Numerical Simulations And Analysis Of The Expanded Mixed Finite Element Method For Convection-Dominated Problem

In this paper,we consider the convection-dominated Sobolev equationand the quasilinear convection-dominated diffusion problemwhich are simulated by Expanded Characteristics-mixed Finite Element Method.The method is a combination of characteristic approximation to handle the con-vection part in time and an expanded mixed finite element approximation todeal with the diffusion part.The scheme is stable since fluid is transported alongthe approximate characteristics on the discrete level. At the same time it ex-pands the standard mixed finite element method in the sense that three variablesare explicitly treated:the scalar unknown,its gradient, and its flux optimally andsimulataneously.Numerical analysis shows that this method is stable and its L~2-error estimate is optimal. Then we consider the convection-dominated problemwhich is simulated by the stablized and Expanded mixed Finite Element Method.The method is a combination of stablized finite element method and Expandedmixed Finite Element Method.Numerical Analysis shows that the method canapproxiate the scalar unknown,its gradient and its flux optimally, also showsthe scheme is much stable than the standard Expanded mixed Finite ElementMethod.