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Anisotropic Meshes (anisotropic + mesh)

Distribution by Scientific Domains

Mathematics and Statistics	83%

Selected Abstracts

Anisotropic meshes and streamline-diffusion stabilization for convection,diffusion problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2005
Torsten Linß
Abstract We study a convection-diffusion problem with dominant convection. Anisotropic streamline aligned meshes with high aspect ratios are recommended to resolve characteristic interior and boundary layers and to achieve high accuracy. We address the question of how the stabilization parameter in the streamline-diffusion FEM (SDFEM) and the Galerkin least-squares FEM (GLSFEM) should be chosen inside the layers. Using a residual free bubbles approach, we show that within the layers the stabilization must be drastically reduced. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Toward anisotropic mesh construction and error estimation in the finite element method

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2002
Gerd Kunert
Abstract Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so-called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well-aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625,648, 2002; DOI 10.1002/num.10023 [source]

A posteriori error estimation for convection dominated problems on anisotropic meshes

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003
Gerd Kunert
Abstract 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. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Anisotropic a posteriori error estimate for an optimal control problem governed by the heat equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
Marco Picasso
Abstract The abstract framework of Becker et al. is considered to solve an optimal control problem governed by a parabolic equation. Existence and uniqueness of a solution are proved using the inf-sup framework and space-time functional spaces. A Crank-Nicolson time discretization is proposed, together with continuous, piecewise linear finite elements in space. Existence and uniqueness of a solution to the discretized problem is also proved using the inf-sup framework. An a posteriori error estimate is proposed, the goal being to control the error between the true and computed cost functional. The error estimate remains valid on strongly anisotropic meshes and an anisotropic error indicator is proposed when the time step is small. Finally, the quality of this error indicator is studied numerically on isotropic and anisotropic meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]