Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem

A time-fractional initial-boundary value problem is considered, where the differential equation has a sum of fractional derivatives of different orders, and the spatial domain lies in $${\mathbb {R}}^d$$ with $$d\in \{1,2,3\}$$. A priori bounds on the solution and its derivatives are stated; these show that typical solutions have a weak singularity at the initial time $$t=0$$. A standard finite element method with mapped piecewise bilinears is used to discretise the spatial derivatives, while for each time derivative we use the L1 scheme on a temporal graded mesh. Our analysis reveals the optimal grading that one should use for this mesh. A novel discrete fractional Gronwall inequality is proved: the statement of this inequality and its proof are different from any previously published Gronwall inequality. This inequality is used to derive an optimal error estimate in $$L^\infty (H^1)$$. It is also used to show that, if each mesh element is rectangular in the case $$d=2$$ or cubical in the case $$d=3$$, with the sides of the element parallel to the coordinate axes, then a simple postprocessing of the computed solution will yield a higher order of convergence in the spatial direction. Numerical results are presented to show the sharpness of our theoretical results.