The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For \emph{isotropic} meshes many estimators are known, but they either fail when used on \emph{anisotropic} meshes, or they were not applied yet. For rectangular (or cuboidal) anisotropic meshes a modified error estimator had already been found. We are investigating error estimators on anisotropic tetrahedral or triangular meshes because such grids offer greater geometrical flexibility. For the Poisson equation a residual error estimator, a local Dirichlet problem error estimator, and an $L_2$ error estimator are derived, respectively. Additionally a residual error estimator is presented for a singularly perturbed reaction diffusion equation. It is important that the anisotropic mesh corresponds to the anisotropic solution. Provided that a certain condition is satisfied, we have proven that all estimators bound the error reliably.

We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For isotropic meshes such estimators are known but they fail when applied to anisotropic meshes. Rectangular (or cuboidal) anisotropic meshes were already investigated. In this paper an error estimator is presented for tetrahedral or triangular meshes which offer a much greater geometrical flexibility.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For \emph{isotropic} meshes many estimators are known, but they either fail when used on \emph{anisotropic} meshes, or they were not applied yet. For rectangular (or cuboidal) anisotropic meshes a modified error estimator had already been found. We are investigating error estimators on anisotropic tetrahedral or triangular meshes because such grids offer greater geometrical flexibility. For the Poisson equation a residual error estimator, a local Dirichlet problem error estimator, and an $L_2$ error estimator are derived, respectively. Additionally a residual error estimator is presented for a singularly perturbed reaction diffusion equation. It is important that the anisotropic mesh corresponds to the anisotropic solution. Provided that a certain condition is satisfied, we have proven that all estimators bound the error reliably.

Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.