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A Novel Unstructured Mesh Finite Element Method for Solving the Time-Space Fractional Wave Equation on a Two-Dimensional Irregular Convex Domain

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Abstract

Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicholson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicholson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analyzed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.

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Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University Jinan, 250100, PR China

    Wenping Fan

  2. School of Mathematical Sciences, Queensland University of Technology GPO, Box 2434, Brisbane, QLD 4001, Australia

    Fawang Liu

  3. School of Mathematics, Shandong University, Jinan, 250100, PR China

    Xiaoyun Jiang

  4. Australian Research Council Centre of Excellence for Mathematics and Statistical Frontiers School of Mathematical Sciences, Queensland University of Technology GPO, Box 2434, Brisbane, QLD 4001, Australia

    Ian Turner

Authors
  1. Wenping Fan
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  2. Fawang Liu
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  4. Ian Turner
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Correspondence to Wenping Fan.

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Fan, W., Liu, F., Jiang, X. et al. A Novel Unstructured Mesh Finite Element Method for Solving the Time-Space Fractional Wave Equation on a Two-Dimensional Irregular Convex Domain. FCAA 20, 352–383 (2017). https://doi.org/10.1515/fca-2017-0019

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  • DOI: https://doi.org/10.1515/fca-2017-0019

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