A Difference Mixed Finite Element Method for the Three Dimensional Poisson Equation

A difference mixed finite element method based on the finite element pair (( P 0 x P 1 ), ( P 0 x P 1 ), ( P 1 x P 0 ))x( P 1 x P 1 ) for the three dimensional Poisson equation is studied. The method combines a mixed finite element discretization based on ( P 0 , P 0 , P 1 )x P 1- element in ( x , y )-plane and a finite difference discretization based on ( P 1 , P 1 , P 0 ) x P 1- element in z-direction. This allows to transmit the finite element solution of the three dimensional Poisson equation in the direction ( x , y , z ) into a series of finite element solution of the two dimensional Poisson equation in the direction ( x , y ). Moreover, error estimates for the discrete approximation are derived. Numerical tests show the accuracy and efficiency for the method proposed.