On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order

Abstract

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 ${\bf R}^n$, or for $C^2$ bounded domains in ${\bf R}^2$, 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 ${\bf R}^n$ or of local degree 5 for $C^2$ domains in ${\bf R}^2$. Better convergence and extensions to ${\bf R}^n$ for $C^2$ domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in ${\bf R}^n$, 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.