## A posteriori $$L^\infty(L^\infty)$$-error estimates for finite-element approximations to parabolic optimal control problems.(English)Zbl 07453248

Summary: We derive space-time a posteriori error estimates of finite-element method for the linear parabolic optimal control problems in a convex bounded polyhedral domain. The variational discretization is used to approximate the state and co-state variables with the piecewise linear and continuous functions, while the control variable is computed using the implicit relation between the control and co-state variables. The temporal discretization is based on the backward Euler method. The key feature of this approach is not to discretize the control variable but to implicitly utilize the optimality conditions for the discretization of the control variable. Our error analysis relies on the elliptic reconstruction technique introduced by C. Makridakis and R. H. Nochetto [SIAM J. Numer. Anal. 41, No. 4, 1585–1594 (2003; Zbl 1052.65088)] in conjunction with heat kernel estimates for linear parabolic problem. The use of elliptic reconstruction technique greatly simplifies the analysis by allowing us to take the advantage of existing elliptic maximum norm error estimate and the heat kernel estimate. We derive a posteriori error estimates for the state, co-state, and control variables in the $$L^\infty (0,T;\, L^\infty (\varOmega))$$-norm. Numerical experiments are conducted to illustrate the performance of the derived estimators.

### MSC:

 49J20 Existence theories for optimal control problems involving partial differential equations 49M05 Numerical methods based on necessary conditions 49M15 Newton-type methods 49M25 Discrete approximations in optimal control 49M29 Numerical methods involving duality 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Zbl 1052.65088
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### References:

