Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method

Abstract

In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability of fully discrete scheme are derived. To linearize the nonlinear system, the two-grid method based on Newton iteration are constructed. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment is presented further confirm the theoretical results.