拟线性双曲方程类Wilson非协调元的高精度分析

High Accuracy Analysis of the Nonconforming Quasi-Wilson Element Solution to Quasi-Linear Hyperbolic Equations

主要研究类Wilson元对拟线性双曲方程的逼近.首先证明了当问题的解u∈H^3(Ω)或u∈H^4(Ω)时,u与其双线性插值之差的梯度与类Wilson元空间任意元素的梯度,在分片意义下的内积可以达到O(h^2)这一重要结论.其次运用能量模意义下该元的非协调误差可以分别达到O(h^2)/O(h^3),即比插值误差高一阶/二阶这一性质,并利用对时间t的导数转移技巧,结合双线性元的高精度结果及插值后处理技术,获得了O(h^2)阶的超逼近性和整体超收敛性,从而进一步拓广了该元的应用范围.

In this paper,quasi-Wilson finite element approximation is mainly studied for quasi-linear hyperbolic equations.First,an important conclusion is proven that the gradient of the difference between u and its bilinear interpolation and the gradient of the any element in quasi-Wilson space can be estimated with the order of O（h^2）in the sense of inner product piecewisely,when the solution u belongs to H^3（Ω） or H^4（Ω）.Then,by using the special property of the element that the consistency error can be estimated with order O（h^2）/O（h^3） in the energy,which is one/two order higher than the interpolation error,by making the transformation of the derivate with respect to time t,and according high accuracy analysis of bilinear element and post-processing techniques,the superclose property and superconvergence with order O（h^2）are derived.Therefore,the applicated scope of the element would be broadened.