Academic literature on the topic 'MSC 65N50, MSC 65N30, MSC 35J05'

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.

The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

In this paper, several modifications of the quasi-interpolation operator of Scott and Zhang (Math. Comp. 54(1990)190, 483--493) are discussed. The modified operators are defined for non-smooth functions and are suited for the application on anisotropic meshes. The anisotropy of the elements is reflected in the local stability and approximation error estimates. As an application, an example is considered where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges.

This paper is concerned with the anisotropic singular behaviour of the solution of elliptic boundary value problems near edges. The paper deals first with the description of the analytic properties of the solution in newly defined, anisotropically weighted Sobolev spaces. The finite element method with anisotropic, graded meshes and piecewise linear shape functions is then investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates in anisotropically weighted spaces are derived. Moreover, it is shown that the condition number of the stiffness matrix is not affected by the mesh grading. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

This paper is concerned with mesh refinement techniques for treating elliptic boundary value problems in domains with re- entrant edges and corners, and focuses on numerical experiments. After a section about the model problem and discretization strategies, their realization in the experimental code FEMPS3D is described. For two representative examples the numerically determined error norms are recorded, and various mesh refinement strategies are compared.

The realization of the finite element method on parallel computers is usually based on a domain decomposition approach. This paper is concerned with the problem of finding an optimal decomposition and an appropriate mapping of the subdomains to the processors. The quality of this partitioning is measured in several metrics but it is also expressed in the computing time for solving specific systems of finite element equations. The software environment is first described. In particular, the data structure and the accumulation algorithm are introduced. Then several partitioning algorithms are compared. Spectral bisection was used with different modifications including Kernighan-Lin refinement, post-processing techniques and terminal propagation. The final recommendations should give good decompositions for all finite element codes which are based on principles similar to ours. The paper is a shortened English version of Preprint SFB393/96-18 (Uwe Reichel: Partitionierung von Finite-Elemente-Netzen), SFB 393, TU Chemnitz-Zwickau, December 1996. To be selfcontained, some material of Preprint SPC95_5 (see below) is included. The paper appeared as Preprint SFB393/96-18a, SFB 393, TU Chemnitz-Zwickau, January 1997.