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In this paper, we propose a novel unstructured mesh control volume method todeal 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 spacefractional diffusion equation with variable coefficients and provide the fullimplementation details for the case where the background interpolation mesh isbased on triangular elements. Secondly, we explore the property of thestiffness matrix generated by the integral of space fractional derivative. Wefind that the stiffness matrix is sparse and not regular. Therefore, we choosea suitable sparse storage format for the stiffness matrix and develop a fastiterative method to solve the linear system, which is more efficient than usingthe Gaussian elimination method. Finally, we present several examples to verifyour method, in which we make a comparison of our method with the finite elementmethod for solving a Riesz space fractional diffusion equation on a circulardomain. The numerical results demonstrate that our method can reduce CPU timesignificantly while retaining the same accuracy and approximation property asthe finite element method. The numerical results also illustrate that ourmethod is effective and reliable and can be applied to problems on arbitrarilyshaped convex domains.

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