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Numerical solution of parabolic equations in high dimensions

Published online by Cambridge University Press:  15 February 2004

Tobias von Petersdorff  and
Christoph Schwab
Tobias von Petersdorff
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH Zentrum, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch.
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Abstract

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ 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 $r = O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many time steps. The linear systems in each time step are solved iteratively by $O(\left|\log h\right|)$ 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 $u_0 \in H^\varepsilon(\Omega)$ 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.

Keywords

Discontinuous Galerkin methodsparse gridwavelets.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 93 - 127
Copyright
© EDP Sciences, SMAI, 2004

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