Geodesic
Mixed methods for optimal control problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 333-343.

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In this paper, we investigate a posteriori error estimates of a mixed finite element method for elliptic optimal control problems with an integral constraint. The gradient for our method belongs to the square integrable space instead of the classical $H(\mathrm{div};\Omega)$ space. The state and co-state are approximated by the $P^2_0$-$P_1$ (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. Using a duality argument method and an energy method, we derive residual a posteriori error estimates for all variables. 
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T. Hou. Mixed methods for optimal control problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 3, pp. 333-343. http://geodesic.mathdoc.fr/item/SJVM_2018_21_3_a6/

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

[2] Brunner H., Yan N., “Finite element methods for optimal control problems governed by integral equations and integro-differential equations”, Numer. Math., 101 (2005), 1–27 | DOI | MR | Zbl

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

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

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

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

[7] Chen Y., Huang Y., Liu W. B., Yan N., “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

[8] Chen S. C., Chen H. R., “New mixed element schemes for a second-order elliptic problem”, Mathematica Numerica Sinica, 32:2 (2010), 213–218 | MR | Zbl

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

[10] Ge L., Liu W. B., Yang D. P., “Adaptive finite element approximation for a constrained optimal control problem via multi-meshes”, J. Sci. Comput., 41:2 (2009), 238–255 | DOI | MR | Zbl

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

[12] Grisvard P., Elliptic Problems in Nonsmooth Domains, Pitman, Boston–London–Melbourne, 1985 | MR | Zbl

[13] Gong W., Yan N., “Adaptive finite element method for elliptic optimal control problems: convergence and optimality”, Numer. Math., 135:4 (2017), 1121–1170 | DOI | MR | Zbl

[14] Hou T., Chen Y., “Mixed discontinuous Galerkin time-stepping method for linear parabolic optimal control problems”, J. Comput. Math., 33:2 (2015), 158–178 | DOI | MR | Zbl

[15] Hou L., Turner J. C., “Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls”, Numer. Math., 71 (1995), 289–315 | DOI | MR | Zbl

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

[17] Kufner A., John O., Fucik S., Function Spaces, Nordhoff, Leiden, 1977 | MR

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

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

[20] Li R., Liu W. B., Yan N., “A posteriori error estimates of recovery type for distributed convex optimal control problems”, J. Sci. Comput., 41:5 (2002), 1321–1349 | MR | Zbl

[21] Liu W., Ma H., Tang T., Yan N., “A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations”, SIAM J. Numer. Anal., 42 (2004), 1032–1061 | DOI | MR | Zbl

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

[23] Liu W., Yan N., “A posteriori error estimates for control problems governed by Stokes equations”, SIAM J. Numer. Anal., 40 (2003), 1850–1869 | DOI | MR

[24] Liu W., Yan N., “A posteriori error estimates for optimal control problems governed by parabolic equations”, Numer. Math., 93 (2003), 497–521 | DOI | MR | Zbl

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

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

[27] Meidner D., Vexler B., “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

[28] Meidner D., Vexler B., “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

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

[30] Scott L. R., Zhang S., “Finite element interpolation of nonsmooth functions satisfying boundary conditions”, Math. Comput., 54 (1990), 483–493 | DOI | MR | Zbl

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

[32] Shen W., Ge L., Yang D., Liu W., “Sharp a posteriori error estimates for optimal control governed by parabolic integro-differential equations”, J. Sci. Comput., 65:1 (2015), 1–33 | DOI | MR | Zbl

[33] Xiong C., Li Y., “A posteriori error estimates for optimal distributed control governed by the evolution equations”, Appl. Numer. Math., 61:2 (2011), 181 | DOI | MR | Zbl