Author : Jung-Han Kimn

Title : Overlapping Schwarz Algorithms using Discontinuous Iterates for Poisson's Equation

Abstract :

A new type of overlapping Schwarz methods, the overlapping Schwarz algorithms using discontinuous
iterates is constructed from the classical overlapping Schwarz algorithm. It allows for discontinuities
at each artificial interface. The new algorithm, for Poisson's equation, can be considered as an overlapping
version of Lions' Robin iteration method for which little is known concerning the convergence. Since overlap
improves the performance of the classical algorithms considerably, the existence of a uniform convergence
factor is the fundamental question for our new algorithm.

The first part of this thesis concerns the formulation of the new algorithm. A variational formulation of the new algorithm is derived from the classical algorithms. The discontinuity of the iterates of the new algorithm is the fundamental distinction from the classical algorithms. To analyze this important property, we use a saddle-point approach. We show that the new algorithm can be interpreted as a block Gauss-Seidel method with dual and primal variables.

The second part of the thesis deals with algebraic properties of the new algorithm. We prove that the fractional steps of the new algorithm are nonsymmetric. The algebraic systems of the primal variables can be reduced to those of the dual variables. We analyze the structure of the dual formulation algebraically and analyze its numerical behavior.

The remaining part of the thesis concerns convergence theory and numerical results for the new algorithm. We first extend the classical convergence theory, without using Lagrange multipliers, in some limited cases. A new theory using Lagrange multiplier is then introduced and we find conditions for the existence of uniform convergence factors of the dual variables, which implies convergence of the primal variables, in the two overlapping subdomain case with any Robin boundary condition. Our condition shows a relation between the given conditions and the artificial interface condition. The numerical results for the general case with cross points are also presented. They indicate possible extensions of our results to this more general case.