Fourier Matrix Decomposition Methods for the Least Squares Solution of Singular Neumann and Periodic Hermite Bicubic Collocation Problems

• Published in: SIAM journal on scientific computing Vol. 16; no. 2; pp. 431 - 451
• Main Authors: Bialecki, Bernard, Remington, Karin A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.03.1995
• Subjects: Algorithms; Approximation; Boundary conditions; Boundary value problems; Decomposition; Exact sciences and technology; Fourier transforms; Laboratories; Mathematics; Methods; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Sciences and techniques of general use; Neumann problem; Parallel algorithm; Fast Fourier transformation; Decomposition method; Poisson equation; Matrix decomposition; Partial differential equation; Implementation; Spline; Finite element method; Boundary value problem; Least squares method; Singular solution; Galerkin method; Finite difference method
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/0916027

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 $N \times N$ 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.