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.Comment: 18 pages, 5 figures, 9 table