Conforming/non-conforming mixed finite element methods for optimal control of velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity
{"title":"Conforming/non-conforming mixed finite element methods for optimal control of velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity","authors":"Harpal Singh, Arbaz Khan","doi":"10.1016/j.camwa.2025.08.005","DOIUrl":null,"url":null,"abstract":"<div><div>This work examines the distributed optimal control of generalized Oseen equations with non-constant viscosity and no-slip boundary conditions. We propose and analyze a new conforming augmented mixed finite element method and a discontinuous Galerkin (DG) scheme for the model written in velocity-vorticity-pressure formulation. The continuous formulation, which incorporates least-squares terms from both the constitutive equation and the incompressibility condition, is well-posed under certain assumptions on the viscosity parameter. The CG method is divergence-conforming and suits any Stokes inf-sup stable velocity-pressure finite element pair, while a generic discrete space approximates vorticity. The DG scheme employs a stabilization technique. We use two different approaches for the control approximation: a variational discretization and approximation through piecewise constant elements. We establish optimal a priori error estimates and residual-based a posteriori error estimates for both the proposed schemes. Finally, we provide numerical experiments to validate the theoretical estimates and showcase the performance and effectiveness of proposed schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"198 ","pages":"Pages 59-92"},"PeriodicalIF":2.5000,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computers & Mathematics with Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S089812212500330X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This work examines the distributed optimal control of generalized Oseen equations with non-constant viscosity and no-slip boundary conditions. We propose and analyze a new conforming augmented mixed finite element method and a discontinuous Galerkin (DG) scheme for the model written in velocity-vorticity-pressure formulation. The continuous formulation, which incorporates least-squares terms from both the constitutive equation and the incompressibility condition, is well-posed under certain assumptions on the viscosity parameter. The CG method is divergence-conforming and suits any Stokes inf-sup stable velocity-pressure finite element pair, while a generic discrete space approximates vorticity. The DG scheme employs a stabilization technique. We use two different approaches for the control approximation: a variational discretization and approximation through piecewise constant elements. We establish optimal a priori error estimates and residual-based a posteriori error estimates for both the proposed schemes. Finally, we provide numerical experiments to validate the theoretical estimates and showcase the performance and effectiveness of proposed schemes.
期刊介绍:
Computers & Mathematics with Applications provides a medium of exchange for those engaged in fields contributing to building successful simulations for science and engineering using Partial Differential Equations (PDEs).