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Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • Mark Ainsworth

    (Brown University, Division of Applied Mathematics
    Oak Ridge National Laboratory, Computer Science and Mathematics Division)

  • Christian Glusa

    (Sandia National Laboratories, Center for Computing Research
    Brown University, Division of Applied Mathematics)

Abstract

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples.

Suggested Citation

  • Mark Ainsworth & Christian Glusa, 2018. "Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 17-57, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_2
    DOI: 10.1007/978-3-319-72456-0_2
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