Domain decomposition methods for the problems with boundary layers - Akimov, Vladimir V. (2002)
In this thesis, we consider efficient preconditioned iterative methods for the numerical solution of large scale systems of linear algebraic equations with symmetric positive definite matrices originated from the finite element discretizations of the elliptic boundary value problems on meshes which are strongly refined in a vicinity of the boundary. We also numerically investigate and compare several typical discretizations of convection-diffusion equations in the case of large values of the Peclet number. The efficiency of iterative methods strongly depends on preconditioning. The approach presented in the thesis is based on the theory of the additive Schwarz methods. We partition the underlying finite element space into two overlapping subspaces and introduce the corresponding additive Schwarz preconditioner. The main idea of such a partitioning is to collect the degrees of freedom, related to the "coarse" part of the mesh in the first subspace, and collect the degrees of freedom, related to the "refined" part of the mesh, which is located in a small size vicinity of the boundary, in the second subspace. We prove that the condition number of the preconditioned matrix is bounded from above by a constant which is independent of the mesh in the vicinity of the boundary and the size of the overlap. We discuss generalizations and applications of the proposed preconditioners as well as their improvements and implementation algorithms. The preconditioners are relevant to parallel computers. Numerical experiments confirm the theoretical conclusions. The proposed approach is easily extendable to the three-dimensional elliptic problems. In the last part of the thesis, we investigate by numerical experiments and compare three finite element approximations of the convection-diffusion equation with realistic input data. We investigate the classical Galerkin, stabilized finite element, and least-squares finite element methods. The results of comparison, especially for large values of the Peclet number, can be used in practical applications.