Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains

TL;DR

This paper develops a finite element method for solving nonlinear Riesz space fractional diffusion equations on irregular convex domains, introducing an algorithm for stiffness matrix formation on unstructured meshes.

Contribution

It presents a novel algorithm for stiffness matrix assembly on unstructured meshes for fractional PDEs, enabling solutions on irregular domains.

Findings

The scheme is stable and convergent.

Numerical examples verify accuracy.

Method effectively handles complex geometries.

Abstract

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.