Numerical solution of parabolic equations in high dimensions
Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
2 Seminar for Applied Mathematics, ETH Zentrum, 8092 Zürich, Switzerland. email@example.com.
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.
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