Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Kassem Mustapha

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arXiv · 1409.6976

Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Kassem Mustapha

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Abstract

Time-stepping $hp$-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $-\alpha$ with $-1<\alpha<0$ will be proposed and analyzed. Generic $hp$-version error estimates are derived after proving the stability of the approximate solution. For $h$-version DG approximations on appropriate graded meshes near$t=0$, we prove that the error is of order$O(k^{\max\{2,p\}+\frac{\alpha}{2}})$, where $k$ is the maximum time-step size and $p\ge 1$ is the uniform degree of the DG solution. For $hp$-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.

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Kassem Mustapha. 2014-09-24. Time-stepping discontinuous Galerkin methods for fractional diffusion problems. https://arxiv.org/abs/1409.6976

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