# Galerkin methods for (fully) nonlinear elliptic equations

• Omar Lakkis (Sussex)
• Monday 24 March 2014, 14:00-15:00
• Watson LRA.

joint work with Tristan Pryer (Reading)

Except some notable cases, most nonlinear elliptic equations are not variational. Traditional finite element methods (FEMs), as a subclass of Galerkin methods, are variational and thus not suitable for the approximation of solutions to fully nonlinear elliptic equations. Finite difference methods are more natural, especially where maximum principle plays a role. However, finite differences have some serious drawbacks, like the need to choose large stencils and the difficulties with handling geometries.

This has lead us to develop a novel numerical technology based on standard finite element spaces, but eschewing the variational approach. We call this method the NVFEM (Nonvariational FEM ). The key is to make a proper use of the distributional Hessian of a FE function by introducing a “recovered” approximation to it. The Newton-Raphson step for a fully nonlinear equation can thus turned into an unbalanced mixed problem, where the dual variable approximates the Hessian of the solution. The resulting method is surprisingly simple yet powerful. Although the analysis of the full method in general has still to be fully understood, the availability of a posteriori error estimates allows adaptive methods to be used in the approximation of singular (continuous but not differentiable) viscosity solutions.

In this talk, I will first review a bit of fully nonlinear elliptic PDEs. I will then present what I’ve learned about the history and the state of the art of numerical methods for fully nonlinear equations and finally I will place our work within the context of other methods. I will close with some numerical examples of solvers for the Monge-Ampère, Pucci and infinite-harmonic equation, including adaptive methods for singular solutions.

This talk is part of the Applied Mathematics Seminar Series series.