The construction of preconditioners for elliptic problems by substructuring. I
Mathematics of Computation
SIAM Journal on Applied Mathematics
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Scientific Computing
SIAM Journal on Numerical Analysis
On Convex Functions and the Finite Element Method
SIAM Journal on Numerical Analysis
Moving Mesh Generation Using the Parabolic Monge-Ampère Equation
SIAM Journal on Scientific Computing
The Monge-Ampère equation: Various forms and numerical solution
Journal of Computational Physics
A Finite Element Method for Second Order Nonvariational Elliptic Problems
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
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The purpose of this paper is twofold. First, we modify a method due to Lakkis and Pryer where the notion of a discrete Hessian is introduced to compute fully nonlinear second order PDEs. The discrete Hessian used in our approach is entirely local, making the resulting linear system within the Newton iteration much easier to solve. The second contribution of this paper is to analyze both Lakkis and Pryer's method and its modification in parallel applied to the two-dimensional Monge-Ampere equation. In both cases we show the well-posedness of the methods as well as derive optimal error estimates. Numerical experiments are presented which (i) back up the theoretical findings and (ii) indicate that the methods are able to capture weak (viscosity) solutions.