Computing Fractional Laplacians on Complex-Geometry Domains: Algorithms and Simulations

We consider a fractional Laplacian defined in bounded domains by the eigen-decomposition of the integer-order Laplacian, and demonstrate how to compute very accurately (using the spectral element method) the eigenspectrum and corresponding eigenfunctions in two-dimensional prototype complex-geometry domains. We then employ these eigenfunctions as trial and test bases to first solve the fractional diffusion equation, and subsequently to simulate two-phase flow based on the Navier--Stokes equations combined with a fractional Allen--Cahn mass-preserving model. A key point to the effectiveness of an exponential convergence of this approach is the use of a weighted Gram--Schmidt orthonormalization of the eigenfunctions that guarantees accurate projection and recovery of spectral accuracy for smooth solutions. We demonstrate that even when only part of the eigenspectrum is computed accurately we can still obtain exponential convergence if we employ the complete set of the eigenvectors of the discrete Laplacian. Accuracy is also verified by computing the eigenfunctions on square, disk, and L-shaped domains and obtaining numerical solutions of the fractional diffusion equation for different fractional orders. This is accomplished without the need of solving any linear systems as the eigenfunction decomposition leads naturally to a system of ODEs, and hence no spatial discretization is employed during time stepping. In the second application of the method, we replace the integer-order Laplacian in the Allen--Cahn model with its fractional counterpart and a similar procedure is followed. However, for the Navier--Stokes equations we need to solve a linear system, which we invert using an efficient ADI scheme. We demonstrate the effectiveness of the fractional Navier--Stokes/Allen--Cahn solver for the rising bubble problem in a square domain, and compare the results with the integer-order system and also with results by a different treatment of the fractional diffusion model using one-dimensional fractional derivatives. The present model yields sharper interface thickness compared to the integer-order model for the same resolution while it preserves the isotropic diffusion, and hence it is a good candidate for phase-field modeling of multiphase fluid flows.