Abstract
In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L2-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.
Similar content being viewed by others
References
Babuška, I., Suri, M.: The h − p version of the finite element method with quasiuniform meshes. Math. Model. Numer. Anal. 21, 199–238 (1987)
Benedix, O., Vexler, B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl. 44(1), 3–25 (2009)
Butt, M. M., Yuan, Y.: A full multigrid method for distributed control problems constrained by Stokes equations. Numer. Math. Theor. Meth. Appl. 10, 639–655 (2017)
Canuto, C., Hussaini, M. Y., Quarteroni, A., et al: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)
Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control. Optimal. 24, 1309–1322 (1986)
Casas, E., Tröltzsch, F.: Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. Appl. Math. Optim. 39, 2524–2550 (1999)
Chen, Y., Huang, F.: Galerkin spectral approximation of elliptic optimal control problems with H1 -norm state constraint. J. Sci. Comput. 67(1), 65–83 (2016)
Chen, Y., Huang, F.: Spectral method approximation of flow optimal control problems with H 1-norm state constraint. Numer. Math. Theory. Meth. Appl. 10(3), 614–638 (2017)
Chen, Y., Lin, Y.: A posteriori error estimates for hp finite element solutions of convex optimal control problems. J. Comput. Appl. Math. 23, 3435–3454 (2011)
Chen, Y., Yi, N., Liu, W.: A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J. Numer. Anal. 46 (5), 2254–2275 (2008)
Chen, Y., Huang, F., Yi, N., Liu, W.: A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations. SIAM J. Numer. Anal. 49(4), 1625–1648 (2011)
Chen, Y., Huang, Y., Liu, W., et al.: A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients. Comput. Math. Appl. 70(4), 297–313 (2015)
Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J Numer. Anal. 45(5), 1937–1953 (2007)
Du, N., Ge, L., Liu, W.: Adaptive finite element approximation for an elliptic optimal control problem with both pointwise and integral control constraints. J. Sci. Comput. 60(1), 160–183 (2014)
Gong, W., Hinze, M.: Error estimates for parabolic optimal control problems with control and state constraints. Comput. Optim. Appl. 56(1), 131–151 (2013)
Gong, W., Liu, W., Yan, N.: A posteriori error estimates of hp-FEM for optimal control problems. Int. J. Numer. Anal. Model. 8(1), 48–69 (2011)
Gottlieb, D., Orszag, S. A.: Numerical analysis of spectral methods: theory and applications. for. Industr. Appl. Math. 45(4), 969–970 (1977)
Guo, B. Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)
Guo, B. Y.: Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68(227), 1067–1078 (1999)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands (2009)
Hou, T.L., Liu, C.M., Chen, H.B.: Fully discrete H1-Galerkin mixed finite element methods for parabolic optimal control problems. Numer. Math. Theor. Meth. Appl. 12, 134–153 (2019)
Huang, F., Chen, Y.: Error estimates for spectral approximation of elliptic control problems with integral state and control constraints. Comput. Math. Appl. 68(8), 789–803 (2014)
Leng, H., Chen, Y.: Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on L2 errors. J. Sci. Comput. 73(1), 1–21 (2017)
Leykekhman, D., Meidner, D., Vexler, B.: Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55(3), 769–802 (2013)
Li, R., Liu, W., Yan, N.: A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 33(2), 155–182 (2007)
Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Liu, W., Yan, N: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15(1–4), 285–309 (2001)
Liu, W., Yan, N: A posteriori error estimates for control problems governed by stokes equations. SIAM J. Numer Anal. 40(5), 1850–1869 (2003)
Liu, W., Yan, N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Scientific Press, Beijing (2008)
Liu, W., Yang, D., Yuan, L., Ma, C.: Finite element approximations of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48, 1163–1185 (2010)
Liu, C.M., Hou, T.L., Yang, Y: Superconvergence of H1-Galerkin mixed finite element methods for elliptic optimal control problems. East. Asia. J. Appl. Math. 9, 87–101 (2019)
Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. SIAM J Control Optim. 47, 1301–1329 (2008)
Melenk, J: hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation. SIAM J. Numer. Anal. 43, 127–155 (2005)
Mozolevski, I., Süli, E., Bösing, P. R.: hp-version a priori error analysis of interior penalty discontinous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30(3), 465–491 (2007)
Neitzel, I., Tröltzsch, F.: On regularization methods for the numerical solution, of parabolic control problems with pointwise state constraints. ESAIM Control Optim. Calc. Var. 15(2), 81–96 (2008)
Neitzel, I., Pfefferer, J., Rösch, A.: Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation. SIAM J. Control Optim. 53(2), 874–904 (2015)
Pozrikidis, C.: Introduction to Finite and Spectral Element Methods Using MATLAB. Chapman Hall/CRC, Boca Raton (2005)
Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)
Shen, J., Wang, L. L.: Spectral approximation of the Helmholtz equation with high wave numbers. SIAM J. Numer. Anal. 43(2), 623–644 (2006)
Shen, J., Tang, T., Wang, L. L.: Spectral Methods, p 41. Springer, Berlin (2011)
Trefethen, L. N: Spectral Methods in MATLAB. SIAM (2000)
Tröltzsch, F.: Optimal control of partial differential equations: theory, methods and applications. SIAM J. Control. Optim. 112(2), 399 (2010)
Xu, C., Lin, Y.: Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation. Int. J. Numer. Meth. Fluids 32(6), 619–646 (2015)
Xu, C., Maday, Y.: A spectral element method for the time-dependent two-dimensional Euler equations: applications to flow simulations. J. Comput. Appl. Math. 91(1), 63–85 (1998)
Xu, C., Pasquetti, R.: Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196(2), 680–704 (2004)
Yan, N.: Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations. Appl. Math. 54(3), 267–283 (2009)
Yang, FW, Venkataraman, C, Styles, V, Madzvamuse, A: A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws. Commun. Comput. Phys. 21, 65–92 (2017)
Yuan, L., Yang, D.: A posteriori error estimate of optimal control problem of PDE with integral constraint for state. J. Comput. Math. 27(4), 525–542 (2009)
Zhou, J., Chen, Y., Dai, Y.: Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint. Appl. Math. Comput. 217(5), 2057–2066 (2010)
Zhou, J., Yang, D.: Legendre-Galerkin spe7ctral methods for optimal control problems with integral constraint for state in one dimension. Comput. Optim. Appl. 61, 135–158 (2015)
Funding
The work is supported by the National Natural Science Foundation of China (Grant No. 11671157,11826212) and Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2018B320)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lin, X., Chen, Y. & Huang, Y. A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint. Numer Algor 83, 1145–1169 (2020). https://doi.org/10.1007/s11075-019-00719-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00719-5