Fourier Matrix Decomposition Methods for the Least Squares Solution of Singular Neumann and Periodic Hermite Bicubic Collocation Problems - Bialecki, Bernard; Remington, Karin A. (SIAM journal on scientific computing, 1995)
The use of orthogonal spline collocation with piecewise Hermite bicubics is examined for the solution of Poisson's equation on a rectangle subject to either pure Neumann or pure periodic boundary conditions. Emphasis is placed on finding a least squares solution of these singular collocation problems. The technique of matrix decomposition is applied and explicit formulas for the requisite eigensystems corresponding to two-point Neumann and periodic collocation boundary value problems are presented. The resulting algorithms use fast Fourier transforms for efficiency and are highly parallel in nature. On an $N × N$ partition, a fourth order accurate least squares solution is computed at a cost of $O(N^2 \log N)$ operations. The results of numerical experiments are provided that demonstrate that the implementations compare very favorably with recent fourth order accurate finite difference and finite element Galerkin codes.