Numerical solution of parabolic equations in high dimensions

Abstract

We consider the numerical solution of diffusion problems in (0, T) × Ω for Ω ⊂ ℝd and for T > 0 in dimension d ≥ 1. We use a wavelet based sparse grid space discretization with meshwidth h and order p ≥ 1, and hp discontinuous Galerkin time-discretization of order r = O(|log h|) on a geometric sequence of O(|log h|) many time steps. The linear systems in each time step are solved iteratively by O(|log h|) 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 ∈ Hε(Ω) 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.