非线性四阶双曲方程一个低阶混合元方法的超收敛和外推  

Superconvergence and extrapolation of a lower order mixed finite method for nonlinear fourth-order hyperbolic equation

作　　者：张厚超[1] 朱维钧[1] 王俊俊[1] 

机构地区：[1]平顶山学院数学与信息科学学院,河南平顶山467000

出　　处：《山东大学学报（理学版）》2015年第12期35-46,共12页Journal of Shandong University(Natural Science)

基　　金：国家自然科学基金资助项目(11271340)

摘　　要：对一类非线性四阶双曲方程利用双线性元Q_(11)及Q_(01)×Q_(10)元给出了一个低阶协调混合元逼近格式。证明了逼近解的存在唯一性。基于上述两个单元的高精度结果,利用对时间t的导数转移技巧,导出了原始变量u和扩散项v=-Δu在H1模及流量p=-▽u在L2模意义下具有O(h2)阶的超逼近结果。进一步地,借助插值后处理技术,得到了整体超收敛性。通过建立Q_(01)×Q_(10)元的一个新的渐近展开式,并构造一个合适的外推格式,得到O(h3)阶的外推解。这里,h表示空间剖分参数。With the help of the bilinear element Q11 and the Q01× Q10 element,a lower order conforming mixed finite element approximation scheme is proposed for nonlinear fourth-order hyperbolic equation. Firstly,the existence and uniqueness of approximation solution are proved. Secondly,Based on the known high accuracy results of the about two elements,by use of derivative delivery techniques,the superclose with order O（ h2） for both scalar unknown u and the diffusion term v =- Δu in H1-norm and the flux p =- ▽u in L2-norm are derived,respectively. M oreover,the global superconvergence is obtained through interpolation post-processing technique. Finally,throught constructing a newasymptotic expansion formula of Q01× Q10 element and a suitable extrapolation scheme,the extrapolation solutions with order O（ h3） are derived. Here,h is the subdivision parameter for the space.

关 键 词：非线性四阶双曲方程 混合元方法 超逼近 超收敛 外推 

分 类 号：O242.21[理学—计算数学]