# Finite Element Methods for Solving Hydrodynamic Dispersion Equations

• Ne-Zheng Sun

## Abstract

Consider the following two-dimensional advection-dispersion equation
$$L(c) \equiv \frac{{\partial C}}{{\partial t}} + {V_x}\frac{{\partial C}}{{\partial x}} + {V_y}\frac{{\partial C}}{{\partial y}} - \frac{\partial }{{\partial x}}({D_{xy}}\frac{{\partial C}}{{\partial x}} + {D_{xy}}\frac{{\partial C}}{{\partial y}}) - \frac{\partial }{{\partial y}}({D_{xy}}\frac{{\partial C}}{{\partial x}} + {D_{yy}}\frac{{\partial C}}{{\partial y}}) + QC - I = 0,$$
(5.1.1)
which is subject to the initial condition
$$C(x,y,0) = f,{\kern 1pt} (x,y) \in (R),$$
(5.1.2)
boundary conditions
$$C(x,y,t) = {g_1},{\kern 1pt} (x,y) \in ({\Gamma _1}),$$
(5.1.3)
and
$$({D_{xx}}\frac{{\partial C}}{{\partial x}} + {D_{xy}}\frac{{\partial C}}{{\partial y}}){n_x} + ({D_{yx}}\frac{{\partial C}}{{\partial x}} + {D_{yy}}\frac{{\partial C}}{{\partial y}}){n_y} = - {g_2}(x,y) \in ({\Gamma _2}),$$
(5.1.4)
where (R) is the flow domain, (Γ1) and (Γ2) are boundary sections of (R), f is a given function in (R), g 1 and g 2 are given functions along (Γ1) and (Γ2), respectively, and n x and n y are components of the unit outer normal vector to the boundary (Γ2). Equation (5.1.3) expresses the boundary condition of given concentration, i.e., the first-type boundary condition, while Eq. (5.1.4) expresses the boundary condition of given dispersion flux, i.e., the second-type boundary condition.

## Keywords

Finite Element Method Basis Function Local Coordinate System Triangular Element Finite Element Equation