Finite element approximation of a time-fractional diffusion problem for a domain with a re-entrant corner

Kim Ngan Le, William McLean, Bishnu Lamichhane

Abstract


An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain 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.

doi:10.1017/S1446181116000365

Keywords


local mesh refinement, non-smooth initial data, Laplace transformation.



DOI: http://dx.doi.org/10.21914/anziamj.v59i0.10940



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.