Jin, B; Duan, B; Lazarov, R; Pasciak, J; Zhou, Z; (2018) Space-Time Petrov–Galerkin FEM for Fractional Diffusion Problems. Computational Methods in Applied Mathematics , 18 (1) pp. 1-20. 10.1515/cmam-2017-0026.

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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 α ∈ (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 derive 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 analysis.

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