Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

Abstract

The anomalous diffusion problem has been played a significant role in many areas. In this paper, we consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. We derive the variational formula of the semi-discrete scheme by using the Galerkin finite element method in space. Existence of the semi-discrete solution for the equation is shown. The stability and the order of convergence of approximate solutions for the semi-discrete equation have been also discussed.

Furthermore, we derive the fully discrete time-space variational formulation using the backward Euler method. Existence of numerical solutions for the backward Euler fully discrete scheme is shown by using the Brouwer fixed point theorem. The stability and error estimates of solutions for the fully discrete approximate solutions are studied along the lines of the semi-discrete analysis. The order of convergence are obtained as $O(k+h^{\tilde{\gamma}})$, where $\tilde \gamma$ is a constant depending on the order of fractional derivative.

Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we discuss in theoretical analysis.

A part of thesis has been published in Abstract and Applied Analysis, 2012, doi:10.1155/2012/596184.