SUPERCONVERGENCE OF A DISCONTINUOUS GALERKIN METHOD FOR FRACTIONAL DIFFUSION AND WAVE EQUATIONS

• Published in: SIAM journal on numerical analysis 2013-01, Vol.51 (1), p.491-515
• Main Authors: MUSTAPHA, KASSEM, MCLEAN, WILLIAM
• Format: Article
• Language: English
• Subjects: Applied mathematics; Cauchy problem; Convergence; Diffusion; Error bounds; Error rates; Errors; Evolution equations; Finite difference methods; Finite element method; Galerkin methods; Interpolation; Mathematical analysis; Mathematical models; Methods; Numerical methods; Orthogonality
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/120880719

We consider an initial-boundary value problem for ${\partial _t}u - \partial _t^{ - \alpha }{\nabla ^2}u = f\left( x \right)$ , that is, for a fractional diffusion (–1 < α < 0) or wave (0 < α < 1) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin (DG) method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near t = 0, but the spatial mesh is quasi-uniform. Previously, we proved that the error, measured in the spatial L₂-norm, is of order k²⁺ α ⁻ + h²ℓ(k), uniformly in t, where k is the maximum time step, h is the maximum diameter of the spatial finite elements, α– = min(α, 0) ≤ 0, and ℓ(k) = max(1, |log k|). Here, we prove convergence of order k³⁺² α ⁻ ℓ(k) + h² at each time level t n for –1 < α < 1. Thus, if –1/2 < α < 1, then the DG solution is superconvergent, which generalizes a known result for the classical heat equation (i.e., the case α = 0). A simple postprocessing step employing Lagrange interpolation leads to superconvergence for any t. Numerical experiments indicate that our theoretical error bound is pessimistic if α < 0. Ignoring logarithmic factors, we observe that the error in the DG solution at t = t n , and after postprocessing at all t, is of order k³⁺ α ⁻ + h² for –1 < α < 1.