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

We study the additive and multiplicative Schwarz domain decomposition metho ds for elliptic boundary value problem of order 2r based on an appropriate spline space of smoothness r - 1. The finite element method reduces-an elli ptic boundary value problem to a linear system of equations. It is well kno wn that as the number of triangles in the underlying triangulation is incre ased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. Th e Schwarz domain decomposition methods will enable us to break the linear s ystem into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domai n 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 me thod. We tested these methods for the biharmonic equation with Dirichlet bo undary condition over an arbitrary polygonal domain using C-1 cubic spline functions over a quadrangulation of the given domain. The computer experime nts agree with our theoretical results.