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September  2020, 10(3): 275-300. doi: 10.3934/naco.2020002

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

Radouen Ghanem and  Billel Zireg

Numerical Analysis, Optimization and Statistical Laboratory (LANOS), Badji-Mokhtar, Annaba University, P.O. Box 12, 23000, Annaba, Algeria

Received  December 2018 Revised  August 2019 Published  February 2020

Fund Project: The authors would like to thank the anonymous referee for careful reading and the suggestions of some improvements in presentation that have been implemented in the final version of the manuscript

This work is deals with the numerical solution of a bilateral obstacle optimal control problem which is similar to the one given in Bergounioux et al [9] with some modifications. It can be regarded as an extension of our previous work [18], where the main feature of the present work is that the controls and the two obstacles are the same. For the numerical resolution we follow the idea of our previous work [18]. We begin by discretizing the optimality system of the underlying problem by using finite differences schemes, then we propose an iterative algorithm. Finally, numerical examples are provides to show the efficiency of the proposed algorithm and the used scheme.

Keywords: Optimal control, Bilateral obstacle problem, Finite differences method.
Mathematics Subject Classification: Primary: 65K15 49K10, 35J87; Secondary: 49J20, 65L12, 49M15.
Citation: Radouen Ghanem, Billel Zireg. Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 275-300. doi: 10.3934/naco.2020002
References:
[1]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods Appl. of Anal., 11 (2004), 533-556. 

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003. 
[3]

D. R. Adams and S. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim., 47 (2002), 79-95.  doi: 10.1007/s00245-002-0739-1.

[4]

G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control Inform., 35 (2016), 107-122.  doi: 10.1093/imamci/dnw040.

[5]

V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984.

[6]

M. Bergounioux, X. Bonnefond, T. Haberkorn and Y. Privat, An optimal control problem in photoacoustic tomography, Math. Models Methods Appl. Sci., 24 (2014), 2525-2548.  doi: 10.1142/S0218202514500286.

[7]

M. Bergounioux and Y. Privat, Shape optimization with Stokes constraints over the set of axisymmetric domains, SIAM J. Control Optim., 51 (2013), 599-628.  doi: 10.1137/100818133.

[8]

M. Bergounioux and S. Lenhart, Optimal control of the obstacle in semilinear variational inequalities, Positivity, 8 (2004), 229–242. doi: 10.1007/s11117-004-5009-9.

[9]

M. Bergounioux and S. Lenhart, Optimal control of the bilateral obstacle problems, SIAM J. Control Optim., 43 (2004), 249-255.  doi: 10.1137/S0363012902416912.

[10]

T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015.

[11]

Bock, Igor and Kečkemétyová, Mária, Regularized optimal control problem for a beam vibrating against an elastic foundation, Tatra Mt. Math. Publ., 63 (2015), 53-71.  doi: 10.1515/tmmp-2015-0020.

[12]

H. Brzis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844.  doi: 10.1512/iumj.1974.23.23069.

[13]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[14]

M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984). doi: 10.1007/978-1-4612-1120-4.

[15]

J. C. De Los Reyes, On the optimal control of some nonsmooth distributed parameter systems arising in mechanics, GAMM-Mitt., 40 (2018), 268-286.  doi: 10.1002/gamm.201740002.

[16]

S. Desong, Z. Zhongding and Y. Fuxin, A variational inequality principle in solid mechanics and application in physically non-linear problems, Communications in Applied Numerical Methods, 6 (1990), 35-45.  doi: 10.1002/cnm.1630060106.

[17]

R. Ghanem, Optimal control of unilateral obstacle problem with a source term, Positivity, 13 (2009), 321-338.  doi: 10.1007/s11117-008-2241-8.

[18]

R. Ghanem and B. Zireg, On the numerical study of an obstacle optimal control problem with source term, J. Appl. Math. Comput., 45 (2014), 375-409.  doi: 10.1007/s12190-013-0728-3.

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[20]

I. Hlaváček, I. Bock and J. Lovíšek, Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports, Appl. Math. Optim., 11 (1984), 111-143.  doi: 10.1007/BF01442173.

[21]

C. U. Huy, P. J. Mckenna and W. Walter, Finite difference approximations to the Dirichlet problem for elliptic systems, Numer. Math., 49 (1986), 227-237.  doi: 10.1007/BF01389626.

[22]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.  doi: 10.1007/s002459911017.

[23]

K. Ito and K. Kunisch, Optimal control of obstacle problems by $H^{1}-$obstacles, Appl. Math. Optim., 56 (2007), 1-17.  doi: 10.1007/s00245-007-0877-6.

