Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Amp\`ere Equation

TL;DR

This paper introduces a novel trivariate spline collocation method for solving the 3D Monge-Ampère equation, demonstrating high accuracy and efficiency through extensive numerical experiments on various solutions and domains.

Contribution

The paper develops a new spline collocation approach for the 3D Monge-Ampère equation, including convergence analysis and extensive computational validation.

Findings

The method achieves high convergence rates in numerical experiments.

It effectively handles both convex and nonconvex solutions.

The approach outperforms several existing numerical methods.

Abstract

We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Amp\'ere equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.