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

  • Mark AinsworthEmail author
  • Christian Glusa
Chapter

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.

Notes

Acknowledgements

This work was supported by the MURI/ARO on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications” (W911NF-15-1-0562).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  3. 3.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA
  4. 4.Division of Applied MathematicsBrown UniversityProvidenceUSA

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