《半线性反应扩散方程扩张混合有限元的高效两层网格方法及理论分析》由会员分享，可在线阅读，更多相关《半线性反应扩散方程扩张混合有限元的高效两层网格方法及理论分析（24页珍藏版）》请在金锄头文库上搜索。
1、湘潭大学 硕士学位论文 半线性反应扩散方程扩张混合有限元的高效两层网格方法及理 论分析 姓名：颜艳华 申请学位级别：硕士 专业：数学与计算科学 指导教师：陈艳萍 20070518 ? ? ? ? Li Wu? ? ? ? ?Wu? ? ? ? ? ? ? ? ? I Abstract Reaction-diff usion equations have received a great deal of application in real life, such as groundwater problem, bio-geochemical phenomena, environment con
2、- tamination and the reasonable exploit of petroleum reservoir and so on. Scientists have done a great deal of research on its numerical methods. Professor Li Wu and Professor Yanping Chen have presented a few two-grid methods for semi-linear reaction diff usion equations with small diff usion coeff
3、i cient by expanded mixed fi nite element method. These two-grid ideas are from Professor Xus work on standard fi nite element method. The basic idea is using Newton iteration to linearize the non-linear algebra systems, then making correction to improve the accuracy. In this paper, we present an im
4、proved two-grid method for two dimensional semi-linear reaction-diff usion equations by expanded mixed fi nite element methods, on the basis of the two-grid methods presented by Wu and Chen by virtue of an interpolation postprocessing technique. Firstly, we solve the original problem on the coarse g
5、rid. Then we postprocess the solution of the coarse grid. Finally, we use Newton iteration on the fi ne grid. In the analysis, we use the property of the interpolation postprocessing operator and use the superconvergence property of mixed fi nite element method. According to the result, we know that
6、 coarse gird can be extremely coarse to improve the effi ciency without aff ecting the accurate. Keywords: reaction-diff usion equations; expanded mixed fi nite element; two-grid methods; an interpolation postprocessing operator II ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?27, 30? ? ? ? ? ? ? ? ? ?
7、 ? ? ? ? ? ?40? ?VH?Vh? ?VH? ?Vh? ? ? ? ? ? ? ? ? ? ?H? ? ?h H? ? ?Wu?Allen ?f? ?36?39?f? ? ?10, 12?10, 12, 36? 1 ? ? ? ? ? ? ? ? ? ? ? 2 ? ? ? ? ?,? ? 2.1? ? ? ? ?R2? = ?D?Dirichlet? ?N?Neumann? = DN? ? spt (Kp) = f(p),(x,t) J(2.1) ? p = pD,on D(2.2) Kp n = gN,on N(2.3) p = p0,for t = 0(2.4) ?pt= p/t,J = (0,T)?T 0?n = (n1,n2)? ?N? ?p?u? ? spt+ u = f(p)(2.5) K1u + p = 0(2.6) ?,?p, u?p?s?k k k? ?L()?u u u? ?Darcy?(?)?f(p)? ?f(p)?x? 3 2.2? ? ? ? 1? ? RN,?p?1 p ? ?Lp() := ?w : kwk Lp() ? ?Banach?,? kwkLp()=(R|w|p) 1 p, 1 p . kwkL()=esssup|f(x)| : x ,p = . ?Sobolev? W m,p() = v : Dv Lp(),|
