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

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.

An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

“…We choose α = β = 0.8 and τ = 10 −3 at t = 1 to observe the running time for different h. We observe that compared to the running time of FEM, CVM can reduce the running time significantly, which illustrates that CVM is more effective for solving the two-dimensional Riesz space fractional diffusion equation on convex domains. This is mainly due to the bilinear form in [41] that involves 8 fractional derivative terms and the approximation of two-fold multiple integrals, which are approximated by Gauss quadrature, while for CVM we only need to calculate 4 fractional derivative terms and the approximation of line integrals. We can see that the numerical solution is in excellent agreement with the exact solution, which demonstrates the effectiveness of our numerical method again.…”

“…Furthermore, according to the property of the stiffness matrix generated by the finite volume method, we chose a suitable sparse matrix format for the stiffness and utilised the Bi-CGSTAB iterative method to solve the linear system, which is more efficient than using Gauss elimination method. In addition, we made a comparison of our method with the finite element method proposed in [41], which demonstrated that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. In future work, we shall investigate the unstructured mesh control volume method applied to other fractional problems on irregular convex domains, such as the twodimensional multi-term time-space fractional diffusion equation with variable coefficients or three-dimensional space fractional diffusion equations with variable coefficients.…”

“…Karaa et al [40] proposed a finite volume element method implemented on an unstructured mesh for approximating the anomalous subdiffusion equations with a temporal fractional derivative. Yang et al [41] established the unstructured mesh finite element method for the nonlinear Riesz space fractional diffusion equations on irregular convex domains. Fan et al [42] extended the unstructured mesh finite element method developed by Yang et al [41] to the time-space fractional wave equation.…”

“…Yang et al [41] established the unstructured mesh finite element method for the nonlinear Riesz space fractional diffusion equations on irregular convex domains. Fan et al [42] extended the unstructured mesh finite element method developed by Yang et al [41] to the time-space fractional wave equation. Feng et al [43] investigated the unstructured mesh finite element method for a two-dimensional time-space Riesz fractional diffusion equation on irregular arbitrarily shaped convex domains and a multiply-connected domain.…”

“…equation (1) can be written as the following Riesz space fractional diffusion equation [38,41] ∂u(x, y, t) ∂t…”