Numerical solution of parabolic equations in high dimensions

Abstract

We consider the numerical solution of diffusion problems in (0,T ) x Ω for and for T > 0 in dimension d d ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order p d ≥ 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 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 with e > 0 and that u(x,t) is smooth in x for t>0 . Numerical experiments in dimension d up to 25 confirm the theory.