[24]

K. Kunisch and D. Wachsmuth, Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var., 18 (2012), 520-547.  doi: 10.1051/cocv/2011105.

[25]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.

[26]

J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968).

[27]

F. Mignot and J. P. Puel, Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.  doi: 10.1137/0322028.

[28]

F. Mignot, Contrôle dans les inéquatons variationelles elliptiques, (French), J. Funct. Anal., 22 (1976), 466-476.  doi: 10.1016/0022-1236(76)90017-3.

[29]

S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172. doi: 10.1155/2013/692879.

[30]

Z. Peng and K. Kunisch, Optimal control of elliptic variational–hemivariational inequalities, J. Optim. Theory Appl., 178 (2018), 1-25.  doi: 10.1007/s10957-018-1303-8.

[31]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987.

[32]

V. Shcherbakov, Shape optimization of rigid inclusions for elastic plates with cracks, Z. Angew. Math. Phys., 67 (2016), 71-76.  doi: 10.1007/s00033-016-0666-7.

[33]

M. Sofonea, A. Benraouda and H. Hechaichi, Optimal control of a two-dimensional contact problem, Appl. Anal., 97 (2018), 1281-1298.  doi: 10.1080/00036811.2017.1337895.

show all references

References:
[1]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods Appl. of Anal., 11 (2004), 533-556. 

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003. 
[3]

D. R. Adams and S. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim., 47 (2002), 79-95.  doi: 10.1007/s00245-002-0739-1.

[4]

G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control Inform., 35 (2016), 107-122.  doi: 10.1093/imamci/dnw040.

[5]

V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984.

[6]

M. Bergounioux, X. Bonnefond, T. Haberkorn and Y. Privat, An optimal control problem in photoacoustic tomography, Math. Models Methods Appl. Sci., 24 (2014), 2525-2548.  doi: 10.1142/S0218202514500286.

[7]

M. Bergounioux and Y. Privat, Shape optimization with Stokes constraints over the set of axisymmetric domains, SIAM J. Control Optim., 51 (2013), 599-628.  doi: 10.1137/100818133.

[8]

M. Bergounioux and S. Lenhart, Optimal control of the obstacle in semilinear variational inequalities, Positivity, 8 (2004), 229–242. doi: 10.1007/s11117-004-5009-9.

[9]

M. Bergounioux and S. Lenhart, Optimal control of the bilateral obstacle problems, SIAM J. Control Optim., 43 (2004), 249-255.  doi: 10.1137/S0363012902416912.

[10]

T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015.

[11]

Bock, Igor and Kečkemétyová, Mária, Regularized optimal control problem for a beam vibrating against an elastic foundation, Tatra Mt. Math. Publ., 63 (2015), 53-71.  doi: 10.1515/tmmp-2015-0020.

[12]

H. Brzis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844.  doi: 10.1512/iumj.1974.23.23069.

[13]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[14]

M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984). doi: 10.1007/978-1-4612-1120-4.

[15]

J. C. De Los Reyes, On the optimal control of some nonsmooth distributed parameter systems arising in mechanics, GAMM-Mitt., 40 (2018), 268-286.  doi: 10.1002/gamm.201740002.

[16]

S. Desong, Z. Zhongding and Y. Fuxin, A variational inequality principle in solid mechanics and application in physically non-linear problems, Communications in Applied Numerical Methods, 6 (1990), 35-45.  doi: 10.1002/cnm.1630060106.

[17]

R. Ghanem, Optimal control of unilateral obstacle problem with a source term, Positivity, 13 (2009), 321-338.  doi: 10.1007/s11117-008-2241-8.

[18]

R. Ghanem and B. Zireg, On the numerical study of an obstacle optimal control problem with source term, J. Appl. Math. Comput., 45 (2014), 375-409.  doi: 10.1007/s12190-013-0728-3.

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[20]

I. Hlaváček, I. Bock and J. Lovíšek, Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports, Appl. Math. Optim., 11 (1984), 111-143.  doi: 10.1007/BF01442173.

[21]

C. U. Huy, P. J. Mckenna and W. Walter, Finite difference approximations to the Dirichlet problem for elliptic systems, Numer. Math., 49 (1986), 227-237.  doi: 10.1007/BF01389626.

[22]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.  doi: 10.1007/s002459911017.

[23]

K. Ito and K. Kunisch, Optimal control of obstacle problems by $H^{1}-$obstacles, Appl. Math. Optim., 56 (2007), 1-17.  doi: 10.1007/s00245-007-0877-6.

[24]

K. Kunisch and D. Wachsmuth, Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var., 18 (2012), 520-547.  doi: 10.1051/cocv/2011105.

