Study of Sparsification Schemes for the FEM Stiffness Matrix of Fractional Diffusion Problems

Anomalous diffusion describes various natural and social phenomena and processes in which the Brownian motion hypothesis is violated. Such problems can be modeled with the fractional Laplace operator. We consider the finite element discretization of the integral formulation of the Fractional Laplacian. The operator is non-local and as a result the stiffness matrix \(K\in {\mathbb {R}}^{N\times N}\) is dense. However, it can be observed that many of the off-diagonal coefficients have very small values relative to the corresponding diagonal elements. In this work, we study sparsification techniques like removing or lumping (summing into the diagonal) the coefficients that fall under a specified threshold. In this way we construct sparse approximations of the stiffness matrix K. Numerical results for a model fractional Laplacian boundary value problem are presented. Based on them, the accuracy of the approximate solutions is analysed.