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arXiv · 1707.08057

Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems

Beiping Duan·Bangti Jin·Raytcho Lazarov·Joseph Pasciak·Zhi Zhou
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Abstract

We present and analyze a space-time Petrov-Galerkin finite element method for a time-fractional diffusion equation involving a Riemann-Liouville fractional derivative of order $\alpha\in(0,1)$ in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness. Further, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and establish error bounds in both energy and $L^2$ norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard $L^2$ stability property of the $L^2$ projection operator plays a key role. We provide extensive numerical examples to verify the convergence of the method.

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Beiping Duan, Bangti Jin, Raytcho Lazarov, Joseph Pasciak, Zhi Zhou. 2017-07-25. Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems. https://arxiv.org/abs/1707.08057

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