[25]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.

[26]

J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968).

[27]

F. Mignot and J. P. Puel, Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.  doi: 10.1137/0322028.

[28]

F. Mignot, Contrôle dans les inéquatons variationelles elliptiques, (French), J. Funct. Anal., 22 (1976), 466-476.  doi: 10.1016/0022-1236(76)90017-3.

[29]

S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172. doi: 10.1155/2013/692879.

[30]

Z. Peng and K. Kunisch, Optimal control of elliptic variational–hemivariational inequalities, J. Optim. Theory Appl., 178 (2018), 1-25.  doi: 10.1007/s10957-018-1303-8.

[31]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987.

[32]

V. Shcherbakov, Shape optimization of rigid inclusions for elastic plates with cracks, Z. Angew. Math. Phys., 67 (2016), 71-76.  doi: 10.1007/s00033-016-0666-7.

[33]

M. Sofonea, A. Benraouda and H. Hechaichi, Optimal control of a two-dimensional contact problem, Appl. Anal., 97 (2018), 1281-1298.  doi: 10.1080/00036811.2017.1337895.

Figure 1.  Left (error $ \epsilon _{n} $, continuous line; $ \omega = 0.25 $, dash line; $ \omega = 0.5 $, dash-dot line; $ \omega = 0.75 $), right (state function $ y $)
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Figure 2.  Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)
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Figure 3.  Left (error $ \epsilon _{n} $, continuous line; $ N = 30 $, dash line; $ N = 35 $, dash-dot line; $ N = 40 $), right (state function $ y $)
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Figure 4.  Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)
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Figure 5.  Left(error $ \epsilon _{n} $, continuous line $ \nu = 0.1 $; dash line $ \nu = 0.5 $; dash-dot line $ \nu = 1 $), right (state function $ y $)
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Figure 6.  Left(obstacle function $ \varphi $), right (obstacle function $ \psi $)
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Figure 7.  Left(error $ \epsilon_{n} $, continuous line; $ \delta = h^{2} $, dash line; $ \delta = h^{3} $, dash-dot line; $ \delta = h^{4} $), right (state function $ y $)
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Figure 8.  Left (obstacle function $ \psi $), right (right(obstacle function $ \varphi $))
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Table202 
Algorithm 2 Implemented continuous algorithm
1: Begin
2: Input :$ \left\{ y_{0}^{\delta }, p_{0}^{\delta }, \varphi _{0}^{\delta }, \lambda _{0}^{\delta },\psi _{0}^{\delta },\delta ,\nu ,\omega_{y},\omega_{\varphi},\omega_{\psi},\varepsilon\right\} $ choose $ \varphi _{0}^{\delta } $ and $ \psi _{0}^{\delta }\in \mathcal{W},\varepsilon $ and $ \delta $ in $ \mathbb{R}_{+}^{\ast }; $
3: Calculate $ J_{n-1} \leftarrow J_{n-1}\left(y^{\delta}_{n-1}, \varphi^{\delta}_{n-1}, \psi^{\delta}_{n-1}\right) $
4:   Solve $ \left( A+\beta _{\delta }^{\prime }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left(\psi _{n-1}^{\delta }- y_{n-1}^{\delta }\right) \right)r_{n}^{\delta }= $ $ -\omega_{y}\left( A y_{n-1}^{\delta }+\beta _{\delta }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta } \right)-\beta _{\delta }\left( \psi _{n-1}^{\delta } -y_{n-1}^{\delta }\right) -f \right) $ on $ r_{n}^{\delta } $.
5: Calculate $ y_{n}^{\delta }=y_{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
6: Solve $ \left( A +\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta } -y_{n}^{\delta }\right) \right) p_{n}^{\delta }=y_{n}^{\delta }-z $ on $ p_{n}^{\delta } $.
7: Calculate $ \lambda_{n}^{\delta } = \nu \Delta \varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta } $.
8: Solve$ \left( \nu \Delta+\beta _{\delta }^{\prime \prime }\left(\psi _{n-1}^{\delta } -y_{n}^{\delta }\right) p_{n}^{\delta }\right)r_{n}^{\delta }=-\omega_{\psi}\left(\nu A_{h}^{d}\psi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta }-y_{n}^{\delta }\right) p_{n}^{\delta }+\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
9: Calculate $ \psi _{n}^{\delta }=\psi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
