Domain decomposition methods for reaction-diffusion systems

[摘要] Domain Decomposition (DD) methods have been successfully used to solve elliptic problems, as they deal with them in a more elegant and efficient way than other existing numerical methods. This is achieved through the division of the domain into subdomains, followed by the solving of smaller problems within these subdomains which leads to the solution. Furthermore DD-techniques can incorporate in their implementation not only the physics of the different phenomena associated with the modeling, but also the enhancement of parallel computing. They can be divided into two major categories: with and without overlapping. The most important factor in both cases is the ability to solve the interface problem referred to as the Steklov-Poincaré problem. There are two existing approaches to solving the interface problem. The first approach consists of approximating the interface problem by solving a sequence of subproblems within the subdomains, while the second approach aims to tackle the interface problem directly. The solution method presented in this thesis falls into the latter category.This thesis presents a non-overlapping domain decomposition (DD) method for solving reaction-diffusion systems. This approach addresses the problem directly on the interface which allows for the presentation and analysis of a new type of interface preconditioner for the arising Schur complement problem. This thesis will demonstrate that the new interface preconditioner leads to a solution technique independent of the mesh parameter. More precisely, the technique, when used effectively, exploits the fact that the Steklov-Poincaré operators arising from a non-overlapping DD-algorithm are coercive and continuous, with respect to Sobolev norms of index 1/2, in order to derive a convergence analysis for a DD-preconditioned GMRES algorithm. This technique is the first of its kind that presents a class of substructuring methods for solving reaction diffusion systems and analyzes their behaviour using fractional Sobolev norms.

[发布机构] University:University of Birmingham;Department:School of Mathematics

[关键词] Q Science;QA Mathematics