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) 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 {L^{2}} norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L 2 {L^{2}} stability property of the L 2 {L^{2}} projection operator plays a key role. We provide extensive numerical examples to verify the convergence analysis.
Original language | English |
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Pages (from-to) | 1-20 |
Number of pages | 20 |
Journal | Computational Methods in Applied Mathematics |
Volume | 18 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Jan 2018 |
Keywords
- Error Estimates
- Fractional Diffusion
- Petrov-Galerkin Method
- Space-Time Finite Element Method
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics