Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallelepipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.

For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient). The optimality of the suggested technique is based on fictitious space and multilevel decom- position methods. Using an additive form of the preconditioners, we intro- duce factors into the preconditioners to optimize the corresponding conver- gence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to two-level $p$-hierarchical precondi- tioners and domain decomposition methods is considered too.

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

The paper gives an overview over local inequalities for anisotropic simplicial Lagrangian finite elements. The main original contributions are the estimates for higher derivatives of the interpolation error, the formulation of the assumptions on admissible anisotropic finite elements in terms of geometrical conditions in the three-dimensional case, and an anisotropic variant of the inverse inequality. An application of anisotropic meshes in the context of a stabilized Galerkin method for a convection-diffusion problem is given.

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.

This paper deals with the application of domain decomposition methods for the parallel solution of boundary value problems for partial differential equations over a domain $Omegabset R^d$, $d=2,3$. The attention is focused on the conception of efficient communication routines for the data exchange which is necessary for example in the preconditioned cg-algorithm for solving the resulting system of algebraic equations. The paper describes the data structure, different algorithms, and computational tests.

Abstract P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re- sults for 1D case, condition number estimates and some inequalities for 2D reference element.

Der Standard des Message Passing Interfaces (MPI) stellt dem Entwickler paralleler Anwendungen ein mächtiges Werkzeug zur Verfügung, seine Softwa- re effizient und weitgehend unabhängig von Details des parallelen Systems zu entwerfen. Im Rahmen einer Projektarbeit erfolgte die Umstellung der Kommunikationsbibliothek eines bestehenden FEM-Programmes auf den MPI-Mechanismus. Die Ergebnisse werden in der hier gegebenen Beschreibung der Cubecom-Implementierung zusammengefasst. In einem zweiten Teil dieser Arbeit wird untersucht, auf welchem Wege mit der in MPI verfügbaren Funktionalität auch die Koppelrandkommunikation mit einem einheitlichen und effizienten Verfahren durchgeführt werden kann. Sowohl fuer die Basisimplementierung als auch die MPI-basierte Koppelrandkommunikation wird die Effizienz untersucht und ein Ausblick auf weitere Anwendungsmoeglichkeiten gegeben.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

The paper gives an overview over local inequalities for anisotropic simplicial Lagrangian finite elements. The main original contributions are the estimates for higher derivatives of the interpolation error, the formulation of the assumptions on admissible anisotropic finite elements in terms of geometrical conditions in the three-dimensional case, and an anisotropic variant of the inverse inequality. An application of anisotropic meshes in the context of a stabilized Galerkin method for a convection-diffusion problem is given.