Superconvergence Analysis for Time-Fractional Diffusion Equations with Nonconforming Mixed Finite Element Method

Abstract

In this paper, a fully discrete scheme based on the $L1$ approximation in temporal direction for the fractional derivative of order in (0, 1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order $O(h^2)$ of $EQ^{rot}_1$ element (see Lemma 2.3). Then, by using the proved character of $EQ^{rot}_1$ element, we present the superconvergent estimates for the original variable $u$ in the broken $H^1$-norm and the flux $\vec{q} = ∇u$ in the $(L^2)^2$-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.