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

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.