FOURIER MATRIX DECOMPOSITION METHODS FOR THE LEAST-SQUARES SOLUTION OF SINGULAR NEUMANN AND PERIODIC HERMITE BICUBIC COLLOCATION PROBLEMS

The use of orthogonal spline collocation with piecewise Hermite bicubi cs is examined for the solution of Poisson's equation on a rectangle s ubject to either pure Neumann or pure periodic boundary conditions. Em phasis is placed on finding a least squares solution of these singular collocation problems. The technique of matrix decomposition is applie d and explicit formulas for the requisite eigensystems corresponding t o two-point Neumann and periodic collocation boundary value problems a re presented. The resulting algorithms use fast Fourier transforms for efficiency and are highly parallel in nature. On an N x N partition, a fourth order accurate least squares solution is computed at a cost o f O(N-2 log N) operations. The results of numerical experiments are pr ovided that demonstrate that the implementations compare very favorabl y with recent fourth order accurate finite difference and finite eleme nt Galerkin codes.