Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian (Communications in Computational Physics, 2025)
In the finite difference approximation of the fractional Laplacian the stiffnessmatrix is typically dense and needs to be approximated numerically. The effect of theaccuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed.While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of thefractional Laplacian, they differ in accuracy, economics to compute, performance ofpreconditioning, and asymptotic decay away from the diagonal line. In addition, twopreconditioners based on sparse and circulant matrices are discussed for the iterativesolution of linear systems associated with the stiffness matrix. Numerical results intwo and three dimensions are presented.