A Domain Decomposition Method for Legendre Spectral Collocation Problems
by
Andreas Karageorghis
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Coauthors: Bernard Bialecki, Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A.

We consider the solution of the Dirichlet boundary value problem for Pois­ son's equation on an L­shaped region using Legendre spectral collocation, that is, polynomial collocation at the Legendre­Gauss nodes. The region is partitioned into three nonoverlapping rectangular subregions with two inter­ faces. On each rectangular subregion, the approximate solution is a poly­ nomial tensor product that satisfies Poisson's equation at the collocation points in the subregion. The approximate solution is continuous throughout the region and its normal derivatives agree at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov­Poincar'e op­ erator corresponding to the interfaces is self­adjoint and positive definite with respect to the discrete inner product associated with the collocation points on the interfaces. The approximate solution on the interfaces is computed using the preconditioned conjugate gradient method. A preconditioner is obtained from two discrete Steklov­Poincar'e operators corresponding to two pairs of the adjacent rectangular subregions. Once the solution of the dis­ crete Steklov­Poincar'e equation is obtained, the collocation solution in each subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N^3 ), where the number of unknowns is proportional to N^2 .

Date received: February 14, 2001