Matrix decomposition algorithms for the C0-quadratic finite element Galerkin method

• Published in: BIT (Nordisk Tidskrift for Informationsbehandling) Vol. 49; no. 3; pp. 509 - 526
• Main Authors: Du, Kui, Fairweather, Graeme, Nguyen, Que N., Sun, Weiwei
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.09.2009; Springer
• Subjects: Computational Mathematics and Numerical Analysis; Exact sciences and technology; Mathematical analysis; Mathematics; Mathematics and Statistics; Numeric Computing; Partial differential equations; Sciences and techniques of general use; Sequences, series, summability; Generalized eigenvalue problem; Finite element Galerkin method; Poisson’s equation; Piecewise quadratic functions; 65N22; 65F05; 65N30; Matrix decomposition algorithms; Polynomial; Poisson's equation; Partition; Fast Fourier transformation; Decomposition method; Poisson equation; Boundary condition; Matrix decomposition; Discretization method; Direct method; Finite element method; Matrix method; Galerkin method; Eigenvalue problem; Fast algorithm
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-009-0233-0

Explicit expressions for the eigensystems of one-dimensional finite element Galerkin (FEG) matrices based on C 0 piecewise quadratic polynomials are determined. These eigensystems are then used in the formulation of fast direct methods, matrix decomposition algorithms (MDAs), for the solution of the FEG equations arising from the discretization of Poisson’s equation on the unit square subject to several standard boundary conditions. The MDAs employ fast Fourier transforms and require O ( N 2 log  N ) operations on an N × N uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms.