一类半线性抛物方程混合有限元方法的超逼近分析  

Superconvergence Analysis of a Mixed Finite Element Method for a Kind of Semilinear Parabolic Equations

作　　者：王俊俊 郭丽娟 WANG Junjun;GUO Lijuan(School of Mathematics and Statistics,Pingdingshan University,Pingdingshan 67000,China)

机构地区：[1]平顶山学院数学与统计学院,河南平顶山467000

出　　处：《应用数学》2019年第1期71-80,共10页Mathematica Applicata

基　　金：国家自然科学基金(11671369)

摘　　要：采用双线性元及零阶Raviart-Thomas元(Q_(11)+Q_(10)×Q_(01))分析了一类半线性抛物方程的H^1-Galerkin格式下的无网格比超逼近性质.首先,引入一个时间离散方程,将误差拆分成时间误差和空间误差两部分.其次,通过时间误差给出时间离散方程解的正则性,再利用空间误差得到了有限元解U_h^n的W^(0,∞)(Ω)模有界,整个过程避免时间步长τ和空间剖分参数h的比值,即网格比的出现.最后,当原始方程右端项f(u)满足局部Lipschitz条件时,有技巧地导出了原始变量u在H^1(Ω)模意义下及流量p=▽u在L^2(Ω)模意义下的O(h^2+τ~2)的无网格比超逼近性质.当f(u)为二阶可导时,给出▽·p在L^2(Ω)模意义下的O(h^2+τ~2)的无网格比超逼近结果.数值算例验证了理论的正确性.An H1-Galerkin mixed finite element method(MFEM) is discussed for a kind of semilinear parabolic equations with the bilinear element and zero-order Raviart-Thomas element(Q11+Q10×Q01).Firstly, a linearized scheme is given and the time discrete equations split the error into two parts, which are called the temporal error and spatial error. Secondly, the regularities of the time discrete equations are reduced by the temporal error and the FE solution ‖Uhn‖0,∞ is bounded unconditionally by the spatial error. Lastly, when the right hand part of the original equation f(u) is local Lipschitz, the unconditional superclose results of u in H1(Ω) and p = ▽u in L2(Ω) are obtained skillfully. When f(u) is twicely continuously differentiable with respective to u, the unconditional superclose results of ▽ ·p in L2(Ω) are gained. Numerical experiment is included to illustrate the feasibility of the proposed method.

关 键 词：半线性抛物方程 H^1-GALERKIN混合有限元方法 时间离散方程 时间误差和空间误差 无网格比超逼近结果 

分 类 号：O242.21[理学—计算数学]