On finite element methods for fully nonlinear elliptic equations of second order. (English) Zbl 1166.35322

Böhmer, Klaus

SIAM J. Numer. Anal. 46, No. 3, 1212-1249 (2008).

Summary: 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 \({\mathbb R}^n\), or for \(C^2\) bounded domains in \({\mathbb R}^2\), 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 [O. Davydov, Smooth finite elements and stable splitting, Bericht Fachbereich Mathematik und Informatik 3, Phillips Universität Marburg, Marburg, Germany (2007)] for \(C^1\) FEs on polyhedral domains in \({\mathbb R}^n\) or of local degree 5 for \(C^2\) domains in \({\mathbb R}^2\). Better convergence and extensions to \({\mathbb R}^n\) for \(C^2\) domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in \({\mathbb R}^n\), 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.

MSC:

35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations
35J60	Nonlinear elliptic equations
35J65	Nonlinear boundary value problems for linear elliptic equations
41A15	Spline approximation
46N40	Applications of functional analysis in numerical analysis
47N40	Applications of operator theory in numerical analysis
65D18	Numerical aspects of computer graphics, image analysis, and computational geometry
65H10	Numerical computation of solutions to systems of equations
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

conforming; nonconforming finite element method; \({\mathbb R}^n\); variational crimes; fully nonlinear elliptic differential equations; order \(2; C^1\) finite elements; quasi-linear elliptic equations not in divergence form; stability; convergence; regularity for finite element solutions; discrete Newton method; locally quadratic convergence