Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Abstract

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order $0<\alpha <1$. As to the conforming scheme, the spatial global superconvergence and temporal convergence order of $O(h^2+\tau ^{2-\alpha })$ for both the original variable u in $H^1$-norm and the flux $\vec {p}=\nabla u$ in $L^2$-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and $\tau$ are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of $\textit{EQ}_1^{\textit{rot}}$ nonconforming element, which manifests that convergence orders of $O(h+\tau ^{2-\alpha })$ and $O(h^2+\tau ^{2-\alpha })$ for the original variable u in broken $H^1$-norm and $L^2$-norm, respectively, and approximation for the flux $\vec {p}$ converging with order $O(h+\tau ^{2-\alpha })$ in $L^2$-norm. Numerical examples are provided to demonstrate the theoretical analysis.