Discretization Methods and Iterative Solvers Based on Domain Decomposition

This chapter concerns domain decomposition methods as discretization techniques for partial differential equations. We present different approaches within the framework of mortar methods [BMP93, BMP94]. Originally introduced as a domain decomposition method for the coupling of spectral elements, these techniques are used in a large class of nonconforming situations. Thus, the coupling of different physical models, discretization schemes, or non-matching triangulations along interior interfaces of the domain can be analyzed by mortar methods. These domain decomposition techniques provide a more flexible approach than standard conforming formulations. They are of special interest for time dependent problems, rotating geometries, diffusion coefficients with jumps, problems with local anisotropies, corner singularities, and when different terms dominate in different regions of the simulation domain. Very often heterogeneous problems can be decomposed into homogeneous subproblems for which efficient discretization techniques are available. To obtain a stable and optimal discretization scheme for the global problem, the information transfer and the communication between the subdomains is of crucial importance; see Fig. 1.1.

This chapter concerns iterative solution techniques for linear systems of equations arising from the discretization of elliptic boundary value problems. Very often huge systems are obtained, with condition numbers which depend on the meshsize h of the triangulation, which typically grow in proportion to h^-2. Then, classical iteration schemes like Jacobi-, Gauß-Seidel or SOR-type methods result in very slow convergence rates. Fig. 2.1 shows the convergence rates and the number of iteration steps versus the number of unknowns, for a simple model problem in 2D. In the left, the convergence rates are given, and in the right the number of iteration steps to obtain an error reduction of 10^-6 are shown. For the Jacobi and the Gauß-Seidel method, the asymptotic convergence rates are 1 — O(h²). The optimal SOR-method is asymptotically better and tends with O(h) to one. However, the optimal damping parameter is, in general, unknown. The number of required iteration steps reflects the order of the method. For the Gauß-Seidel and the Jacobi method, the number of required iteration steps grows quadratically with one over the meshsize. In case of the optimal SOR-method, the increase is linear. Moreover, the numerical results show that the Jacobi method requires two times the number of Gauß-Seidel iteration steps.