Numerical Algorithms for Partially Segregated Elliptic Systems

TL;DR

This paper introduces two novel numerical algorithms for solving elliptic systems with partial segregation constraints, effectively handling the nonconvex admissible set and producing accurate phase separation patterns.

Contribution

It develops and analyzes two computational frameworks—penalty and projected gradient methods—for elliptic systems with nonconvex segregation constraints, including convergence and stability results.

Findings

Both algorithms successfully resolve segregated phase patterns.

The penalty method demonstrates exponential improvement in the strong-competition regime.

Numerical experiments confirm the effectiveness of the proposed methods on benchmark problems.

Abstract

We develop numerical methods for elliptic systems governed by partial segregation constraints, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This constraint enforces that at least one component must be zero at every spatial location, leading to a highly nonconvex admissible set that prevents the use of standard convex optimization techniques. We propose two complementary computational frameworks. The first is a strong-competition penalty method, solved via damped Gauss-Seidel/Picard iterations with a continuation strategy on the penalty parameter, for which we establish compactness results, Lipschitz estimates, and interior exponential improvement in the strong-competition regime. The second is a projected gradient method, together with an accelerated variant, that exploits an explicit pointwise projection onto the three-phase segregation set. Numerical experiments on a suite of benchmark boundary configurations confirm that both algorithms resolve segregated phase patterns.