Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

Abstract

AbstractWe present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on theL1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.