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

TL;DR

This work tackles solving two-dimensional space-fractional diffusion equations with variable coefficients on irregular convex domains. It introduces an unstructured mesh control volume method (CVM) on triangular grids to discretize space-fractional derivatives of orders $0<\alpha<1$ and $0<\beta<1$, transforming the problem into a sparse linear system via Green's theorem and linear basis functions. A key contribution is the observation that the resulting stiffness matrix $\mathbf{M}$ is sparse and non-regular, enabling efficient storage and solution with the Bi-CGSTAB solver, achieving significant CPU-time savings while preserving accuracy relative to finite element methods on circular domains. The results demonstrate that CVM on unstructured triangular meshes is robust for arbitrarily shaped convex domains and provide a foundation for extending to more complex fractional models in higher dimensions.

Abstract

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.

Figures (5)

• Figure 1: The illustration of a solution domain with curved boundary
• Figure 2: The illustration of a control volume
• Figure 3: The illustration of line $y=y_r$ intersecting $n_q$ points with the support domain $\Omega_{e_k}$ of $l_k(x,y)$, where $(x_r,y_r)$ locates out of $\Omega_{e_k}$
• Figure 4: The illustration of line $y=y_r$ intersecting $n_q$ points with the support domain $\Omega_{e_k}$ of $l_k(x,y)$, where $(x_r,y_r)$ locates in $\Omega_{e_k}$
• Figure 5: The unstructured meshes with control volumes for $h\approx 8.6550\times 10^{-2}, 4.5873\times 10^{-2}$, respectively

Theorems & Definitions (4)

• Remark 1.1
• Remark 2.1
• Example 3.1
• Example 3.2