Computing optimal partition problems via Lagrange multiplier approach

TL;DR

This paper introduces efficient numerical schemes based on Lagrange multipliers for solving optimal partition problems, ensuring properties like orthogonality and energy dissipation, validated through extensive 2D and 3D experiments.

Contribution

It develops novel Lagrange multiplier-based numerical schemes for optimal partition problems that preserve key properties and are computationally efficient.

Findings

Schemes preserve orthogonality, positivity, and energy dissipation.

Only linear Poisson equations are solved per time step.

Numerical results confirm effectiveness and accuracy in 2D and 3D.

Abstract

In this paper, we consider numerical approximations for the optimal partition problem using Lagrange multipliers. By rewriting it into constrained gradient flows, three and four steps numerical schemes based on the Lagrange multiplier approach \cite{ChSh22,ChSh_II22} are proposed to solve the constrained gradient system. Numerical schemes proposed for the constrained gradient flows satisfy the nice properties of orthogonality-preserving, norm-preserving, positivity-preserving and energy dissipating. The proposed schemes are very efficient in which only linear Poisson equations are solved at each time step. Extensive numerical results in 2D and 3D for optimal partition problem are presented to validate the effectiveness and accuracy of the proposed numerical schemes.