Abstract
A mixed finite element discretization of an optimal control problem for the linear wave equation with homogeneous Dirichlet boundary condition is considered. For the temporal discretization, a Petrov–Galerkin scheme is utilized and the Raviart–Thomas finite elements for spatial discretization is used. A priori error analysis is proved for this numerical scheme. A hybridized formulation is proposed and if the Arnold–Brezzi post-processing method is applied, better convergence rates with respect to space are observed. The interchangeability of discretization and optimization holds both for mixed and hybrid formulations. Numerical experiments illustrating the theoretical results are presented using the lowest-order Raviart–Thomas elements.
Similar content being viewed by others
References
Anselmann, M., Bause, M., Becher, S., Matthies, G.: Galerkin-collocation approximation in time for the wave equation and its post-processing. ESAIM Math. Model. Numer. Anal. 54, 2099–2123 (2020)
Apel, T., Flaig, T.G.: Crank-Nicolson Schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50, 1484–1512 (2012)
Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, post-processing and error estimates. RAIRO Model. Math. Anal. Numer. 19, 7–32 (1985)
Bahriawati, C., Carstensen, C.: Three Matlab implementations of the lowest-order Raviart–Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5, 333–361 (2005)
Bangerth, W., Rannacher, R.: Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoustics 1, 1–17 (1999)
Baker, G.A.: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 564–576 (1976)
Barzilai, J., Borwein, J.: Two-point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988)
Bause, M., Köcher, U., Radu, F.A., Schieweck, F.: Post-processed Galerkin approximation of improved order for wave equations. Math. Comput. 89, 595–627 (2020)
Bècache, E., Joly, P., Tsogka, C.: An analysis of new mixed finite elements for the approximation of wave propagation problems. SIAM J. Numer. Anal. 37, 1053–1084 (2000)
Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68, 311–324 (1994)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Chen, Y., Lu, Z.: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 46, 957–965 (2010)
Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element methods for the wave equations. Comput. Methods Appl. Mech. Eng. 8, 205–222 (1990)
Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions. SIAM J. Numer. Anal. 33, 492–504 (1996)
Dai, Y.H., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box constrained quadratic programming. Numer. Math. 100, 21–47 (2005)
Dörfler, W., Findeisen, S., Wieners, C., Ziegler, D.: Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves. In: Langer, U., Steinbach, O. (eds.) Space-Time Methods: Applications to Partial Differential Equations. De Gruyter, Berlin (2019)
Egger, H., Radu, B.: Super-convergence and post-processing for mixed finite element approximations of the wave equation. B. Numer. Math. 140, 427–447 (2018)
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)
Ernesti, J., Wieners, C.: Space-time discontinuous Petrov–Galerkin methods for linear wave equations in heterogeneous media. Comput. Methods Appl. Math. 19, 465–481 (2019)
Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–61 (2005)
Hou, T.: Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations. Numer. Anal. Appl. 5, 348–362 (2012)
Jenkins, E.W., Rivière, B., Wheeler, M.F.: A priori error estimates for mixed finite element approximations fo the acoustic wave equation. SIAM J. Numer. Anal. 40, 1698–1715 (2002)
Kröner, A., Kunisch, K., Vexler, B.: Semismooth Newton methods for optimal control of the wave equation with control constraints. SIAM J. Control Optim. 49, 830–858 (2011)
Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA (1998)
Meidner, D., Vexler, B.: A priori error estimates for the space-time finite element discretization of parabolic optimal control problems. Part I: Problems without control constraints. SIAM J. Control Optim. 47, 1150–1177 (2008)
Meidner, D., Vexler, B.: A priori error analysis of the Petrov–Galerkin Crank–Nicholson Scheme for parabolic optimal control problems. SIAM J. Control. Optim. 49, 2183–2211 (2011)
Makridakis, Ch.G.: On mixed finite element methods for linear elastodynamics. Numer. Math. 61, 235–260 (1992)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Peralta, G.: Error estimates for mixed and hybrid FEM for elliptic optimal control problems with penalizations, submitted
Quarteroni, A., Valli, A.: Numerical Approximations of Partial Differential Equations. Springer, Heidelberg (2008)
Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. Lect. Notes Math 606, 292–315 (1977)
Sellitto, A., Cimmelli, V.A., Jou, D.: Mesoscopic Theories of Heat Transport in Nanosystems. Spinger, New York (2016)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory. American Mathematical Society, Providence, Rhode Island, Methods and Applications (2010)
Xing, X., Chen, Y.: Error estimates of mixed methods for optimal control problems governed by parabolic equations. Int. J. Numer. Method Eng. 75, 735–754 (2008)
Xing, X., Chen, Y., Yi, N.: Error estimates of mixed finite element methods for quadratic control problems. J Comput. Appl. Math. 233, 1812–1820 (2010)
Acknowledgements
The authors are grateful to the reviewers for the careful reading and valuable suggestions that led to the improvement of the paper.
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
Peralta, G., Kunisch, K. Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation. Numer. Math. 150, 591–627 (2022). https://doi.org/10.1007/s00211-021-01258-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-021-01258-9