using a priori error estimation techniques, we prove the convergence of the method. The a priori estimates are independent of the local Peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth. Moreover, we show how this method may be cast in the framework of variational multiscale methods. We indicate what modeling assumptions must be made to use the method for large eddy simulations.

Research paper thumbnail of Multigrid techniques for finite elements on locally refined meshes

Numerical Linear Algebra with Applications, 2000

ABSTRACT We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast... more ABSTRACT We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast to a standard multigrid algorithm, where the hierarchy of meshes is generated by global refinement, we suppose that the finest mesh results from an adaptive refinement algorithm using bisection and ‘hanging nodes’. We discuss the additional difficulties introduced by these meshes and investigate two different algorithms. The first algorithm uses merely the local refinement regions per level, leading to optimal solver complexity even on strongly locally refined meshes, whereas the second one constructs the lower level meshes by agglomeration of cells. In this note, we are mainly interested in implementation details and practical performance of the two multigrid schemes. Copyright © 2000 John Wiley & Sons, Ltd.

Research paper thumbnail of Solution of a stationary benchmark problem for natural convection with large temperature difference

International Journal of Thermal Sciences, 2002

Computation of compressible flows at low Mach number is often performed by modifications of compr... more Computation of compressible flows at low Mach number is often performed by modifications of compressible flow solvers to the low Mach number limit. In particular, time stepping schemes are used where a Poisson problem is solved for pressure updates. In this paper, we propose a different approach for fast computation of stationary solutions. The discretization is made by bilinear and biquadratic finite elements with an elaborated stabilization. It allows the analytical construction of the Jacobian for the whole coupled system. It is used in Newton iterations for solving the nonlinear problems. No time stepping is needed for a wide class of stationary problems. Numerical computations of a benchmark problem for natural convection show the accuracy of the presented method. Different kinds of discretization are compared quantitatively. Furthermore, the aspect of local mesh refinement is addressed. The local refinement is controled by measuring the discretization error of the sought quantity by a posteriori error estimates. This concept is based on solving an adjoint problem which enters in the error estimator as weighting factors of local residuals.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.