Space-Time Petrov–Galerkin FEM for Fractional Diffusion Problems
| Autor: | Joseph E. Pasciak, Zhi Zhou, Bangti Jin, Beiping Duan, Raytcho D. Lazarov |
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| Rok vydání: | 2017 |
| Předmět: |
Numerical Analysis
Diffusion equation Applied Mathematics Space time Petrov–Galerkin method 010103 numerical & computational mathematics Weak formulation Bilinear form 01 natural sciences Finite element method Fractional calculus 010101 applied mathematics Computational Mathematics Convergence (routing) Applied mathematics 0101 mathematics Mathematics |
| Zdroj: | Computational Methods in Applied Mathematics. 18:1-20 |
| ISSN: | 1609-9389 1609-4840 |
| DOI: | 10.1515/cmam-2017-0026 |
| Popis: | 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 ) {\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 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. |
| Databáze: | OpenAIRE |
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