Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian

By Adam M. Oberman. 2008

170 citations recorded by the source corpus.

Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampere equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c3cd31a0ba91c2827366e9b18bded32df64a4f12

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Finite element approximations of the three dimensional Monge-Ampère equation (2012, methodology): “In [16, 17], Froese and Oberman generalized the two dimensional finite difference scheme given in [25] by constructing wide-stencil finite difference methods for the Monge-Ampère equation in dimensions greater than two.”
Two Numerical Methods for the elliptic Monge-Ampère equation (2010, methodology): “While one of the authors has previously introduced a convergent finite difference method [21], the increased attention by the numerical PDE community has prompted us to write this paper.”
A review of numerical methods for nonlinear partial differential equations (2012, methodology): “The theory of viscosity solutions for nonlinear Hamilton–Jacobi and fully nonlinear elliptic equations was developed in the L∞-framework [57, 30], as are the corresponding convergence theories [188, 158, 31].”
A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two (2013, methodology): “Note that, in the case of curved boundaries, the use of mixed piecewise linear finite elements provides a substantial simplification compared to using high order finite elements (as in [24, 42] for instance), or finite differences [26, 27, 43].”