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A FINITE ELEMENT METHOD FOR NONLINEAR ELLIPTIC PROBLEMS
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DOI：10.1137/120887655.png)
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Abstract
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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 by product 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-Ampere equation and the Pucci equation.
Keywords:
fully nonlinear PDE
finite element method
Monge-Ampere equation
Pucci's equation
finite element Hessian
nonvariational finite element method
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