Space-Fractional Diffusion with Variable Order and Diffusivity: Discretization and Direct Solution Strategies

We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these due to kernel singularity in integral operator resulting dense discretized operators, which quickly become prohibitively expensive handle because of their memory arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes singular integrals through subtraction technique adapted spatial variability This regularization strategy is conveniently formulated as sparse matrix correction added operator, applicable different formulations equations. also block low rank representation representations, by exploiting ability approximate blocks formally factorizations. A Cholesky factorization solver operates directly on using its atomic tiles, achieves high performance multicore hardware. Numerical results show treatment robust, substantially reduces errors, attains first-order convergence rate allowed regularity solutions. They considerable savings obtained storage ( $$O(N^{1.5})$$ ) cost $$O(N^2)$$ compared translates orders-of-magnitude time problems, shows proposed methods offer practical tools for tackling large nonlocal simulations.