Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A two-grid Adaptive Finite Element Method for the Dirichlet Boundary Control Problem Governed by Stokes Equation (2401.15582v1)

Published 28 Jan 2024 in math.NA and cs.NA

Abstract: In this article, we derive \textit{a posteriori} error estimates for the Dirichlet boundary control problem governed by Stokes equation. An energy-based method has been deployed to solve the Dirichlet boundary control problem. We employ an inf-sup stable finite element discretization scheme by using $\mathbf{P}_1$ elements(in the fine mesh) for the velocity and control variable and $P_0$ elements(in the coarse mesh) for the pressure variable. We derive an \textit{a posteriori} error estimator for the state, adjoint state, and control error. The control error estimator generalizes the standard residual type estimator of the unconstrained Dirichlet boundary control problems, by additional terms at the contact boundary addressing the non-linearity. We prove the reliability and efficiency of the estimator. Theoretical results are illustrated by some numerical experiments.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (33)
  1. On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim., 53(6):3620–3641, 2015.
  2. The mathematical theory of finite element methods. Texts in Applied Mathematics. Springer, New York, third edition, 2008.
  3. A review of unified a posteriori finite element error control. Numer. Math. Theory Methods Appl., 5(4):509–558, 2012.
  4. E. Casas and V. Dhamo. Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations. Comput. Optim. Appl., 52(3):719–756, 2012.
  5. E. Casas and M. Mateos. Error estimates for the numerical approximation of Neumann control problems. Comput. Optim. Appl., 39(3):265–295, 2008.
  6. E. Casas and J.-P. Raymond. Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim., 45(5):1586–1611, 2006.
  7. Penalization of Dirichlet optimal control problems. ESAIM Control Optim. Calc. Var., 15(4):782–809, 2009.
  8. L. Chen and C. Zhang. A coarsening algorithm on adaptive grids by newest vertex bisection and its applications. J. Comput. Math., 28(6):767–789, 2010.
  9. A framework for the error analysis of discontinuous finite element methods for elliptic optimal control problems and applications to C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT IP methods. Numer. Funct. Anal. Optim., 36(11):1388–1419, 2015.
  10. Error bounds for a Dirichlet boundary control problem based on energy spaces. Math. Comp., 86(305):1103–1126, 2017.
  11. P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4.
  12. An error analysis of discontinuous finite element methods for the optimal control problems governed by Stokes equation. Numer. Funct. Anal. Optim., 40(4):421–460, 2019.
  13. W. Dörfler. A convergent adaptive algorithm for poisson’s equation. SIAM Journal on Numerical Analysis, 33(3):1106–1124, 1996.
  14. F. Fierro and A. Veeser. A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal., 41(6):2032–2055, 2003.
  15. Optimal Dirichlet control and inhomogeneous boundary value problems for the unsteady Navier-Stokes equations. In Control and partial differential equations (Marseille-Luminy, 1997), volume 4 of ESAIM Proc., pages 97–116. Soc. Math. Appl. Indust., Paris, 1998.
  16. V. Girault and P.-A. Raviart. Finite element approximation of the Navier-Stokes equations. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1979.
  17. Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system. ESAIM Math. Model. Numer. Anal., 54(6):2229–2264, 2020a.
  18. Analysis and approximations of Dirichlet boundary control of Stokes flows in the energy space. https://arxiv.org/abs/2011.08221., 2020b.
  19. T. Gudi and R. C. Sau. Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem. ESAIM Control Optim. Calc. Var., 26:Paper No. 78, 19, 2020.
  20. T. Gudi and R. C. Sau. A two level finite element method for Stokes constrained Dirichlet boundary control problem. Comput. Math. Appl., 129:126–135, 2023.
  21. A. Günther and M. Hinze. Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls. Comput. Optim. Appl., 49(3):549–566, 2011.
  22. Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO Modél. Math. Anal. Numér., 25(6):711–748, 1991.
  23. The primal-dual active set strategy as a semismooth newton method. SIAM Journal on Optimization, 13(3):865–888, 2002.
  24. M. Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl., 30(1):45–61, 2005.
  25. Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comput., 22/23:347–370, 2005.
  26. M. Karkulik. A finite element method for elliptic Dirichlet boundary control problems. Comput. Methods Appl. Math., 20(4):827–843, 2020.
  27. An efficient and reliable residual-type a posteriori error estimator for the Signorini problem. Numer. Math., 130(1):151–197, 2015.
  28. C. Meyer and A. Rösch. Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43(3):970–985, 2004.
  29. Boundary element methods for Dirichlet boundary control problems. Math. Methods Appl. Sci., 33(18):2187–2205, 2010.
  30. An energy space finite element approach for elliptic Dirichlet boundary control problems. Numer. Math., 129(4):723–748, 2015.
  31. A. Veeser and R. Verfürth. Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal., 47(3):2387–2405, 2009.
  32. R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Wiley-Teubner, Chichester, 1995.
  33. M. Winkler. Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization. Numer. Math., 144(2):413–445, 2020.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets