Abstract
In this paper, we investigate the superconvergence property of the numerical solution to a quadratic elliptic control problem by using mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We prove the superconvergence error estimate of \(h^{\tfrac{3} {2}} \) in L 2-norm between the approximated solution and the average L 2 projection of the control. Moreover, by the postprocessing technique, a quadratic superconvergence result of the control is derived.
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References
Arada N, Casas E, Tröltzsch F. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput Optim Appl, 2002, 23: 201–229
Becker R, Kapp H, Rancher R. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J Control Optim, 2000, 39: 113–132
Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag, 1991
Chen Y. Superconvergence of mixed finite element methods for optimal control problems. Math Comp, 2008, 77: 1269–1291
Chen Y. Superconvergence of quadratic optimal control problems by triangular mixed finite elements. Inter J Numer Meths Eng, 2008, 75: 881–898
Chen Y, Huang Y, Liu W B, et al. Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J Sci Comput, 2009, 42: 382–403
Chen Y, Dai Y. Superconvergence for optimal control problems governed by semi-linear elliptic equations. J Sci Comput, 2009, 39: 206–221
Douglas J, Roberts J E. Global estimates for mixed finite element methods for second order elliptic equations. Math Comp, 1985, 44: 39–52
Falk F S. Approximation of a class of optimal control problems with order of convergence estimates. J Math Anal Appl, 1973, 44: 28–47
Grisvard P. Elliptic Problems in Nonsmooth Domains. Boston-London-Melbourne: Pitman, 1985
Geveci T. On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO Anal Numer, 1979, 13: 313–328
Li R, Liu W B, Ma H P, et al. Adaptive finite element approximation of elliptic optimal control. SIAM J Control Optim, 2002, 41: 1321–1349
Li R, Liu W B. http://circus.math.pku.edu.cn/AFEPack
Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971
Liu W B, Yan N. A posteriori error analysis for convex distributed optimal control problems. Adv Comp Math, 2001, 15: 285–309
Meyer C, Rösch A. Superconvergence properties of optimal control problems. SIAM J Control Optim, 2004, 43: 970–985
Raviart P A, Thomas J M. A mixed finite element method for 2nd order elliptic problems, Aspecs of the finite element method. In: Lecture Notes in Math, vol. 606. Berlin: Springer, 1977, 292–315
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Hou, T., Chen, Y. Superconvergence of RT1 mixed finite element approximations for elliptic control problems. Sci. China Math. 56, 267–281 (2013). https://doi.org/10.1007/s11425-012-4461-4
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DOI: https://doi.org/10.1007/s11425-012-4461-4
Keywords
- elliptic equations
- optimal control problems
- superconvergence
- mixed finite element methods
- postprocessing