A Finite Element Method for Nonlinear Elliptic Problems

(2013) A finite element method for nonlinear elliptic problems. This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available. Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Abstract. We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Ampère equation and the Pucci equation. 1. Introduction. Fully nonlinear partial differential equations (PDEs) arise in many areas, including differential geometry (the Monge–Ampère equation), mass transportation (the Monge–Kantorovich problem), dynamic programming (the Bell-man equation), and fluid dynamics (the geostrophic equations). The numerical approximation of the solutions of such equations is thus an important scientific task. There are at least three main difficulties apparent when attempting to derive numerical methods for fully nonlinear equations. The first is the strong nonlinearity on the highest order derivative which generally precludes a variational formulation. The second is that a fully nonlinear equation does not always admit a classical solution, even if the problem data is smooth, and the solution has to sought in a generalized sense (e.g., viscosity solutions), which is …