Numerical and Computational Challenges
in Science and Engineering Program
LECTURE
Tuesday, November 13, 2001, 11:00 am, room 230
Graeme Fairweather
Department of Mathematical and Computer Sciences
Colorado School of Mines
Matrix Decomposition Algorithms in Finite Element Methods
In recent years, several matrix decomposition algorithms have been
developed for the efficient solution of the linear algebraic systems
arising when finite difference, finite element Galerkin (FEG), orthogonal
spline collocation and spectral methods are applied to Poisson problems
in the unit square. These algorithms depend on knowledge of the eigensystems
of discrete second derivative operators subject to certain boundary
conditions. When such an eigensystem is known for a particular method,
fast Fourier transforms can be employed to solve the corresponding linear
system in O(N^2 log N) operations on an N x N uniform partition of the
unit square.
In this talk, we describe new matrix decomposition algorithms for the
FEG method with piecewise Hermite bicubics and for modified spline collocation
with C^2 bicubic splines, for various boundary conditions. For modified
spline collocation, we present the results of numerical experiments
which confirm the published analysis of Dirichlet
and Neumann problems and indicate that similar results hold for mixed
and periodic boundary conditions. These results also exhibit superconvergence
phenomena not reported in earlier studies.
This is joint work with Bernard Bialecki, Andreas Karageorghis, D.
Abram Lipman and Gadalia M. Weinberg.
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