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

TL;DR

This paper develops and analyzes Galerkin finite element methods for solving fractional diffusion equations with non-smooth initial data in negative Sobolev spaces, providing optimal error estimates and numerical validation.

Contribution

It introduces and theoretically justifies optimal error estimates for Galerkin FEM schemes applied to fractional diffusion problems with initial data in $H^{-s}$, including singular cases like Dirac delta functions.

Findings

Optimal error estimates in $L_2$ and $H^1$ norms for non-smooth initial data.

Numerical tests confirm theoretical error bounds, including cases with Dirac delta initial data.

Methods are applicable to convex polygonal and polyhedral domains in 1-3 dimensions.

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.