Superconvergence Of Finite Element Methods With Interpolated Coefficients

For the semilinear differential equations with nonlinear term f(u), this paper introduces a simplified finite element method, called as interpolated coefficient finite element method, which requests that /(?/,) is replaced by interpolation Ihf(uh) in numerical computation. It has the same superconvergence as that classical finite element method, but more economic and efficient. For the first time this paper makes a systematic study about superconvergence of interpolated coefficients finite element method for many semilinear problems and obtains rather integrated results. Applying element analytical method and structuring superclose interpolated polynomials this paper proves that interpolated coefficients finite elements has the same superconvergence as that linear finite elements for one order nonlinear initial value problem of ordinary differential equation, semilinear elliptic problem, semilinear parabolic problems and semilinear hyperbolic problem. These facts are also shown by numerical examples in the paper.The following is an outline of the main results of the paper:1. By application of adjustment orthogonal approximate technique in an element, superconvergence of continuous finite elements for one order nonlinear initial value problem of ordinary differential equation is studied and ultracon-vergence of its finite element derivative recovery is deduced. As application of this method new numerical scheme for a nonlinear oscillating equations is obtained and the computational results proclaims that interpolated coefficient finite element methods has higher benefit than popular perturbation method.2. For two order semilinear elliptic problem with first boundary value superconvergence of triangular quadratic finite elements and rectangular finite element is studied respectively. These better properties are shown by corre-...