Efficient Implementation of Finite Element Methods for Spatial Fractional PDEs in Three Dimensions.

In this paper, we address the main challenges of the implementation of finite element methods for solving spatial fractional problems on three dimensional irregular convex regions. Different from the integer case, the non-locality of fractional derivative operators makes the assembly of fractional stiffness matrix much more difficult, mainly in two aspects: one is the search of the integral path of Gaussian points, and the other is the calculation of fractional derivatives of basis functions at Gaussian points. By introducing the ray-simplex intersection algorithm, we present an efficient method for finding integration paths of Gaussian points. By analyzing the expression of the fractional derivative of finite element basis functions, we give a method for efficiently calculating the fractional derivative. In order to speed up the procedure in MATLAB, some implementation techniques are introduced. We demonstrate our method by applying it to different problems, including steady and transient spatial fractional problems.