Space time stabilized finite element methods for a unique continuation problem subject to the wave equation. (English) Zbl 1487.65131
Summary: We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the approximation order of the finite element residual. Numerical examples are provided that illustrate the methodology.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 76Q05 | Hydro- and aero-acoustics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 93B27 | Geometric methods |
| 35R30 | Inverse problems for PDEs |
Keywords:
unique continuation; data assimilation; wave equation; finite element method; geometric control; condition; observability estimateSoftware:
FreeFem++References:
| [1] | S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed. Inverse Prob. 31 (2015) 065009. · Zbl 1320.65135 |
| [2] | D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems. C. R. Math. Acad. Sci. Paris 340 (2005) 873-878. · Zbl 1074.34006 |
| [3] | C. Bardos, G. Lebeau and J. Rauch, Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue) (1988) 11-31. · Zbl 0673.93037 |
| [4] | C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. · Zbl 0786.93009 |
| [5] | M. Bernadou and K. Hassan, Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced. Int. J. Numer. Methods Eng. 17 (1981) 784-789. · Zbl 0478.73050 |
| [6] | L. Bourgeois, D. Ponomarev and J. Dardé, An inverse obstacle problem for the wave equation in a finite time domain. Inverse Prob. Imaging 13 (2019) 377-400. · Zbl 1410.35283 |
| [7] | S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York (2008). · Zbl 1135.65042 |
| [8] | E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput. 35 (2013) A2752-A2780. · Zbl 1286.65152 |
| [9] | E. Burman, Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris 352 (2014) 655-659. · Zbl 1297.65148 |
| [10] | E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem. Math. Comput. 86 (2017) 75-96. · Zbl 1351.65074 |
| [11] | E. Burman and L. Oksanen, Data assimilation for the heat equation using stabilized finite element methods. Numer. Math. 139 (2018) 505-528. · Zbl 1412.65111 |
| [12] | E. Burman, J. Ish-Horowicz and L. Oksanen, Fully discrete finite element data assimilation method for the heat equation. ESAIM: M2AN 52 (2018) 2065-2082. · Zbl 1420.65124 |
| [13] | E. Burman, A. Feizmohammadi and L. Oksanen, A fully discrete numerical control method for the wave equation. SIAM J. Control Optim.. Preprint arXiv:1903.02320 (2019). · Zbl 1444.35110 |
| [14] | E. Burman, A. Feizmohammadi and L. Oksanen, A finite element data assimilation method for the wave equation. Math. Comput. 89 (2020) 1681-1709. · Zbl 1436.65132 |
| [15] | C. Castro, N. Cndea and A. Münch, Controllability of the linear one-dimensional wave equation with inner moving forces. SIAM J. Control Optim. 52 (2014) 4027-4056. · Zbl 1320.35185 |
| [16] | O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for photoacoustic tomography., Inverse Prob. 32 (2016) 125004. · Zbl 1362.35329 |
| [17] | N. Cîndea and A. Münch, Inverse problems for linear hyperbolic equations using mixed formulations. Inverse Prob. 31 (2015) 075001. · Zbl 1372.35360 |
| [18] | N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal L^2-norm for linear type wave equations. Calcolo 52 (2015) 245-288. · Zbl 1350.65067 |
| [19] | C. Clason and M.V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30 (2007/2008) 1-23. · Zbl 1159.65346 |
| [20] | R. Codina, Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput. Methods Appl. Mech. Eng. 190 (2000) 1579-1599. · Zbl 0998.76047 |
| [21] | G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei and R.L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191 (2002) 3669-3750. · Zbl 1086.74038 |
| [22] | A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. In: Vol. 159 of Applied Mathematical Sciences. Springer, New York (2004). · Zbl 1059.65103 |
