The Analysis And Application Of Two Mixed Finite Element Methods

In the first part of the paper, we consider the regularized long wave equationand the Sobolev equationwhich are simulated by Galerkin mixed finite element method. This method first split the initial problem into a first order system and then propose a nonsymmetric version of a least square method that is an Hl-Ga\erkin prosedure for the solution and its flux. Compared to the standard '-Galerkin, -continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method. Moreover, the approximating finite element spaces and Wh are allowed to be of differing polynomial degrees. Hence, estimations have been obtained which ditinguish the better approximation properties of Vh and W. We obtain the optimal order of convergence theoreticaly. Numerical examples conform the efficiency of our method.Then we consider the numerical simulation of the convection-dominated transport nrohlemThe governing equation is uniformly parabolic, but in many applications, the convection dominates diffusion, the equation is nearly hyperbolic in nature. It is especially difficult to approximate well the sharp fronts and conserve the material or mass in the system. In this paper, we propose an expanded characteristics-mixed finite element method for approximating the solution to convection dominated transport problem.The method is a combination of characteristic approximation to handle the convection part in time and anexpanded mixed finite element spatial approximation to deal with the diffusion part. The scheme is stable since fluid is transported along the approximate characteristics on the discrete level. At the same time it expands the standard mixed finite element method in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux. Our analysis show the method approximate the scalar unknown, its gradient, and its flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the scheme is of high performance.