Issue |
ESAIM: M2AN
Volume 38, Number 1, January-February 2004
|
|
---|---|---|
Page(s) | 93 - 127 | |
DOI | https://doi.org/10.1051/m2an:2004005 | |
Published online | 15 February 2004 |
Numerical solution of parabolic equations in high dimensions
1
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
2
Seminar for Applied Mathematics, ETH Zentrum, 8092 Zürich, Switzerland.
schwab@sam.math.ethz.ch.
Received:
18
April
2003
We consider the numerical solution of diffusion problems in (0,T) x Ω for and for T > 0 in
dimension dd ≥ 1. We use a wavelet based sparse grid
space discretization with mesh-width h and order pd ≥ 1, and
hp discontinuous Galerkin time-discretization of order
on a geometric sequence of
many time
steps. The linear systems in each time step are solved iteratively
by
GMRES iterations with a wavelet preconditioner.
We prove that this algorithm gives an L2(Ω)-error of
O(N-p) for u(x,T) where N is the total number of operations,
provided that the initial data satisfies
with ε > 0
and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm the
theory.
Mathematics Subject Classification: 65N30
Key words: Discontinuous Galerkin method / sparse grid / wavelets.
© EDP Sciences, SMAI, 2004
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