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.

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.

In this paper, we investigate the question of the spectrally equivalence of the so- called Pseudo-Laplacian to the usual discrete Laplacian in order to use hierarchical preconditioners for this more complicate matrix. The spectral equivalence is shown to be equivalent to a Brezzi-type inequality, which is fulfilled for the finite element spaces considered here.

A parallel realisation of the Conjugate Gradient Method with Schur-Complement preconditioning, based on a domain decomposition approach, is described in detail. Special kinds of solvers for the resulting interiour and coupling systems are presented. A large range of numerical results is used to demonstrate the properties and behaviour of this solvers in practical situations.

Adaptive finite element methods for the solution of partial differential equations require effective methods of mesh refinement and coarsening, fast multilevel solvers for the systems of FE equations need a hierarchical structure of the grid. In the paper a technology is presented for the application of irregular hierarchical triangular meshes arising from refinement by only dividing elements into four congruent triangles. The paper describes the necessary data structures and data structure management, the principles and algorithms of refining and coarsening the mesh, and also a specific assembly technique for the FE equations system. Aspects of the parallel implementation on MIMD computers with a message passing communication are included.

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.

In this paper, we investigate the question of the spectrally equivalence of the so- called Pseudo-Laplacian to the usual discrete Laplacian in order to use hierarchical preconditioners for this more complicate matrix. The spectral equivalence is shown to be equivalent to a Brezzi-type inequality, which is fulfilled for the finite element spaces considered here.

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.