A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909883971928064 |
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| author | Jeong, SeongHee Lee, Seulip Liu, Sijing |
| author_facet | Jeong, SeongHee Lee, Seulip Liu, Sijing |
| contents | We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation Jeong, SeongHee Lee, Seulip Liu, Sijing Numerical Analysis 49M41, 65N12, 65N30 We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes. |
| title | A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation |
| topic | Numerical Analysis 49M41, 65N12, 65N30 |
| url | https://arxiv.org/abs/2510.27167 |