Nonoverlapping domain decomposition with cross points for orthogonal spline collocation

Abstract

A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log²N, the total cost of the algorithm is O(N² log²N), where the number of unknowns in the collocation solution is O(N²). The approach presented in this paper, along with that in [B.Bialecki and M.Dryja, A nonoverlapping domain decomposition method for orthogonal spline collocation problems. SIAM J. Numer. Anal. (2003) 41, 1709 – 1728.], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation.