A two-grid method for the $C^0$ interior penalty discretization of the Monge-Amp\`{e}re equation

TL;DR

This paper introduces a two-grid method for efficiently solving the nonlinear system from a $C^0$ interior penalty discretization of the Monge-Ampère equation, combining coarse mesh solutions with a single Newton iteration on a finer mesh.

Contribution

It proposes a novel two-grid approach that reduces computational cost while maintaining accuracy for discretized Monge-Ampère equations.

Findings

Quasi-optimal $W^{1,inity}$ error estimates are established.

The two-grid method is computationally more efficient than standard Newton's method.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Amp\`ere equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.