Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912977963188224 |
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| author | Shaikh, Tooba M. Dond, Asha K. |
| author_facet | Shaikh, Tooba M. Dond, Asha K. |
| contents | This paper focuses on the quasi-optimality of an adaptive nonconforming finite element method for a distributed optimal control problem governed by the Stokes equation. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretise the velocity and pressure variables, respectively. The control variable is discretised using both variational and discretised approach. The error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates are presented under minimal regularity assumption on optimal solutions. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality. The theoretical findings are validated through numerical experiments on convex as well as nonconvex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00555 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework Shaikh, Tooba M. Dond, Asha K. Numerical Analysis 49M25, 49M41, 65N15, 65N30 This paper focuses on the quasi-optimality of an adaptive nonconforming finite element method for a distributed optimal control problem governed by the Stokes equation. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretise the velocity and pressure variables, respectively. The control variable is discretised using both variational and discretised approach. The error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates are presented under minimal regularity assumption on optimal solutions. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality. The theoretical findings are validated through numerical experiments on convex as well as nonconvex domains. |
| title | Quasi-Optimality of AFEM for Distributed Optimal Control Problems of Stokes Equations: An Axiomatic Framework |
| topic | Numerical Analysis 49M25, 49M41, 65N15, 65N30 |
| url | https://arxiv.org/abs/2307.00555 |