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Two numerical methods for the elliptic Monge-Ampère equation. (English) Zbl 1192.65138

From the authors’ summary: The numerical solution of the elliptic Monge-Ampère partial differential equation has been a subject of increasing interest recently. There are already two methods available [V. I. Oliker and L. D. Prussner, Numer. Math. 54, No. 3, 271–293 (1988; Zbl 0659.65116); A. M. Oberman, Discrete Contin. Dyn. Syst., Ser. B 10, No. 1, 221–238 (2008; Zbl 1145.65085)] which converge even for singular solutions. However, many of the newly proposed methods lack numerical evidence of convergence on singular solutions, or are known to break down in this case. In this article we present and study the performance of two methods. The first method, which is simply the natural finite difference discretization of the equation, is demonstrated to be the best performing method (in terms of convergence and solution time) currently available for generic (possibly singular) problems, in particular when the right hand side touches zero. The second method, which involves the iterative solution of a Poisson equation involving the Hessian of the solution, is demonstrated to be the best performing (in terms of solution time) when the solution is regular, which occurs when the right hand side is strictly positive.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
35R35 Free boundary problems for PDEs
35K65 Degenerate parabolic equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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