The Effect Of Numerical Integration For Expanded Mixed Finite Element Methods

It is well-known that the treatment of approximations to the solution of a partial differential equations using numerical analysis techniques results in a linear system of equations. Finally, we need compute the coefficients of the linear system of equations by numerical integration. The effect of numerical integration in finite element methods for solving elliptic equations and parabolic equations has been analyzed by others. In this paper we study the effect of numerical integration in expanded mixed finite element methods for linear elliptic problems. To the best of author's acknowledge, there is little results on this aspect.In Chapter one. we consider the effect of numerical integration in expanded mixed finite element methods for linear elliptic problems as follows:we first write this problem in weak form and then give a general description of its corresponding formulation when numerical integration is present. And we prove the existance and exclusive of the formulation, then achieve the optimal and suboptimal error estimate of the scalar unknown, its gradient, and its flux. Finally, we give some sufficient conditions to ensure that the order of convergence is unaltered when numerical integration is used.In Chapter two, we consider the effect of numerical integration in expanded mixed finite element methods for quasilinear elliptic problems as follows:we first write this problem in weak form and then give a general description of its corresponding formulation when numerical integration is present. And we prove the existance and exclusive of the formulation, then achieve the optimal and suboptimal error estimate of the scalar unknown, its gradient, and its flux. Finally, we give some sufficient conditions to ensure that the order of convergence is unaltered when numerical integration is used.In Chapter three, we present numerical results for the problemwhere f = -ae~x (x~2 + 3x), a = 1/20. u(x) = e~xx(x-1).