Figure 1.
Regular meshes for $ Q_T $ ; Left: uniform mesh - $ h = 1.41\times10^{-1} $ . Right: non uniform mesh - $ h = 1.52\times 10^{-1} $
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doi: 10.3934/mcrf.2019030
Approximation of controls for linear wave equations: A first order mixed formulation
Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, UMR CNRS 6620, Campus des Cézeaux, 63178 Aubière, France |
This paper deals with the numerical approximation of null controls for the wave equation posed in a bounded domain of $ \mathbb{R}^n $. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & Münch, A mixed formulation for the direct approximation of the control of minimal $ L^2 $-norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized observability inequality, requires $ H^2 $-finite element approximation both in time and space. Following a similar approach, we present and analyze a variational method still leading to strong convergent results but using simpler $ H^1 $-approximation. The main point is to preliminary restate the second order wave equation into a first order system and then prove an appropriate observability inequality.
Keywords:
Wave equation,
first order system,
null controllability,
space-time discretization,
finite element.
Mathematics Subject Classification:
Primary: 35B37, 93B05; Secondary: 65N06.
Citation:
Santiago Montaner, Arnaud Münch.
Approximation of controls for linear wave equations: A first order mixed formulation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019030
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References:
| [1] |
H. J. C. Barbosa and T. J. R. Hughes,
The finite element method with Lagrange multipliers on the boundary: Circumventing the Babuška-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85 (1991), 109-128.
doi: 10.1016/0045-7825(91)90125-P. |
| [2] |
L. Baudouin, M. De Buhan and S. Ervedoza,
Global Carleman estimates for waves and applications, Comm. Partial Differential Equations, 38 (2013), 823-859.
doi: 10.1080/03605302.2013.771659. |
| [3] |
E. Bécache, P. Joly and C. Tsogka,
An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J. Numer. Anal., 37 (2000), 1053-1084.
doi: 10.1137/S0036142998345499. |
| [4] |
____, A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal., 39 (2002), 2109-2132.
doi: 10.1137/S0036142999359189. |
| [5] |
D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013.
doi: 10.1007/978-3-642-36519-5. |
| [6] |
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.
doi: 10.1007/s00211-005-0651-0. |
| [7] |
D. Chapelle and K.-J. Bathe,
The inf-sup test, Comput. & Structures, 47 (1993), 537-545.
doi: 10.1016/0045-7949(93)90340-J. |
| [8] |
P. G. Ciarlet, The finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
doi: 10.1137/1.9780898719208. |
| [9] |
N. Cîndea and A. Münch, Inverse problems for linear hyperbolic equations using mixed formulations, Inverse Problems, 31 (2015), 075001, 38pp.
doi: 10.1088/0266-5611/31/7/075001. |
| [10] |
____, A mixed formulation for the direct approximation of the control of minimal L2-norm for linear type wave equations, Calcolo, 52 (2015), 245-288.
doi: 10.1007/s10092-014-0116-x. |
| [11] |
R Codina,
Finite element approximation of the hyperbolic wave equation in mixed form, Comput. Methods Appl. Mech. Engrg., 197 (2008), 1305-1322.
doi: 10.1016/j.cma.2007.11.006. |
| [12] |
S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, SpringerBriefs in Mathematics, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5808-1. |
| [13] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. |
| [14] |
R. Glowinski,
Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., 103 (1992), 189-221.
doi: 10.1016/0021-9991(92)90396-G. |
| [15] |
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.
doi: 10.1002/nme.1620270313. |
| [16] |
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.
doi: 10.1007/BF03167859. |
| [17] |
F. Hecht,
New development in freefem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
| [18] |
V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. |
| [19] |
A. Kröner, K. Kunisch and B. Vexler,
Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858.
doi: 10.1137/090766541. |
| [20] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. |
| [21] |
A. Münch,
A uniformly controllable and implicit scheme for the 1-D wave equation, M2AN Math. Model. Numer. Anal., 39 (2005), 377-418.
