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Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids

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Version 2 2023-06-12, 08:35
Version 1 2023-06-09, 04:22
journal contribution
posted on 2023-06-12, 08:35 authored by Xiaobing Feng, Max Jensen
This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Ampère type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Ampère Dirichlet problem. A superlinearly convergent Howard's algorithm, which is a Newton--type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.

History

Publication status

  • Published

File Version

  • Published version

Journal

SIAM Journal on Numerical Analysis

ISSN

0036-1429

Publisher

Society for Industrial and Applied Mathematics

Issue

2

Volume

55

Page range

691-712

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Numerical Analysis and Scientific Computing Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2016-12-07

First Open Access (FOA) Date

2017-02-24

First Compliant Deposit (FCD) Date

2016-12-07

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