Convergent filtered scheme for the Monge-Amp\`ere Equation

TL;DR

This paper introduces a filtered scheme that enhances the accuracy of monotone methods for solving the Monge-Ampère equation while maintaining convergence to the viscosity solution.

Contribution

It extends existing monotone methods by integrating filtered schemes, improving accuracy without losing convergence guarantees.

Findings

Significant accuracy improvement demonstrated

Convergence to viscosity solution preserved

Applicable in dimensions d ≥ 2

Abstract

We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate non-monotone modification, using an appropriately chosen filter. This results in a remarkable improvement of accuracy, but without sacrificing the convergence to the unique viscosity solution.