7. Concluding remarks
In this paper we utilize the DPG framework of Demkowicz and Gopalakrishnan [11–14] to develop a Petrov– Galerkin finite element method for one-dimensional variable-coefficient fractional diffusion equations, which arise in many physical and engineering applications. We note that the DPG framework applies to multidimensional spacefractional diffusion equations. Hence, in principle, we can apply the same idea in the current paper to develop Petrov–Galerkin finite element methods for multidimensional space-fractional diffusion equations. However, there exist major obstacles that need to be overcome in the development: (i) The proof of Theorem 2 in [9] relies heavily on the particular form of (5). (ii) Because of the weakly singular kernel and non-locality of fractional differential operators, the multidimensional trial to test operator can be expensive. (iii) The stiffness matrix of the Petrov–Galerkin finite element is full, for which traditional direct solver requires O(N 3 ) computational complexity and O(N 2 ) memory for a problem with N unknowns. A careful study needs to be carried out to investigate whether the stiffness matrix has certain Toeplitz-like structure as in the case of finite difference methods, so that a fast Fourier transform based fast Krylov subspace iterative method can be developed which has an almost linear computational complexity and memory requirement [33–35]. (iv) Efficient (usually full matrix) preconditioners need to be developed to significantly reduce the number of iterations in the Krylov subspace iterative method [36–38].
