Fast Calculation of Linear Finite Element Method for the Stationary Fractional Advection Dispersion Equations

Abstract

Appropriate variational formulation and detailed implementation of linear finite element for the stationary fractional advection dispersion equation(FADE) are discussed. Since fractional derivative is nonlocal operator, the stiffness matrix of finite element on traditional variational formulation for FADE is no longer sparse and the computation becomes costly. In this paper, we establish some fractional order integral and differential formulas for linear interpolation basis functions, and then design a special variational formulation which makes the stiffness matrix possess some good properties, such as quasi-symmetry, quasi-sparseness and strictly diagonally domination. These properties are very important in reducing the computational cost and guaranteeing the stability of finite element equations. Numerical examples demonstrating these properties are presented and the applications in contaminant transport in groundwater flow are given.