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Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation. (English) Zbl 1206.65242

Summary: The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail.
In this article, we build a finite difference solver for the Monge-Ampère equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton’s method.
Computational results in two and three-dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J93 Quasilinear elliptic equations with mean curvature operator
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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