on a geometric sequence of many timesteps. The linear systems in each time step are solved iterativelyby GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2( > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm thetheory." name="description" /> Cambridge Journals Online - ESAIM: Mathematical Modelling and Numerical Analysis - Abstract - Numerical solution of parabolic equations in high dimensions

ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Numerical solution of parabolic equations in high dimensions

von Petersdorff, Tobiasa1 and Schwab, Christopha2

a1 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.

a2 Seminar for Applied Mathematics, ETH Zentrum, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch.

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 L 2(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.

(Received April 18 2003)

(Online publication February 15 2004)

Key Words:

  • Discontinuous Galerkin method;
  • sparse grid;
  • wavelets.

Mathematics Subject Classification:

  • 65N30
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