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A Petrov-Galerkin Finite Element Method for Solving the Time-fractional Diffusion Equation with Interface

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  • Liwei Shi

Abstract

Time-fractional partial differential equation is widely applied in a variety of disciplines, its numerical solution has attracted much attention from researchers in recent years. Time-fractional differential equations with interfaces is a more challenging problem because the governing equation has discontinuous coefficients at interfaces and sometimes singular source term exists. In this paper, we propose a Petrov-Galerkin finite element method for solving the two-dimensional time-fractional diffusion equation with interfaces. In this method, a finite difference scheme is employed in time and a Petrov-Galerkin finite element method is employed in space. Extensive numerical experiments show that for a fractional diffusion equation of order $\alpha$ with interfaces, our method gets to $(2-\alpha)$-order accurate in the $L^2$ and $L^{\infty}$ norm.

Suggested Citation

  • Liwei Shi, 2018. "A Petrov-Galerkin Finite Element Method for Solving the Time-fractional Diffusion Equation with Interface," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 10(4), pages 136-150, August.
    • Handle: RePEc:ibn:jmrjnl:v:10:y:2018:i:4:p:136
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    References listed on IDEAS

    as
    1. Baeumer, B. & Benson, D.A. & Meerschaert, M.M., 2005. "Advection and dispersion in time and space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 245-262.
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    More about this item

    Keywords

    time-fractional partial differential equation; sharp-edged interface; Petrov-Galerkin finite element method; finite difference method;
    All these keywords.

    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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