Fast Direct Solvers for Piecewise Hermite Bicubic Orthogonal Spline Collocation Equations

Abstract

Fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise Hermite bicubics is employed for the approximate solution of Poisson’s equation in a rectangle. The new methods, which are matrix decomposition algorithms involving fast Fourier transforms, require $O(N^2 \log N)$ arithmetic operations on an $N \times N$ uniform partition and are highly parallel in nature. The treatment of more general elliptic problems is also discussed and the results of numerical experiments using parallel implementations of the new algorithms on several shared memory machines are presented.