The Superconvergence Of Finiteelement Method For Fractional Diffusion Equation

In this paper, we investigate time-fractional diffusion equation and mainly discussed finite element method as a whole superconvergent.In chapter one, we give a brief description on the history of development and research Status of fractional calculus. The classic definitions together with their properties of fractional derivatives, namely Grünwald-Letnikov, Riemann-Liouville, Caputo derivatives are listed. In addition, the overview on numercal methods for fractional differential equations is introduced.In chapter two, we discuss the error estimates about one-dimensional fractional diffusion, using hybrid quqadrature formual to replace the integral so as to build up the corresponding variational equation. Thus we obtain the error estimates.In chapter three, we consider two-dimensional fractional diffusion equation for semidiscrete space, getting semi-discrete equation. Then we use the properties of integral indentitties and fractional derivative discuss the superclose property. We can immediately have superconvergence from the superclose and improve the precision of the solution in the whole area through we construst the interpolation postprocessing operator. Finally this paper gives concrete numerical examples validated.