Hybrid Finite Element--Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver

Journal Article · Thu Aug 02 00:00:00 EDT 2018 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/17M1144696· OSTI ID:1465192

Ainsworth, Mark ^[1]; Glusa, Christian ^[2]

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1. Brown Univ., Providence, RI (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
2. Brown Univ., Providence, RI (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Here, a numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $$\Omega$$ on the extruded domain $$\mathcal{C}=\Omega\times[0,\infty)$$ following. The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain $$\Omega$$, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $$\Omega$$. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions suggest that the approach is quasi-optimal in terms of complexity.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: ARO/MURI; USDOE
• Grant/Contract Number: AC04-94AL85000
• OSTI ID: 1465192
• Report Number(s): SAND--2018-5601J; 663488
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 4 Vol. 40; ISSN 1064-8275
• Publisher: SIAM
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
finite element--spectral method
fractional laplacian
numerical approximation