Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H~(-s), 0 ≤ s ≤ 1

摘要

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω (is contained in) R~d, d = 1, 2, 3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L_(2-) and H~1-norms for initial data in H~(-s) (Ω), 0 ≤ s ≤ 1. We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac δ-function supported on a (d-1)-dimensional manifold.