A Finite Element Method for Nonlinear Elliptic Problems

Abstract

We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis 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.

The Monge–Ampère equation

“…To reduce the number of unknowns in the mixed system (4.28), continuity constraints can be added in the matrix-valued space Σ h . This is the idea of the method proposed in [75]. There, the auxiliary space is defined as the matrixvalued Lagrange space, i.e.,…”

“…In addition to (4.3)-(4.5), we define the jump of a matrix-valued function τ across for all τ h ∈ Σ c h . This definition leads to a finite element method proposed in [75] which similar to (4.26), but with the continuous version of the discrete Hessian.…”

Spline element method for Monge–Ampère equations

“…The latter has been recently shown to be valid for strictly convex radial viscosity solutions [26]. The formal limit of the vanishing moment methodology turns out, in the case of the Monge-Ampère equation, to be the method recently proposed by Lakkis and Pryer in [39]. Neilan analyzed the method of Lakkis and Pryer for the two-dimensional problem under the assumption that the solution is smooth in [42].…”

“…The analysis for non smooth solutions of the mixed methods discussed in [10,39,43] will be discussed in [7] 1.2 Organization of the paper…”

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

“…In [16,34,35] Oberman uses techniques from Barles and Souganidis [11] for the approximation of fully nonlinear PDEs to construct wide stencil difference schemes for the ∞-Laplacian. See also [33] where the authors construct a local mesh refinement (h-adaptive) finite element scheme based on a residual error indicator and the method derived in [32]. Herein we report on numerical experiments that provide further understanding of ∞-harmonic mappings.…”