# American Institute of Mathematical Sciences

December  2019, 9(4): 729-758. 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

* Corresponding author: Arnaud Münch

Received  May 2018 Published  April 2019

Fund Project: This work has been sponsored by the French government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

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.

Citation: Santiago Montaner, Arnaud Münch. Approximation of controls for linear wave equations: A first order mixed formulation. Mathematical Control and Related Fields, 2019, 9 (4) : 729-758. doi: 10.3934/mcrf.2019030
##### 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.

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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.
Regular meshes for $Q_T$; Left: uniform mesh - $h = 1.41\times10^{-1}$. Right: non uniform mesh - $h = 1.52\times 10^{-1}$
Non uniform mesh - Evolution of $\sqrt{hr}\delta_h$ with respect to $h$ (see Table 3) for $r = 1$ $({\bigcirc})$, $r = 10^{-1}$ $(\bigtriangledown)$, $r = 10^{-2}$ $({\bigtriangleup})$, $r = h$ $({\star})$, $r = h^2$ $({\circ)}$
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 $r = h$ $(\bigtriangledown)$
Iterative refinement of the triangular mesh over $Q_T$ with respect to the variable $\lambda_h$: $110$, $2\, 880$ and $8\, 636$ triangles
The primal variable $\lambda_h$ in $Q_T$ - Third adapted mesh in Figure 4, $r = 10^{-6}$.
Control of minimal $L^2$-norm $v$ (dashed blue line) and its approximation $\lambda_h(1, \cdot)$ (red line) on $(0, T)$. Third adapted mesh in Figure 4, $r = 10^{-6}$
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 $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $0.264$ $0.197$ $0.132$ $0.099$ $0.070$ $r=10^{-1}$ $0.751$ $0.569$ $0.412$ $0.310$ $0.222$ $r=10^{-2}$ $1.881$ $1.478$ $1.112$ $0.839$ $0.627$ $r=h$ $0.652$ $0.660$ $0.660$ $0.679$ $0.661$ $r=h^2$ $1.397$ $1.934$ $2.642$ $3.636$ $5.031$
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $0.264$ $0.197$ $0.132$ $0.099$ $0.070$ $r=10^{-1}$ $0.751$ $0.569$ $0.412$ $0.310$ $0.222$ $r=10^{-2}$ $1.881$ $1.478$ $1.112$ $0.839$ $0.627$ $r=h$ $0.652$ $0.660$ $0.660$ $0.679$ $0.661$ $r=h^2$ $1.397$ $1.934$ $2.642$ $3.636$ $5.031$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh.
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$ $r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$ $r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$ $r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$ $r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$ $r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$ $r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$ $r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$ $r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$ $r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$ $r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
$\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $\widetilde{W_h}$-$M_h$ finite elements and uniform mesh
 $h$ $1.41\times 10^{-1}$ $7.01\times 10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $5.77\times 10^{-5}$ $1.41\times 10^{-10}$ $1.8\times 10^{-10}$ $4.07\times 10^{-9}$ $1.97\times 10^{-10}$ $r=10^{-1}$ $2.45\times 10^{-9}$ $5.17\times 10^{-10}$ $2.23\times 10^{-10}$ $2.05\times 10^{-9}$ $1.63\times 10^{-9}$ $r=10^{-2}$ $1.51\times 10^{-8}$ $1.71\times 10^{-9}$ $3.88\times 10^{-9}$ $1.77\times 10^{-8}$ $8.11\times 10^{-9}$ $r=h$ $2.4\times 10^{-9}$ $1.05\times 10^{-9}$ $7.77\times 10^{-10}$ $6.73\times 10^{-9}$ $1.64\times 10^{-9}$ $r=h^2$ $4.92\times 10^{-9}$ $4.19\times 10^{-9}$ $2.6\times 10^{-9}$ $3.33\times 10^{-9}$ $1.44\times 10^{-9}$
 $h$ $1.41\times 10^{-1}$ $7.01\times 10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $8.83\times 10^{-3}$ $r=1$ $5.77\times 10^{-5}$ $1.41\times 10^{-10}$ $1.8\times 10^{-10}$ $4.07\times 10^{-9}$ $1.97\times 10^{-10}$ $r=10^{-1}$ $2.45\times 10^{-9}$ $5.17\times 10^{-10}$ $2.23\times 10^{-10}$ $2.05\times 10^{-9}$ $1.63\times 10^{-9}$ $r=10^{-2}$ $1.51\times 10^{-8}$ $1.71\times 10^{-9}$ $3.88\times 10^{-9}$ $1.77\times 10^{-8}$ $8.11\times 10^{-9}$ $r=h$ $2.4\times 10^{-9}$ $1.05\times 10^{-9}$ $7.77\times 10^{-10}$ $6.73\times 10^{-9}$ $1.64\times 10^{-9}$ $r=h^2$ $4.92\times 10^{-9}$ $4.19\times 10^{-9}$ $2.6\times 10^{-9}$ $3.33\times 10^{-9}$ $1.44\times 10^{-9}$
$r = h$ - non uniform mesh
