半线性粘弹性方程非常规Hermite型矩形元的高精度分析

High Accuracy Analysis of Unconventional Hermite-type Finite Element for Semi-linear Viscoelasticity Equations

利用非常规的Hermite元对一类半线性粘弹性方程进行了有限元分析.首先给出了半离散格式下解的存在唯一性证明,同时利用插值和投影相结合的方法,借助于该元已有的高精度结果、平均值技巧和插值后处理技术,得到了H1模意义下的超逼近和超收敛性质.最后给出了一种该方程的全离散逼近格式,在不需要网格比的情况下,得到了O(h^3+τ~2)的结果.

This paper studies a new unconventional Hermite-type FEMs for the semi-linear viscoelasticity equations.Firstly,it proves the existence and uniqueness of solution under the semi-discrete schemes.Based on the known high accuracy analysis,mean-value and postprocessing techniques,the superclose properties and the global superconvergence result in H1-norm that are deduced by means of interpolation and projection technology combination.Finally,a fully-discrete schemes are constructed,and a lomplexity O（h^3＋τ^2）is obtained without the meshes ratio in these schemes.