Geodesic
A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 63-76.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we discuss a priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by a $P_0^2-P_1$ (velocity-pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables and the co-state variables. Then we obtain a superconvergence result for the control variable by using postprocessing projection operator. 
@article{SJVM_2021_24_1_a5,
     author = {C. Xu},
     title = {A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {63--76},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/}
}
TY  - JOUR
AU  - C. Xu
TI  - A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2021
SP  - 63
EP  - 76
VL  - 24
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/
LA  - ru
ID  - SJVM_2021_24_1_a5
ER  - 
%0 Journal Article
%A C. Xu
%T A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2021
%P 63-76
%V 24
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/
%G ru
%F SJVM_2021_24_1_a5
C. Xu. A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 63-76. http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a5/

[1] J. F. Bonnans, E. Casas, “An extension of Pontryagin's principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities”, SIAM J. Control Optim., 33:1 (1995), 274–298 | DOI | MR | Zbl

[2] Y. Chen, “Superconvergence of mixed finite element methods for optimal control problems”, Math. Comp., 77:263 (2008), 1269–1291 | DOI | MR | Zbl

[3] Y. Chen, “Superconvergence of quadratic optimal control problems by triangular mixed finite element methods”, Int. J. Numer. Meths. Eng., 75:8 (2008), 881–898 | DOI | MR | Zbl

[4] Y. Chen, Y. Dai, “Superconvergence for optimal control problems governed by semi-linear elliptic equations”, J. Sci. Comput., 39:2 (2009), 206–221 | DOI | MR | Zbl

[5] Y. Chen, Y. Huang, W. B. Liu, N. Yan, “Error estimates and superconvergence of mixed finite element methods for convex optimal control problems”, J. Sci. Comput., 42:3 (2010), 382–403 | DOI | MR | Zbl

[6] S. C. Chen, H. R. Chen, “New mixed element schemes for second order elliptic problem”, Math. Numer. Sin., 32:2 (2010), 213–218 | MR | Zbl

[7] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam, 1978 | MR | Zbl

[8] K. Deckelnick, A. Günther, M. Hinze, “Finite element approximation of Dirichlet boundary control for elliptic PDEs on two and three-dimensional curved domains”, SIAM J. Control Optim., 48:4 (2009), 2798–2819 | DOI | MR | Zbl

[9] M. D. Gunzburger, L. S. Hou, “Finite-dimensional approximation of a class of constrained nonlinear optimal control problems”, SIAM J. Control Optim., 34:3 (1996), 1001–1043 | DOI | MR | Zbl

[10] M. D. Gunzburger, H. Lee, J. Lee, “Error estimates of stochastic optimal Neumann boundary control problems”, SIAM J. Numer. Anal., 49:3/4 (2011), 1532–1552 | DOI | MR | Zbl

[11] H. Guo, H. Fu, J. Zhang, “A splitting positive definite mixed finite element method for elliptic optimal control problem”, Appl. Math. Comput., 219:24 (2013), 11178–11190 | MR | Zbl

[12] W. Gong, N. Yan, “Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs”, SIAM J. Control Optim., 49:3 (2011), 984–1014 | DOI | MR | Zbl

[13] G. Knowles, “Finite element approximation of parabolic time optimal control problems”, SIAM J. Control Optim., 20 (1982), 414–427 | DOI | MR | Zbl

[14] R. Li, W. B. Liu, H. Ma, T. Tang, “Adaptive finite element approximation for distributed elliptic optimal control problems”, SIAM J. Control Optim., 41:5 (2002), 1321–1349 | DOI | MR | Zbl

[15] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971 | MR | Zbl

[16] W. Liu, N. Yan, “A posteriori error estimates for some model boundary control problems”, J. Comput. Appl. Math., 120:1 (2000), 159–173 | DOI | MR | Zbl

[17] W. Liu, N. Yan, “A posteriori error estimates for convex boundary control problems”, SIAM J. Numer. Anal., 39 (2001), 73–99 | DOI | MR | Zbl

[18] H. Liu, N. Yan, “Superconvergence and a posteriori error estimates for boundary control of Stokes problems”, J. Comput. Math., 24 (2006), 343–356 | MR | Zbl

[19] C. Meyer, A. Rösch, “Superconvergence properties of optimal control problems”, SIAM J. Control Optim., 43:3 (2004), 970–985 | DOI | MR | Zbl

[20] C. Meyer, A. Rösch, “$L^\infty$-estimates for approximated optimal control problems”, SIAM J. Control Optim., 44:5 (2005), 1636–1649 | DOI | MR

[21] D. Meidner, B. Vexler, “A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints”, SIAM J. Control Optim., 47:3 (2008), 1150–1177 | DOI | MR | Zbl

[22] D. Meidner, B. Vexler, “A priori error estimates for space-time finite element discretization of parabolic optimal control problems part II: problems with control constraints”, SIAM J. Control Optim., 47:3 (2008), 1301–1329 | DOI | MR | Zbl

[23] R. S. McKinght, W. E. Borsarge, “The Ritz–Galerkin procedure for parabolic control problems”, SIAM J. Control., 11:3 (1973), 510–542 | DOI | MR

[24] F. Shi, J. P. Yu, K. T. Li, “A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair”, Int. J. Comput. Math., 88 (2011), 2293–2305 | DOI | MR | Zbl

[25] D. Yang, Y. Chang, W. Liu, “A priori error estimate and superconvergence analysis for an optimal control problem of bilinear type”, J. Comput. Math., 26:4 (2008), 471–487 | MR | Zbl

[26] M. Yan, W. Gong, N. Yan, “Finite element methods for elliptic optimal control problems with boundary observations”, Appl. Numer. Math., 90 (2015), 190–207 | DOI | MR | Zbl