doi: 10.1051/m2an:2005012. |
| [22] |
A. Münch and D. A. Souza,
Inverse problems for linear parabolic equations using mixed formulations-Part 1: Theoretical analysis, J. Inverse Ill-Posed Probl., 25 (2017), 445-468.
doi: 10.1515/jiip-2015-0112. |
| [23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
show all references
References:
| [1] |
H. J. C. Barbosa and T. J. R. Hughes,
The finite element method with Lagrange multipliers on the boundary: Circumventing the Babuška-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85 (1991), 109-128.
doi: 10.1016/0045-7825(91)90125-P. |
| [2] |
L. Baudouin, M. De Buhan and S. Ervedoza,
Global Carleman estimates for waves and applications, Comm. Partial Differential Equations, 38 (2013), 823-859.
doi: 10.1080/03605302.2013.771659. |
| [3] |
E. Bécache, P. Joly and C. Tsogka,
An analysis of new mixed finite elements for the approximation of wave propagation problems, SIAM J. Numer. Anal., 37 (2000), 1053-1084.
doi: 10.1137/S0036142998345499. |
| [4] |
____, A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal., 39 (2002), 2109-2132.
doi: 10.1137/S0036142999359189. |
| [5] |
D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013.
doi: 10.1007/978-3-642-36519-5. |
| [6] |
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.
doi: 10.1007/s00211-005-0651-0. |
| [7] |
D. Chapelle and K.-J. Bathe,
The inf-sup test, Comput. & Structures, 47 (1993), 537-545.
doi: 10.1016/0045-7949(93)90340-J. |
| [8] |
P. G. Ciarlet, The finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
doi: 10.1137/1.9780898719208. |
| [9] |
N. Cîndea and A. Münch, Inverse problems for linear hyperbolic equations using mixed formulations, Inverse Problems, 31 (2015), 075001, 38pp.
doi: 10.1088/0266-5611/31/7/075001. |
| [10] |
____, A mixed formulation for the direct approximation of the control of minimal L2-norm for linear type wave equations, Calcolo, 52 (2015), 245-288.
doi: 10.1007/s10092-014-0116-x. |
| [11] |
R Codina,
Finite element approximation of the hyperbolic wave equation in mixed form, Comput. Methods Appl. Mech. Engrg., 197 (2008), 1305-1322.
doi: 10.1016/j.cma.2007.11.006. |
| [12] |
S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, SpringerBriefs in Mathematics, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5808-1. |
| [13] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. |
| [14] |
R. Glowinski,
Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation, J. Comput. Phys., 103 (1992), 189-221.
doi: 10.1016/0021-9991(92)90396-G. |
| [15] |
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.
doi: 10.1002/nme.1620270313. |
| [16] |
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.
doi: 10.1007/BF03167859. |
| [17] |
F. Hecht,
New development in freefem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
| [18] |
V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. |
| [19] |
A. Kröner, K. Kunisch and B. Vexler,
Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858.
doi: 10.1137/090766541. |
| [20] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. |
| [21] |
A. Münch,
A uniformly controllable and implicit scheme for the 1-D wave equation, M2AN Math. Model. Numer. Anal., 39 (2005), 377-418.
doi: 10.1051/m2an:2005012. |
| [22] |
A. Münch and D. A. Souza,
Inverse problems for linear parabolic equations using mixed formulations-Part 1: Theoretical analysis, J. Inverse Ill-Posed Probl., 25 (2017), 445-468.
doi: 10.1515/jiip-2015-0112. |
| [23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
Figure 3.
Evolution of $ \|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)} $ w.r.t $ h $ for uniform mesh with $ r = 1 $ $ (\circ) $ , $ r = h $ $ (\star) $ and non uniform mesh with $ r = 1 $ $ (\bigtriangleup) $ and $ r = h $ $ (\bigtriangledown) $
Figure 4.