 [1] Aronson, D. G., Non-negative solutions of linear parabolic equations, Ann Scuola Norm Sup Pisa (3), 22, 607-694 (1968) · Zbl 0182.13802 [2] Barbu V (1984) Optimal control for variational inequalities, research notes in mathematics, Vol 100, Pitman, London · Zbl 0574.49005 [3] Becker R, Kapp H (1997) Optimization in PDE models with adaptive finite element discretization. In: Proc. ENUMATH97, Heidelberg, Sep. 29-Oct.3, World Scientific Publ., 1998 · Zbl 0968.65035 [4] Becker R, Kapp H, Rannacher R (1998) Adaptive Finite element methods for optimal control of partial differential equations: basic concept, SFB359, University Heidelberg · Zbl 0967.65080 [5] Boman M (2000) On a posteriori error analysis in the maximum norm. Phd thesis, Chalmers University of Technology and Göteborg University [6] Brenner, SC; Scott, LR, The mathematical theory of finite element methods, texts in applied mathematics (2008), New York: Springer, New York [7] Dari, E.; Duran, RG; Padra, C., Maximum norm error estimators for three dimensional elliptic problems, SIAM J Numer Anal, 37, 683-700 (2000) · Zbl 0945.65120 [8] Demlow, A., Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems, SIAM J Numer Anal, 44, 494-514 (2006) · Zbl 1116.65114 [9] Demlow, A., Local a posteriori estimates for gradient errors in finite element methods for elliptic problems, Math Comp, 76, 19-42 (2007) · Zbl 1144.65068 [10] Demlow, A.; Lakkis, O.; Makridakis, C., A posteriori error estimates in the maximum norm for parabolic problems, SIAM J Numer Anal, 47, 2157-2176 (2009) · Zbl 1196.65153 [11] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems II: Optimal estimates in $$L^\infty (L^2)$$ and $$L^\infty (L^\infty )$$, SIAM J Numer Anal, 32, 706-740 (1995) · Zbl 0830.65094 [12] Haslinger, J.; Neittaanmaki, P., Finite element approximation for optimal shape design (1989), Chichester: Wiley, Chichester · Zbl 0713.73062 [13] Hetch, F., New development in free Fem++, J Numer Math, 20, 251-265 (2012) [14] Hinze, M., A variational discretization concept in control constrained optimization: the linear quadratic case, Comput Optim Appl, 30, 45-63 (2005) · Zbl 1074.65069 [15] Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J Control Optim, 20, 414-427 (1982) · Zbl 0481.49026 [16] Lakkis, O.; Makridakis, C., Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math Comp, 75, 1627-1658 (2006) · Zbl 1109.65079 [17] Langer, U.; Repin, S.; Wolfmayr, M., Functional a posteriori error estimates for parabolic time-periodic parabolic optimal control problems, Numer Funct Anal Optim, 37, 1267-1294 (2016) · Zbl 1377.49029 [18] Lions, JL, Optimal control of systems governed by partial differential equations (1971), Berlin: Springer-Verlag, Berlin [19] Liu, W.; Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer Math, 93, 497-521 (2003) · Zbl 1049.65057 [20] Luskin, M.; Rannacher, R., On the smoothing property of the Galerkin method for parabolic equations, SIAM J Numer Anal, 19, 93-113 (1982) · Zbl 0483.65064 [21] Makridakis, C.; Nochetto, RH, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J Numer Anal, 41, 1585-1594 (2003) · Zbl 1052.65088 [22] Meyer, C.; Rösch, A., Superconvergence properties of optimal control problems, SIAM J Control Optim, 43, 970-985 (2004) · Zbl 1071.49023 [23] Morin, P.; Nochetto, RH; Siebert, KG, Data oscillation and convergence of adaptive FEM, SIAM J Numer Anal, 38, 466-488 (2000) · Zbl 0970.65113 [24] Neittaanmaki, P.; Tiba, D., Optimal control of nonlinear parabolic systems: theory, algorithms and applications (1994), New York: M. Dekker, New York · Zbl 0812.49001 [25] Nochetto, RH, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math Comp, 64, 1-22 (1995) · Zbl 0920.65063 [26] Nochetto, RH; Schmidt, A.; Siebert, KG; Veeser, A., Pointwise a posteriori error control for elliptic obstacle problems, Numer Math, 95, 163-195 (2003) · Zbl 1027.65089 [27] Nochetto, RH; Schmidt, A.; Siebert, KG; Veeser, A., Fully localized a posteriori error estimators and barrier sets for contact problems, SIAM J Numer Anal, 42, 2118-2135 (2005) · Zbl 1095.65099 [28] Nochetto, RH; Schmidt, A.; Siebert, KG; Veeser, A., Pointwise a posteriori error estimates for monotone semilinear problems, Numer Math, 104, 515-538 (2006) · Zbl 1104.65107 [29] Nietzel, I.; Vexler, B., A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer Math, 120, 345-386 (2012) · Zbl 1245.65074 [30] Otárola, E.; Rankin, R.; Salgado, AJ, Maximum-norm a posteriori error estimates for an optimal control problem, Comput Optim Appl, 73, 997-1017 (2019) · Zbl 1430.49033 [31] Pironneau, O., Optimal shape design for elliptic systems (1984), Berlin: Springer-Verlag, Berlin · Zbl 0496.93029 [32] Rösch, A., Error estimates for parabolic optimal control problems with control constraints, Z Anal Anwend, 23, 353-376 (2004) · Zbl 1052.49031 [33] Sun, T.; Ge, L.; Liu, W., Equivalent a posteriori error estimates for a constrained optimal control problem governed by parabolic equations, Int J Numer Anal Model, 10, 1-23 (2013) · Zbl 1266.65198 [34] Tang Y, Chen Y (2012a) Recovery type a posteriori error estimates of fully discrete finite element methods for general convex parabolic optimal control problems. Numer Math Theoret Math Appl 5:573-591 · Zbl 1289.65148 [35] Tang, Y.; Chen, Y., Variational discretization for optimal parabolic optimal control problems with control constraints, J Syst Sci Complex, 25, 880-895 (2012) · Zbl 1269.49054 [36] Tang, Y.; Hua, Y., Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems, J Appl Anal Comput, 4, 295-306 (2014) · Zbl 1311.49071 [37] Tiba D (1995) Lectures on the optimal control of elliptic equations. University of Jyväskylä, Department of Mathematics [38] Wheeler, MF, A priori $$L^2$$-error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J Numer Anal, 10, 723-759 (1973) · Zbl 0232.35060 [39] Winther, R., Error estimates for a Galerkin approximation of a parabolic control problem, Ann. Mat. Pura Appl., 117, 173-206 (1978) · Zbl 0434.65092 [40] Xiong, C.; Li, Y., A posteriori error estimates for optimal distributed control governed by the evaluation equations, Appl Numer Math, 61, 181-200 (2011) · Zbl 1208.65091
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