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Numerical Algorithms for Partially Segregated Elliptic Systems

Farid Bozorgnia, Avetik Arakelyan, Vyacheslav Kungurtsev, Jan Valdman

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.

Numerical Algorithms for Partially Segregated Elliptic Systems

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.
Paper Structure (11 sections, 14 theorems, 115 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 11 sections, 14 theorems, 115 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Background and Previous Works
  3. Proposed Numerical Constrained Optimization Algorithms
  4. Penalization Method
  5. Projected Gradient Method.
  6. Simulations
  7. Boundary data.
  8. Initialization.
  9. FISTA iteration with projection.
  10. Parameters and stopping criterion.
  11. Conclusion and Impact

Key Result

Proposition 3.1

Let $(u_1^\varepsilon, u_2^\varepsilon, u_3^\varepsilon)$ be minimizer to the penalized energy subject to $u_i^\varepsilon|_{\partial\Omega} = \phi_i$ and $u_i^\varepsilon \geq 0$. Then $u_{i}^{\varepsilon} \rightharpoonup \overline{u}_{i}$ weakly in $H^{1}(\Omega)$ up to a subsequence for $i = 1,2,3.$ Moreover $(\overline{u}_{1}, \overline{u}_{2}, \overline{u}_{3}) \in S.$

Figures (5)

  • Figure 1: Surface plots of $u_1^\varepsilon$ (left) and $u_2^\varepsilon$ (right).
  • Figure 2: Surface plot of $u_3^\varepsilon$ (left) and a $\varepsilon$ contour(right) with penalization.
  • Figure 3: Projected gradient method for Example 4.1 on $\Omega = [-1,1]^2$. Top row: filled contour plots of $u_1$, $u_2$, and $u_3$; the dashed rectangle marks the inner square $[-0.3,\,0.3]^2$. Bottom row: energy convergence history (left), maximum constraint violation $\max|u_1 u_2 u_3|$ (center), and mean values of $u_3$ by region (right).
  • Figure 4: Gradient projected method on $\Omega=[-1,1]^2$.
  • Figure 5: Penalization Method: segregation interfaces on $\Omega=[-1,1]^2$.

Theorems & Definitions (37)

  • Definition 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 27 more