Numerical Algorithms for Partially Segregated Elliptic Systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912947685556224 |
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| author | Bozorgnia, Farid Arakelyan, Avetik Kungurtsev, Vyacheslav Valdman, Jan |
| author_facet | Bozorgnia, Farid Arakelyan, Avetik Kungurtsev, Vyacheslav Valdman, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Numerical Algorithms for Partially Segregated Elliptic Systems Bozorgnia, Farid Arakelyan, Avetik Kungurtsev, Vyacheslav Valdman, Jan Numerical Analysis 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. |
| title | Numerical Algorithms for Partially Segregated Elliptic Systems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.05991 |