Numerical Algorithms for Partially Segregated Elliptic Systems

We develop numerical methods for elliptic systems governed by a ternary partial-segregation constraint, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This condition forces at least one component to vanish at every spatial location while still permitting coexistence of the other two components, and therefore produces a highly nonconvex admissible set. In particular, the limiting constrained Dirichlet minimization problem need not have a unique minimizer, as we show by an explicit one-dimensional example with two distinct global minimizers. We propose two complementary computational frameworks. The first is a strong-competition penalty method solved by a relaxed Picard iteration and a sequential semi-implicit variant, combined with continuation in the penalty parameter. For this formulation, we establish existence of penalized minimizers and weak solutions for every fixed penalty parameter, monotone bracketing properties for the fixed-point iteration, geometric convergence under an explicit contraction condition, and interior exponential estimates in regions of uniformly positive competition. The second framework is a discrete projected-gradient method, together with a safeguarded inertial variant, based on an explicit pointwise projection onto the ternary partial-segregation set. Numerical experiments for several discrete boundary configurations show that, when started on a common solution branch, the two methods produce the same interfaces, and that apparent differences between them reflect the selection of different branches by their default initializations.

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