Petrov-galerkin Approximation Model To Semi-linear Elliptic Boundary Value Problems And A New Scheme For The Mixed Finite Element Method For The Biharmonic Equation

In this paper we introduce a Petrov-Galerkin approximation model to semi-linear elliptic boundary value problems and a new scheme for the mixed finite element method for the bihar-monic equation .To semi-linear elliptic boundary value problems the biquadratic polynomial space and bilinear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the semi-linear finite element solution converge to the proposed Petrov-Galerkin approximate solution. To the mixed finite element method for the biharmonic equation the flow function is approximated by biquadratic polynomial and the vortex function by bilinear polynomial. Assuming that the rectangular element partition is quasi-uniform,then proposed scheme can acheive the same approximation order as the Ciarlet-Raviart mixed finite element with the flow funtion and the vortex functions by piecewise quadratic polynomials.