A Nonoverlapping Domain Decomposition Method for Legendre Spectral Collocation Problems

We consider the Dirichlet boundary value problem for Poisson’s equation in an L-shaped region or a rectangle with a cross-point. In both cases, we approximate the Dirichlet problem using Legendre spectral collocation, that is, polynomial collocation at the Legendre–Gauss nodes. The L-shaped region is partitioned into three nonoverlapping rectangular subregions with two interfaces and the rectangle with the cross-point is partitioned into four rectangular subregions with four interfaces. In each rectangular subregion, the approximate solution is a polynomial tensor product that satisfies Poisson’s equation at the collocation points. The approximate solution is continuous on the entire domain and its normal derivatives are continuous at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. At the cross point, we require continuity of the normal derivative in the vertical direction. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov–Poincaré operator corresponding to the interfaces is self-adjoint 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 the discrete Steklov–Poincaré operators corresponding to pairs of the adjacent rectangular subregions. Once the solution of the discrete Steklov–Poincaré equation is obtained, the collocation solution in each rectangular subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N3), where the number of unknowns is proportional to N2.