Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain

TL;DR

This paper develops a finite element method for a time-fractional diffusion problem in non-convex polygons, showing that local mesh refinement near re-entrant corners restores second-order accuracy, extending classical results.

Contribution

It introduces a localized mesh refinement strategy for fractional diffusion equations in non-convex domains, improving convergence rates over uniform meshes.

Findings

Local mesh refinement restores second-order convergence.

Error bounds are established for non-convex polygonal domains.

Extension of classical heat equation results to fractional diffusion problems.

Abstract

An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer $H^2$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation due to Chatzipantelidis, Lazarov, Thom\'ee and Wahlbin.