Galerkin FEM for fractional order parabolic equations with initial data in $H^{-s},0<s\le 1$

Abstract

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 $\Omega \subset {\mathbb{R}}^d$, $d=1,2,3$ is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(\Omega ),0\le s\le 1$. We confirm our theoretical findings with a number of numerical tests that include initial data $v$ being a Dirac $\delta$-function supported on a $(d-1)$-dimensional manifold.

arXiv comment13 pages, 3 figures