Computing optimal partition problems via Lagrange multiplier approach

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.