Wavelet discretizations of parabolic integrodifferential equations

摘要

We consider parabolic problems. (u) over dot + Au = f in (0, T) x Omega, T < &INFIN;, where &UOmega; &SUB; R-d is a bounded domain and A is a strongly elliptic classical pseudodifferential operator of order ρ &ISIN; [0, 2] in <(H)over tilde>(rho/2)(Omega). We use theta-scheme for time discretization and a Galerkin method with N degrees of freedom for space discretization. The full Galerkin matrix for A can be replaced with a sparse matrix using a wavelet basis, and the linear systems for each time step are solved approximatively with GMRES. We prove that the total cost of the algorithm for M time steps is bounded by O(M N(log N)(beta)) operations and O(N(log N)(beta)) memory. We show that the algorithm gives optimal convergence rates ( up to logarithmic terms) for the computed solution with respect to L-2 in time and the energy norm in space. [References: 18]