A Fast Domain Decomposition Poisson Solver on a Rectangle for Hermite Bicubic Orthogonal Spline Collocation

Abstract

A fast domain decomposition solver is presented for the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The rectangle is divided into parallel strips and the collocation solution is first obtained on the interfaces by solving a collection of independent tridiagonal linear systems. A recently developed fast Fourier transform solver for piecewise Hermite bicubic orthogonal spline collocation is then used to compute the collocation solution on each strip. On an $N \times N$ uniform partition, the proposed domain decomposition solver requires $O(N^2 \log \log N)$ arithmetic operations, assuming that the strips have the same width and that their number is proportional to ${N / {\log N}}$. The solver is also highly parallel in nature.