American Institute of Mathematical Sciences

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doi: 10.3934/mcrf.2019030

Approximation of controls for linear wave equations: A first order mixed formulation

Santiago Montaner and  Arnaud Münch ,

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.

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
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. doi: 10.1515/jnum-2012-0013. Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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Figure 2.  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)} $
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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) $
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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
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Figure 5.  The primal variable $ \lambda_h $ in $ Q_T $ - Third adapted mesh in Figure 4, $ r = 10^{-6} $.
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Figure 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} $
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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 $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
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$N$ $10$ $20$ $40$ $80$ $160$
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
$ 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 $
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$ 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 $
Table 3.  $\delta_h$ w.r.t. $r$ and $h$, $T = 2$, for the $W_h$-$M_h$ finite elements and non uniform mesh.
$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
$r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$
$r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$
$r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$
$r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$
$r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
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$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$ $9.50\times 10^{-3}$
$r=1$ $0.426$ $0.316$ $0.229$ $0.155$ $0.106$
$r=10^{-1}$ $0.991$ $0.868$ $0.698$ $0.489$ $0.339$
$r=10^{-2}$ $2.269$ $1.738$ $1.373$ $1.099$ $0.896$
$r=h$ $0.885$ $0.927$ $0.929$ $0.921$ $0.908$
$r=h^2$ $1.612$ $2.154$ $2.974$ $4.115$ $5.733$
Table 4.  $ \delta_h $ w.r.t. $ r $ and $ h $, $ T = 2 $, for the $ \widetilde{W_h} $-$ M_h $ finite elements and uniform mesh
$ 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} $
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$ 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} $
Table 5.  $ 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 $
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$ 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 $
Table 6.  $r = h$ - uniform mesh
$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
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$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.535$ $0.549$ $0.559$ $0.566$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $3.43\times 10^{-2}$ $2.22\times 10^{-2}$ $1.45\times 10^{-2}$ $9.43\times 10^{-3}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.545$ $0.558$ $0.566$ $0.57$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $3.89\times 10^{-2}$ $2.3\times 10^{-2}$ $1.46\times 10^{-2}$ $9.35\times 10^{-3}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.561$ $0.388$ $0.265$ $0.184$
Table 7.  $r = 10^{-6}$ - 4 adaptive meshes. Figure 4 displays the $1$st, $3$rd and $4th$ adaptive meshes used.
$\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
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$\sharp\ \text{triangles}$ $110$ $1197$ $2880$ $8636$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.46$ $0.57$ $0.574$ $0.577$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $8.24\times10^{-2}$ $1.55\times10^{-2}$ $3.72\times10^{-3}$ $5.18\times10^{-4}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.451$ $0.569$ $0.574$ $0.577$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $8.04\times10^{-2}$ $1.52\times10^{-2}$ $3.88\times10^{-3}$ $4.48\times10^{-4}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $1.13\times10^5$ $4.45\times10^4$ $1.48\times10^4$ $2.86\times10^3$
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} $
$ \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} $
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$ 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} $
Table 9.  $ 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 $
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$ 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 $
Table 10.  $ r = h $ - non uniform mesh - stabilized formulation with $ \alpha = h^2 $
$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
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$h$ $1.52\times 10^{-1}$ $7.60\times 10^{-2}$ $3.80\times 10^{-2}$ $1.90\times 10^{-2}$
$\|\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $0.456$ $0.498$ $0.523$ $0.54$
$\|v-\mathbf{q}_h(1, \cdot)\|_{L^2(0, T)}$ $6.99\times 10^{-2}$ $4.99\times 10^{-2}$ $3.56\times 10^{-2}$ $2.54\times 10^{-2}$
$\|\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $0.511$ $0.536$ $0.55$ $0.559$
$\|v-\lambda_h(1, \cdot)\|_{L^2(0, T)}$ $7.2\times 10^{-2}$ $5.06\times 10^{-2}$ $3.59\times 10^{-2}$ $2.55\times 10^{-2}$
$\|M(w_h, \mathbf{q}_h)\|_{L^2(Q_T)}$ $0.474$ $0.363$ $0.29$ $0.238$
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