Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems

In this paper, we analyze the spatially semi-discrete piecewise linear finite volume element method for the time fractional sub-diffusion problem in two dimensions, and give an approximate solution of this problem. At first, we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods. Furthermore, we use the standard finite element Ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Finally, some numerical examples are included to illustrate the effectiveness of the new technique.