Galerkin finite element approximation of symmetric space-fractional partial differential equations

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• 作者：H. Zhang ; F. Liu ; V. Anh
• 关键词：Galerkin finite element approximation ; Space-fractional partial differential equations ; Stability ; Convergence
• 刊名：Applied Mathematics and Computation
• 出版年：2010
• 出版时间：15 November 2010
• 年：2010
• 卷：217
• 期：6
• 页码：2534-2545
• 全文大小：312 K

文摘

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β  (0, 1) and 2α  (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax–Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.