A novel approach of high accuracy analysis of anisotropic bilinear finite element for time-fractional diffusion equations with variable coefficient

In this paper, by using bilinear finite element and L1 approximation, a fully-discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes. Unconditionally stable analysis of the proposed scheme are presented in both L2-norm and H1-norm. Moreover, convergence, superclose and superconvergence results are derived by combining interpolation with projection, which is the key technique for the numerical analysis. Specifically, by defining a novel projection operator, the error estimate between the projection and the exact solution is obtained on anisotropic meshes. Furthermore, high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique. Based on the related results about projection and interpolation, the optimal error estimate in L2-norm and superclose of interpolation in H1-norm are deduced by skillfully dealing with fractional derivative. At the same time, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.