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

## TL;DR

This paper introduces a novel unstructured mesh control volume method for two-dimensional space fractional diffusion equations on convex domains, offering improved efficiency and applicability over traditional finite element methods.

## Contribution

The paper presents a new finite volume scheme for space fractional diffusion equations on arbitrary convex domains, including implementation details and an efficient solver for the resulting linear system.

## Key findings

- The method reduces CPU time significantly compared to finite element methods.
- The stiffness matrix is sparse and irregular, requiring specialized storage and solvers.
- Numerical experiments confirm the method's accuracy, efficiency, and applicability to complex domains.

## 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.