Space–time finite element method for the multi-term time–space fractional diffusion equation on a two-dimensional domain

Abstract

In this paper, we propose a space–time finite element method for the multi-term time–space fractional diffusion equation. First, some fractional derivative spaces are listed, some properties of the fractional derivatives and fractional derivative spaces are given, some definitions and properties of finite element spaces are introduced, and a space–time finite element fully discrete scheme for the considered problem is developed. Second, the existence, uniqueness and stability of the obtained numerical scheme are discussed. Third, under the hypothesis about singular behavior of exact solution near t=0, the convergence is investigated in detail based on the suitable graded time mesh. At last, some numerical tests are given to verify the rationality and effectiveness of our method.