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.543$ $0.556$ $0.564$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.85\times 10^{-2}$ $2.49\times 10^{-2}$ $1.63\times 10^{-2}$ $1.06\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.538$ $0.555$ $0.564$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $5.27\times 10^{-2}$ $3.37\times 10^{-2}$ $2.18\times 10^{-2}$ $1.41\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.645$ $0.462$ $0.331$ $0.239$
 $h$ $1.41\times10^{-1}$ $7.01\times10^{-2}$ $3.53\times 10^{-2}$ $1.76\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.543$ $0.556$ $0.564$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.85\times 10^{-2}$ $2.49\times 10^{-2}$ $1.63\times 10^{-2}$ $1.06\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.538$ $0.555$ $0.564$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $5.27\times 10^{-2}$ $3.37\times 10^{-2}$ $2.18\times 10^{-2}$ $1.41\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.645$ $0.462$ $0.331$ $0.239$
$r = h$ - uniform mesh
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
$r = 10^{-6}$ - 4 adaptive meshes. Figure 4 displays the $1$st, $3$rd and $4th$ adaptive meshes used.
 $\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
 $\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
Non uniform mesh - Conjugate gradient method - Number of iterates for $r = 1$ (top), $r = 10^{-2}$ and $r = h$ (bottom)
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $31$ $41$ $54$ $77$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.469$ $0.576$ $0.589$ $0.586$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.21\times 10^{-1}$ $1.72\times 10^{-1}$ $1.43\times 10^{-1}$ $1.25\times 10^{-1}$ $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $46$ $103$ $125$ $133$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.55$ $0.566$ $0.569$ $0.571$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.05\times 10^{-1}$ $1.47\times 10^{-1}$ $1.12\times 10^{-1}$ $8.71\times 10^{-2}$ $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $36$ $43$ $56$ $80$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.566$ $0.574$ $0.573$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.39\times 10^{-1}$ $1.46\times 10^{-1}$ $1.19\times 10^{-1}$ $9.54\times 10^{-2}$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $31$ $41$ $54$ $77$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.469$ $0.576$ $0.589$ $0.586$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.21\times 10^{-1}$ $1.72\times 10^{-1}$ $1.43\times 10^{-1}$ $1.25\times 10^{-1}$ $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $46$ $103$ $125$ $133$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.55$ $0.566$ $0.569$ $0.571$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.05\times 10^{-1}$ $1.47\times 10^{-1}$ $1.12\times 10^{-1}$ $8.71\times 10^{-2}$ $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\sharp$ iterates $36$ $43$ $56$ $80$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.523$ $0.566$ $0.574$ $0.573$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $2.39\times 10^{-1}$ $1.46\times 10^{-1}$ $1.19\times 10^{-1}$ $9.54\times 10^{-2}$
$r = h$ - non uniform mesh - stabilized formulation with $\alpha = 0.5$.
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.444$ $0.494$ $0.522$ $0.539$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $7.47\times 10^{-2}$ $5.21\times 10^{-2}$ $3.65\times 10^{-2}$ $2.56\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.525$ $0.543$ $0.554$ $0.561$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $1.26\times 10^{-1}$ $6.5\times 10^{-2}$ $4.2\times 10^{-2}$ $2.79\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.423$ $0.343$ $0.281$ $0.235$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.444$ $0.494$ $0.522$ $0.539$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $7.47\times 10^{-2}$ $5.21\times 10^{-2}$ $3.65\times 10^{-2}$ $2.56\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.525$ $0.543$ $0.554$ $0.561$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $1.26\times 10^{-1}$ $6.5\times 10^{-2}$ $4.2\times 10^{-2}$ $2.79\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.423$ $0.343$ $0.281$ $0.235$
$r = h$ - non uniform mesh - stabilized formulation with $\alpha = h^2$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
 $h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$ $\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$ $\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$ $\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$ $\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
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