Iterative refinement of the triangular mesh over $ Q_T $ with respect to the variable $ \lambda_h $ : $ 110 $ , $ 2\, 880 $ and $ 8\, 636 $ triangles
Table 1.
Number of elements for the uniform(u)/non uniform(nu) meshes and value of $h$ for each type of mesh w.r.t. $N$ with $T = 2$.
| $N$ | $10$ | $20$ | $40$ | $80$ | $160$ |
| card$(\mathcal{T}_h)$-u | $400$ | $1\, 600$ | $6\, 400$ | $25\, 600$ | $102\, 400$ |
| card$(\mathcal{T}_h)$-nu | $446$ | $1\, 784$ | $7\, 136$ | $28\, 544$ | $114\, 176$ |
| $\sharp$ nodes-u | $861$ | $3\, 321$ | $13\, 041$ | $56\, 681$ | $205\, 761$ |
| $\sharp$ nodes-nu | $953$ | $3\, 689$ | $14\, 513$ | $57\, 569$ | $229\, 313$ |
| $h$-u | $1.41\times 10^{-1}$ | $7.01\times 10^{-2}$ | $3.53\times 10^{-2}$ | $1.76\times 10^{-2}$ | $8.83\times 10^{-3}$ |
| $h$-nu | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ | $9.50\times 10^{-3}$ |
| $N$ | $10$ | $20$ | $40$ | $80$ | $160$ |
| card$(\mathcal{T}_h)$-u | $400$ | $1\, 600$ | $6\, 400$ | $25\, 600$ | $102\, 400$ |
| card$(\mathcal{T}_h)$-nu | $446$ | $1\, 784$ | $7\, 136$ | $28\, 544$ | $114\, 176$ |
| $\sharp$ nodes-u | $861$ | $3\, 321$ | $13\, 041$ | $56\, 681$ | $205\, 761$ |
| $\sharp$ nodes-nu | $953$ | $3\, 689$ | $14\, 513$ | $57\, 569$ | $229\, 313$ |
| $h$-u | $1.41\times 10^{-1}$ | $7.01\times 10^{-2}$ | $3.53\times 10^{-2}$ | $1.76\times 10^{-2}$ | $8.83\times 10^{-3}$ |
| $h$-nu | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ | $9.50\times 10^{-3}$ |
Table 2.
$ \delta_h $ w.r.t. $ r $ and $ h $ , $ T = 2 $ , for the $ W_h $ -$ M_h $ finite elements and non uniform mesh
Table 3.
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh.
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ | $9.50\times 10^{-3}$ |
| $r=1$ | $0.426$ | $0.316$ | $0.229$ | $0.155$ | $0.106$ |
| $r=10^{-1}$ | $0.991$ | $0.868$ | $0.698$ | $0.489$ | $0.339$ |
| $r=10^{-2}$ | $2.269$ | $1.738$ | $1.373$ | $1.099$ | $0.896$ |
| $r=h$ | $0.885$ | $0.927$ | $0.929$ | $0.921$ | $0.908$ |
| $r=h^2$ | $1.612$ | $2.154$ | $2.974$ | $4.115$ | $5.733$ |
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ | $9.50\times 10^{-3}$ |
| $r=1$ | $0.426$ | $0.316$ | $0.229$ | $0.155$ | $0.106$ |
| $r=10^{-1}$ | $0.991$ | $0.868$ | $0.698$ | $0.489$ | $0.339$ |
| $r=10^{-2}$ | $2.269$ | $1.738$ | $1.373$ | $1.099$ | $0.896$ |
| $r=h$ | $0.885$ | $0.927$ | $0.929$ | $0.921$ | $0.908$ |
| $r=h^2$ | $1.612$ | $2.154$ | $2.974$ | $4.115$ | $5.733$ |
Table 4.
$ \delta_h $ w.r.t. $ r $ and $ h $ , $ T = 2 $ , for the $ \widetilde{W_h} $ -$ M_h $ finite elements and uniform mesh
Table 5.
