Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation

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• 作者：B.D. Froese ; A.M. Oberman
• 关键词：Fully nonlinear elliptic Partial Differential Equations ; Monge– ; Amp&#xe8 ; re equations ; Nonlinear finite difference methods ; Viscosity solutions ; Monotone schemes ; Convexity constraints
• 刊名：Journal of Computational Physics
• 出版年：2011
• 出版时间：1 February 2011
• 年：2011
• 卷：230
• 期：3
• 页码：818-834
• 全文大小：627 K

文摘

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.