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Parallel BURA Based Numerical Solution of Fractional Laplacian with Pure Neumann Boundary Conditions

  • Gergana Bencheva
  • Nikola Kosturski
  • Yavor VutovEmail author
Conference paper
  • 88 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

Abstract

The study is motivated by the increased usage of fractional Laplacian in the modeling of nonlocal problems like anomalous diffusion. We present a parallel numerical solution method for the nonlocal elliptic problem: \(-\varDelta ^\alpha u = f\), \(0<\alpha < 1\), \(-\partial u(x)/\partial n=g(x)\) on \(\partial \varOmega \), \(\varOmega \subset \mathrm{I\!R}^d\). The Finite Element Method (FEM) is used for discretization leading to the linear system \(A^\alpha {\mathbf{u}} = {\mathbf{f}}\), where A is a sparse symmetric and positive semidefinite matrix. The implemented method is based on the Best Uniform Rational Approximation (BURA) of degree k, \(r_{\alpha ,k}\), of the scalar function \(t^{\alpha }\), \(0\le t \le 1\). The related approximation of \(A^{-\alpha }{\mathbf{f}}\) can be written as a linear combination of the solutions of k local problems. The latter are found using the preconditioned conjugate gradient method. The method is applicable to computational domains with general geometry. Linear finite elements on unstructured tetrahedral meshes with local refinements are used in the presented numerical tests. The behavior of the relative error, the number of Preconditioned Conjugate Gradient (PCG) iterations, and the parallel time is analyzed varying the parameter \(\alpha \in \{0.25, 0.50, 0.75\}\), the BURA degree \(k \in \{5, 6,\dots ,12\}\), and the mesh size.

Keywords

BURA Fractional diffusion Neumann boundary conditions Unstructured meshes Parallel algorithm 

Notes

Acknowledgments

We acknowledge the provided access to the e-infrastructure and support of the Centre for Advanced Computing and Data Processing, with the financial support by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014–2020) and co-financed by the European Union through the European structural and Investment funds.

The presented work is partially supported by the Bulgarian National Science Fund under grant No. DFNI-DN12/1.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Gergana Bencheva
    • 1
  • Nikola Kosturski
    • 1
  • Yavor Vutov
    • 1
    Email author
  1. 1.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria

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