Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation (Advances in Applied Mathematics and Mechanics, 2012)
We present the finite difference/element method for atwo-dimensional modified fractional diffusion equation. The analysisis carried out first for the time semi-discrete scheme, and then forthe full discrete scheme. The time discretization is based on the$L1$-approximation for the fractional derivative terms and thesecond-order backward differentiation formula for the classicalfirst order derivative term. We use finite element method for thespatial approximation in full discrete scheme. We show that both thesemi-discrete and full discrete schemes are unconditionally stableand convergent. Moreover, the optimal convergence rate is obtained.Finally, some numerical examples are tested in the case of one andtwo space dimensions and the numerical results confirm ourtheoretical analysis.