05625002

j

05625002 Du, Kui Fairweather, Graeme Nguyen, Que N. Sun, Weiwei Matrix decomposition algorithms for the $C^{0}$-quadratic finite element Galerkin method. BIT 49, No. 3, 509-526 (2009). 2009 Springer, Dordrecht EN finite element Galerkin method fast direct methods matrix decomposition algorithms Poisson's equation fast Fourier transforms numerical results

doi:10.1007/s10543-009-0233-0

Summary: 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 $N\times N$ uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms.