On finite element methods for fully nonlinear elliptic equations of second order
On finite element methods for fully nonlinear elliptic equations of second order
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DOI：
10.1137/040621740
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发表时间：
2008-01-01
影响因子：
2.9
通讯作者：
Boehmer, Klaus
中科院分区：
文献类型：
--
作者：
Boehmer, Klaus
首次给出了二阶完全非线性椭圆型方程的一般情况下的非标准C-1有限元方法。在整篇文章中，我们同时考虑两种情况：对于R-n中的凸、有界、多面体区域，或者对于R-2中的C-2有界域，我们分别证明了相应的协调或非协调C-1有限元的稳定性和收敛性。对于2阶和2m阶方程和系统以及求积近似的结果出现在别处。将经典的离散化理论应用于微分算子或微分与边界算子的组合。对于多面体或曲线域上满足或违反的边界条件，必须估计一致性误差。这种稳定性必须以一种不同寻常的方式来证明。这是本文的核心所在。基本工具是线性化，紧致性论证，线性化算子的弱形式和强形式之间的相互作用，以及有限元方程解的新的正则性结果。我们证明的一个基本基础是Davydov关于R-n中多面体区域上的C-1 FES或R-2中C-2区域的局部5次FES的结果。从他即将得到的关于曲线域的结果可望得到更好的收敛和对C-2域的R-n的扩张。我们在R-n中对第二种情况的证明，基本上包括了第一种情况作为特例。该方法同样适用于非散度形式的拟线性椭圆型问题。根据网格无关原理，证明了离散牛顿法是局部二次收敛的，基本上与实际网格大小无关。
For the first time, we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard C-1 finite element method (FEM). We consider, throughout the paper, two cases in parallel: For convex, bounded, polyhedral domains in R-n, or for C-2 bounded domains in R-2, we prove stability and convergence for the corresponding conforming or nonconforming C-1 FEM, respectively. The results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere. The classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator. The consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated. The stability has to be proved in an unusual way. This is the hard core of the paper. Essential tools are linearization, a compactness argument, the interplay between the weak and strong form of the linearized operator, and a new regularity result for solutions of finite element equations. An essential basis for our proofs are Davydov's results for C-1 FEs on polyhedral domains in R-n or of local degree 5 for C-2 domains in R-2. Better convergence and extensions to R-n for C-2 domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in R-n, includes the first essentially as a special case. The method applies to quasi-linear elliptic problems not in divergence form as well. A discrete Newton method is shown to converge locally quadratically, essentially independently of the actual grid size by the mesh independence principle.