Open Access
Issue |
ESAIM: COCV
Volume 29, 2023
|
|
---|---|---|
Article Number | 41 | |
Number of page(s) | 40 | |
DOI | https://doi.org/10.1051/cocv/2023028 | |
Published online | 09 June 2023 |
- P.F. Antonietti, I. Mazzieri and F. Migliorini, A space-time discontinuous Galerkin method for the elastic wave equation. J. Comput. Phys. 419 (2020) 109685. [CrossRef] [MathSciNet] [Google Scholar]
- E. Aranda and P. Pedregal, A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations. J. Franklin Inst. 351 (2014) 3865–3882. [CrossRef] [MathSciNet] [Google Scholar]
- M. Asch and A. Münch, Uniformly controllable schemes for the wave equation on the unit square. J. Optim. Theory Appl. 143 (2009) 417–438. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- F. Bourquin, Approximation theory for the problem of exact controllability of the wave equation with boundary control. In Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993). SIAM, Philadelphia, PA (1993), pages 103–112. [Google Scholar]
- S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, third edition (2008). [CrossRef] [Google Scholar]
- E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem. Math. Comp. 86 (2017) 75–96. [Google Scholar]
- E. Burman, A. Feizmohammadi, A. Munch and L. Oksanen, Space time stabilized finite element methods for a unique continuation problem subject to the wave equation. ESAIM: Math. Model. Numer. Anal. 55 (2021) S969–S991. [CrossRef] [EDP Sciences] [Google Scholar]
- E. Burman, A. Feizmohammadi and L. Oksanen, A finite element data assimilation method for the wave equation. Math. Comp. 89 (2020) 1681–1709. [CrossRef] [MathSciNet] [Google Scholar]
- E. Burman, A. Feizmohammadi and L. Oksanen, A fully discrete numerical control method for the wave equation. SIAM J. Control Optim. 58 (2020) 1519–1546. [CrossRef] [MathSciNet] [Google Scholar]
- C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413–462. [CrossRef] [MathSciNet] [Google Scholar]
- C. Castro, S. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28 (2008) 186–214. [Google Scholar]
- N. Cîndea, E. Fernández-Cara and A. Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates. ESAIM: COCV 19 (2013) 1076–1108. [CrossRef] [EDP Sciences] [Google Scholar]
- N. Cîndea, S. Micu and M. Tucsnak, An approximation method for exact controls of vibrating systems. SIAM J. Control Optim. 49 (2011) 1283–1305. [CrossRef] [MathSciNet] [Google Scholar]
- N. Cîndea and A. Münch, A mixed formulation for the direct approximation of the control of minimal L2-norm for linear type wave equations. Calcolo 52 (2015) 245–288. [Google Scholar]
- W. Dörfler, S. Findeisen and C. Wieners, Space-time discontinuous galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl. Math. 16 (2016) 409–428. [CrossRef] [MathSciNet] [Google Scholar]
- S. Dumont, F. Jourdan and T. Madani, 4D remeshing using a space-time finite element method for elastodynamics problems. Math. Comput. Appl. 23 (2018) Paper No. 29, 18. [Google Scholar]
- A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [CrossRef] [Google Scholar]
- A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation. ESAIM Math. Model. Numer. Anal. 51 (2017) 1367–1385. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401. [Google Scholar]
- S. Ervedoza and E. Zuazua, Numerical approximation of exact controls for waves. SpringerBriefs in Mathematics. Springer, New York (2013). [CrossRef] [Google Scholar]
- L.C. Evans, Partial differential equations. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition (2010). [CrossRef] [Google Scholar]
- D.A. French, A space-time finite element method for the wave equation. Comput. Methods Appl. Mech. Eng. 107 (1993) 145–157. [CrossRef] [Google Scholar]
- R. Glowinski, W. Kinton and M.F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation. Internat. J. Numer. Methods Engrg. 27 (1989) 623–635. [CrossRef] [MathSciNet] [Google Scholar]
- R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods. Japan J. Appl. Math. 7 (1990) 1–76. [CrossRef] [MathSciNet] [Google Scholar]
- M. Gunzburger, L.S. Hou and L. Ju, A numerical method for exact boundary controllability problems for the wave equation. Comput. Math. Appl. 51 (2006) 721–750. [CrossRef] [MathSciNet] [Google Scholar]
- F. Hecht, New development in Freefem++. J. Numer. Math. 20 (2012) 251–265. [Google Scholar]
- L. Hörmander, The analysis of linear partial differential operators. III. Classics in Mathematics. Springer, Berlin (2007). [CrossRef] [Google Scholar]
- G.M. Hulbert and T.J.R. Hughes, Space-time finite element methods for second-order hyperbolic equations. Comput. Methods Appl. Mech. Engrg. 84 (1990) 327–348. [CrossRef] [MathSciNet] [Google Scholar]
- L.I. Ignat and E. Zuazua, Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. (JEMS) 11 (2009) 351–391. [CrossRef] [MathSciNet] [Google Scholar]
- C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129. [CrossRef] [MathSciNet] [Google Scholar]
- U. Langer and O. Steinbach, editors, Space-Time Methods. De Gruyter (2019). [CrossRef] [Google Scholar]
- I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9) 65 (1986) 149–192. [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- G. Lebeau and M. Nodet, Experimental study of the HUM control operator for linear waves. Exp. Math. 19 (2010) 93–120. [CrossRef] [Google Scholar]
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. [Google Scholar]
- P. Lissy and I. Rovenţa, Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method. Math. Comp. 88 (2019) 273–291. [Google Scholar]
- P. Loreti and M. Mehrenberger, An Ingham type proof for a two-grid observability theorem. ESAIM: COCV 14 (2008) 604–631. [CrossRef] [EDP Sciences] [Google Scholar]
- S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation. Numer. Math. 91 (2002) 723–768. [CrossRef] [MathSciNet] [Google Scholar]
- S. Micu, I. Roventa and L.E. Temereancă, Approximation of the controls for the wave equation with a potential. Numer. Math. 144 (2020) 835–887. [CrossRef] [MathSciNet] [Google Scholar]
- L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Control Optim. 41 (2002) 1554–1566. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. M2AN Math. Model. Numer. Anal. 39 (2005) 377–418. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- A. Münch and E. Trélat, Constructive exact control of semilinear 1-D wave equations by a least-squares approach (2020). [Google Scholar]
- P. Pedregal, F. Periago and J. Villena, A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems. Stud. Appl. Math. 121 (2008) 27–47. [CrossRef] [MathSciNet] [Google Scholar]
- L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. [CrossRef] [MathSciNet] [Google Scholar]
- O. Steinbach and M. Zank, A Stabilized Space–Time Finite Element Method for the Wave Equation. Springer International Publishing, Cham (2019), pp. 341–370. [Google Scholar]
- N.J. Walkington, Combined DG-CG time stepping for wave equations. SIAM J. Numer. Anal. 52 (2014) 1398–1417. [CrossRef] [MathSciNet] [Google Scholar]
- X. Zhang, Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000) 1101–1115. [Google Scholar]
- E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. [CrossRef] [MathSciNet] [Google Scholar]
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