ON multigrid methods for solving boundary value problems

. The multigrid iterative technique is a powerful method for solving the system of equations associated with discretized boundary value problems.The aim of this thesis is to develope and present the details of several multigrid algorithms which are applied to solve various types of boundary value problems.Basic methods, theoretical approaches and practical aspects of the multigridtechnique are described in a systematic way. Furthermore, some simple applications are discussed, and examplary multigrid subroutines are presented based on finite difference techniques.Although our methodic approaches is quite general, the concrete consideration inthe begining of this study refer to a limited class of problems; we explicitly treat only scalar partial differential equations in two dimensions; the underlying discretizations arebased on finite difference methods. Mostly, we are concerned with second or fourth order Direchlet boundary value problems on rectangular domains. Most of these restrictions are mainly for the sake of technical simplifications.The most important and new application in this thesis is the use of multigridtechniques in conjunction with the direct automated method to solve multiphase contact problems of plates and deep beams on continuous or settled elastic foundation. These problems are nonlinear boundary value problems. In these applications, the discretization is made by finite elements.Finite elements require an assembly process so costly as to make the use of the fast multigrid solver almost pointless. To overcome this difficulty in using finite elements with multigrid solvers, where applications are restricted to boundary value problems onuniform rectangular domains, a new method ” stencil representation ” is applied to assemble the elementary matrices in stencil patterns rather than in overall coefficient matrix.