| [23] | J. Ernesti and C. Wieners, Space-time discontinuous Petrov-Galerkin methods for linear wave equations in heterogeneous media. Comput. Methods Appl. Math. 19 (2019) 465-481. · Zbl 1420.65100 |
| [24] | S. Ervedoza and E. Zuazua, The wave equation: control and numerics. In: Vol. 2048 of Lecture Notes in Mathematics. Control of Partial Differential Equations. Berlin-Heidelberg, Springer (2012) 245-339. |
| [25] | S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves. Springer Briefs in Mathematics. Springer, New York (2013). · Zbl 1282.93001 |
| [26] | S. Ervedoza, A. Marica and E. Zuazua, Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis. IMA J. Numer. Anal. 36 (2016) 503-542. · Zbl 1433.65157 |
| [27] | R. Glowinski and J.-L. Lions, Exact and Approximate Controllability for Distributed Parameter Systems. In: Acta Numerica. Cambridge University Press, Cambridge (1994) 269-378. · Zbl 0838.93013 |
| [28] | G. Haine and K. Ramdani, Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations. Numer. Math. 120 (2012) 307-343. · Zbl 1235.65082 |
| [29] | F. Hecht, New development in Freefem++. J. Numer. Math. 20 (2012) 251-265. · Zbl 1266.68090 |
| [30] | J.A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation. M2AN 33 (1999) 407-438. · Zbl 0947.65101 · doi:10.1051/m2an:1999123 |
| [31] | M.V. Klibanov and J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data. Inverse Prob. 7 (1991) 577-596. · Zbl 0744.35065 |
| [32] | M. Klibanov and Rakesh, Numerical solution of a time-like Cauchy problem for the wave equation. Math. Methods Appl. Sci. 15 (1992) 559-570. · Zbl 0759.65061 |
| [33] | P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography. Eur. J. Appl. Math. 19 (2008) 191-224. · Zbl 1185.35327 |
| [34] | R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. In: Vol. 15 of Travaux et Recherches Mathématiques. Dunod, Paris (1967). · Zbl 0159.20803 |
| [35] | J. Le Rousseau, G. Lebeau, P. Terpolilli and E. Trélat, Geometric control condition for the wave equation with a time-dependent observation domain. Anal. PDE 10 (2017) 983-1015. · Zbl 1391.35254 |
| [36] | L. Miller, Resolvent conditions for the control of unitary groups and their approximations. J. Spectr. Theory 2 (2012) 1-55. · Zbl 1239.93012 |
| [37] | S. Montaner and A. Münch, Approximation of controls for linear wave equations: a first order mixed formulation. Math. Control Relat. Fields 9 (2019) 729-758. · Zbl 1443.93034 |
| [38] | A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. M2AN 39 (2005) 377-418. · Zbl 1130.93016 · doi:https://www.esaim-m2an.org/articles/m2an/abs/2005/02/m2an0479/m2an0479.html |
| [39] | L.V. Nguyen and L.A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity. SIAM J. Imaging Sci. 9 (2016) 748-769. · Zbl 1515.35354 |
| [40] | J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. · Zbl 0229.65079 |
| [41] | K. Ramdani, M. Tucsnak and G. Weiss, Recovering and initial state of an infinite-dimensional system using observers. Automatica J. IFAC 46 (2010) 1616-1625. · Zbl 1204.93023 |
| [42] | L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions., Math. Comput. 54 (1990) 483-493. · Zbl 0696.65007 |
| [43] | P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed. Inverse Prob. 25 (2009) 075011. · Zbl 1177.35256 |
| [44] | P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: averaged sharp time reversal. Inverse Prob. 31 (2015) 065007. · Zbl 1320.35191 |
| [45] | O. Steinnach and M. Zank, A stabilized space-time finite element method for the wave equation. Technische Universität Graz Report 2018/5 (2018) 1-27. |
| [46] | V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. In: Vol. 25 of Springer Series in Computational Mathematics. Springer, Berlin-Heidelberg (1997). · Zbl 0884.65097 |
| [47] | L.V. Wang, Photoacoustic Imaging and Spectroscopy. CRC Press (2009). |
| [48] | E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. · Zbl 1077.65095 |
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