An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates

doi:10.1007/s42967-019-00037-6

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (1): 147-162.doi: 10.1007/s42967-019-00037-6

An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates

Xiangcheng Zheng¹, V. J. Ervin², Hong Wang¹

1 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;
2 Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA

Received:2019-04-16 Revised:2019-06-09 Online:2020-03-20 Published:2020-02-19
Contact: Hong Wang, Xiangcheng Zheng, V. J. Ervin E-mail:hwang@math.sc.edu;xz3@math.sc.edu;vjervin@clemson.edu
Supported by:
This work was funded by the OSD/ARO MURI Grant W911NF-15-1-0562 and by the National Science Foundation under Grant DMS-1620194.

Abstract

Abstract: We study an indirect fnite element approximation for two-sided space-fractional difusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefcient models in terms of v(x), the solutions to the constant coeffcient analogues, we apply fnite element methods for the constant coefcient fractional difusion equations to solve for the approximations v_h(x) to v(x) and then obtain the approximations u_h(x) of u(x) by plugging v_h(x) into the representation of u(x). Optimal-order convergence estimates of u(x)-u_h(x) are proved in both L² and H^α∕2 norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.

Key words: Fractional difusion equation, Finite element method, Convergence estimate

CLC Number:

Xiangcheng Zheng, V. J. Ervin, Hong Wang. An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 147-162.

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An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates

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