Parallel scalability of sparse solvers for sparsified fractional diffusion problems discretized with FEM

Abstract

Anomalous diffusion is a non-local process that describes a wide spectrum of natural processes and phenomena with many applications in science and technology. It is described mathematically by the fractional Laplace operator. In this work we examine the integral definition of the fractional Laplacian modeled with the Riesz potential and discretized with the finite element method. The thus obtained system of linear algebraic equations is dense due to the non-local nature of the fractional Laplacian and is computationally complex to solve. With LU factorization, for example, solving the problem has an O(N 3) computational complexity, where N is the number of unknowns. However, it can be observed that a large amount of the off-diagonal coefficients have very small absolute values compared to the diagonal coefficients. Those small off-diagonal coefficients can be lumped (set to zero and added to the diagonal coefficient) without significant loss of accuracy. In this work we employ a direct sparse parallel solver to the resultant sparse matrix. We analyze the parallel performance and speed-up, as well as the accuracy, varying the fractional power and the lumping threshold.