On Schwarz's domain decomposition methods for elliptic boundary value problems

We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2r based on an appropriate spline space of smoothness r - 1. The finite element method reduces-an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using C-1 cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results.