$ r = h $ - non uniform mesh
Table 6.
$r = h$ - uniform mesh
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.535$ | $0.549$ | $0.559$ | $0.566$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $3.43\times 10^{-2}$ | $2.22\times 10^{-2}$ | $1.45\times 10^{-2}$ | $9.43\times 10^{-3}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.545$ | $0.558$ | $0.566$ | $0.57$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $3.89\times 10^{-2}$ | $2.3\times 10^{-2}$ | $1.46\times 10^{-2}$ | $9.35\times 10^{-3}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $0.561$ | $0.388$ | $0.265$ | $0.184$ |
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.535$ | $0.549$ | $0.559$ | $0.566$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $3.43\times 10^{-2}$ | $2.22\times 10^{-2}$ | $1.45\times 10^{-2}$ | $9.43\times 10^{-3}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.545$ | $0.558$ | $0.566$ | $0.57$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $3.89\times 10^{-2}$ | $2.3\times 10^{-2}$ | $1.46\times 10^{-2}$ | $9.35\times 10^{-3}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $0.561$ | $0.388$ | $0.265$ | $0.184$ |
| $\sharp\ \text{triangles}$ | $110$ | $1197$ | $2880$ | $8636$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.46$ | $0.57$ | $0.574$ | $0.577$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $8.24\times10^{-2}$ | $1.55\times10^{-2}$ | $3.72\times10^{-3}$ | $5.18\times10^{-4}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.451$ | $0.569$ | $0.574$ | $0.577$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $8.04\times10^{-2}$ | $1.52\times10^{-2}$ | $3.88\times10^{-3}$ | $4.48\times10^{-4}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $1.13\times10^5$ | $4.45\times10^4$ | $1.48\times10^4$ | $2.86\times10^3$ |
| $\sharp\ \text{triangles}$ | $110$ | $1197$ | $2880$ | $8636$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.46$ | $0.57$ | $0.574$ | $0.577$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $8.24\times10^{-2}$ | $1.55\times10^{-2}$ | $3.72\times10^{-3}$ | $5.18\times10^{-4}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.451$ | $0.569$ | $0.574$ | $0.577$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $8.04\times10^{-2}$ | $1.52\times10^{-2}$ | $3.88\times10^{-3}$ | $4.48\times10^{-4}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $1.13\times10^5$ | $4.45\times10^4$ | $1.48\times10^4$ | $2.86\times10^3$ |
Table 8.
Non uniform mesh - Conjugate gradient method - Number of iterates for $ r = 1 $ (top), $ r = 10^{-2} $ and $ r = h $ (bottom)
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.456$ | $0.498$ | $0.523$ | $0.54$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $6.99\times 10^{-2}$ | $4.99\times 10^{-2}$ | $3.56\times 10^{-2}$ | $2.54\times 10^{-2}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.511$ | $0.536$ | $0.55$ | $0.559$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $7.2\times 10^{-2}$ | $5.06\times 10^{-2}$ | $3.59\times 10^{-2}$ | $2.55\times 10^{-2}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $0.474$ | $0.363$ | $0.29$ | $0.238$ |
| $h$ | $1.52\times 10^{-1}$ | $7.60\times 10^{-2}$ | $3.80\times 10^{-2}$ | $1.90\times 10^{-2}$ |
| $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $0.456$ | $0.498$ | $0.523$ | $0.54$ |
| $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ | $6.99\times 10^{-2}$ | $4.99\times 10^{-2}$ | $3.56\times 10^{-2}$ | $2.54\times 10^{-2}$ |
| $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $0.511$ | $0.536$ | $0.55$ | $0.559$ |
| $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ | $7.2\times 10^{-2}$ | $5.06\times 10^{-2}$ | $3.59\times 10^{-2}$ | $2.55\times 10^{-2}$ |
| $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ | $0.474$ | $0.363$ | $0.29$ | $0.238$ |
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