非线性抛物积分微分方程的类Wilson非协调元分析

Analysis of Quasi-Wilson Nonconforming Element for Nonlinear Parabolic Integro-differential Equation

在半离散和全离散格式下讨论非线性抛物积分微分方程的类Wilson非协调有限元逼近.当问题的精确解u∈H3(Ω)/H4(Ω)时,利用该元的相容误差在能量模意义下可以达到O(h2)/O(h3)比其插值误差高一阶和二阶的特殊性质,再结合协调部分的高精度分析及插值后处理技术,并借助于双线性插值代替传统有限元分析中不可缺少的Ritz-Volterra投影导出了半离散格式下的O(h2)阶超逼近和超收敛结果.同时分别得到了向后Euler全离散格式下的超逼近性和Crank-Nicolson全离散格式下的最优误差估计.

A nonconforming quasi-Wilson finite element approximation for nonlinear par- abolic integro-differential equation is discussed under the semi-discrete and fully-discrete schemes. By use of the special property of the element,i, e. ,the consistence error estimate in energy norm when the exact solution u of the problem belongs to H3（Ω）/H4（Ω） can reach to O（h2）/O（h3）, one/two order higher than the interpolation error, then combination it with the higher accuracy analysis of its conforming part and the interpolated postprocessing tech- nique,the superclose and superconvergence results with order O（h2） are obtained for semi- discrete scheme through interpolation instead of the Ritz-Volterra projection which is an in- dispensable tool in traditional finite element analysis. The superclose property and the opti- mal error estimate for backward Euler and Crank-Nicolson fully-discrete schemes are de- rived, respectively.