We present a spectral element algorithm and open-source code for computing the fractional Laplacian defined by the eigenfunction expansion on finite 2D/3D complex domains with both homogeneous and nonhomogeneous boundaries. We demonstrate the scalability of the spectral element algorithm on large clusters by constructing the fractional Laplacian based on computed eigenvalues and eigenfunctions using up to thousands of CPUs. To demonstrate the accuracy of this eigen-based approach for computing the factional Laplacian, we approximate the solutions of the fractional diffusion equation using the computed eigenvalues and eigenfunctions on a 2D quadrilateral, and on a 3D cubic and cylindrical domain, and compare the results with the contrived solutions to demonstrate fast convergence. Subsequently, we present simulation results for a fractional diffusion equation on a hand-shaped domain discretized with 3D hexahedra, as well as on a domain constructed from the Hanford site geometry corresponding to nonzero Dirichlet boundary conditions. Finally, we apply the algorithm to solve the surface quasi-geostrophic (SQG) equation on a 2D square with periodic boundaries. Simulation results demonstrate the accuracy, efficiency, and geometric flexibility of our algorithm and that our algorithm can capture the subtle dynamics of anomalous diffusion modeled by the fractional Laplacian on complex geometry domains. The included open-source code is the first of its kind.

Program title: Nektarpp_EigenMM

CPC Library link to program files: https://doi.org/10.17632/whtc75rj55.1

Developer’s repository link: https://github.com/paralab/Nektarpp_EigenMM

Licensing provisions: MIT License

Programming language: C/C++, MPI

Nature of problem: An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains.

Solution method: A distributed, sparse, iterative algorithm is developed to solve an associated integer-order Laplace eigenvalue problem for use in computing approximate solutions to the fractional diffusion equation.

Additional comments including restrictions and unusual features: The code is implemented on CPUs with super-linear parallel efficiency at extreme scale.