10: Solve$ \left( \nu \Delta-\beta _{\delta }^{\prime \prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta } \right) p_{n}^{\delta }\right) r_{n}^{\delta }=-\omega_{\varphi}\left(\nu A_{h}^{d}\varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta }-\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
11: Calculate $ \varphi _{n}^{\delta }=\varphi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
12: Calculate $ J_{n} \leftarrow J_{n-1}\left(y^{\delta}_{n}, \varphi^{\delta}_{n},\psi^{\delta}_{n}\right) $.
13: If $ |J_{n}-J_{n-1}| \leq \varepsilon $ Stop.
14: Ensure : $ s_{n}^{\delta }=\left( y_{n}^{\delta },\varphi_{n}^{\delta },\psi_{n}^{\delta },p_{n}^{\delta }\right) $ is a solution.
15:      Else; $ n\leftarrow n+1 $, go to Begin.
16: End if
17: End
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Algorithm 2 Implemented continuous algorithm
1: Begin
2: Input :$ \left\{ y_{0}^{\delta }, p_{0}^{\delta }, \varphi _{0}^{\delta }, \lambda _{0}^{\delta },\psi _{0}^{\delta },\delta ,\nu ,\omega_{y},\omega_{\varphi},\omega_{\psi},\varepsilon\right\} $ choose $ \varphi _{0}^{\delta } $ and $ \psi _{0}^{\delta }\in \mathcal{W},\varepsilon $ and $ \delta $ in $ \mathbb{R}_{+}^{\ast }; $
3: Calculate $ J_{n-1} \leftarrow J_{n-1}\left(y^{\delta}_{n-1}, \varphi^{\delta}_{n-1}, \psi^{\delta}_{n-1}\right) $
4:   Solve $ \left( A+\beta _{\delta }^{\prime }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left(\psi _{n-1}^{\delta }- y_{n-1}^{\delta }\right) \right)r_{n}^{\delta }= $ $ -\omega_{y}\left( A y_{n-1}^{\delta }+\beta _{\delta }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta } \right)-\beta _{\delta }\left( \psi _{n-1}^{\delta } -y_{n-1}^{\delta }\right) -f \right) $ on $ r_{n}^{\delta } $.
5: Calculate $ y_{n}^{\delta }=y_{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
6: Solve $ \left( A +\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta } -y_{n}^{\delta }\right) \right) p_{n}^{\delta }=y_{n}^{\delta }-z $ on $ p_{n}^{\delta } $.
7: Calculate $ \lambda_{n}^{\delta } = \nu \Delta \varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta } $.
8: Solve$ \left( \nu \Delta+\beta _{\delta }^{\prime \prime }\left(\psi _{n-1}^{\delta } -y_{n}^{\delta }\right) p_{n}^{\delta }\right)r_{n}^{\delta }=-\omega_{\psi}\left(\nu A_{h}^{d}\psi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta }-y_{n}^{\delta }\right) p_{n}^{\delta }+\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
9: Calculate $ \psi _{n}^{\delta }=\psi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
10: Solve$ \left( \nu \Delta-\beta _{\delta }^{\prime \prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta } \right) p_{n}^{\delta }\right) r_{n}^{\delta }=-\omega_{\varphi}\left(\nu A_{h}^{d}\varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta }-\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
11: Calculate $ \varphi _{n}^{\delta }=\varphi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
12: Calculate $ J_{n} \leftarrow J_{n-1}\left(y^{\delta}_{n}, \varphi^{\delta}_{n},\psi^{\delta}_{n}\right) $.
13: If $ |J_{n}-J_{n-1}| \leq \varepsilon $ Stop.
14: Ensure : $ s_{n}^{\delta }=\left( y_{n}^{\delta },\varphi_{n}^{\delta },\psi_{n}^{\delta },p_{n}^{\delta }\right) $ is a solution.
15:      Else; $ n\leftarrow n+1 $, go to Begin.
16: End if
17: End
Table 1.  Numerical results for two dimensional space while varying $ \omega $
$ \omega $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
0.25 62 29.460551e-4 9.601694e-16 3.982865e-11
0.5 28 29.460551e-4 6.392456e-16 1.180507e-11
0.75 15 29.460551e-4 5.212844e-16 5.253112e-12
1 5 29.460551e-4 6.878178e-16 9.642406e-10
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$ \omega $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
0.25 62 29.460551e-4 9.601694e-16 3.982865e-11
0.5 28 29.460551e-4 6.392456e-16 1.180507e-11
0.75 15 29.460551e-4 5.212844e-16 5.253112e-12
1 5 29.460551e-4 6.878178e-16 9.642406e-10
Table 2.  Numerical results for two dimensional space while varying $ N $
$ N $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
30 26 1.632e-3 5.193328e-16 7.355250e-12
35 27 2.241e-3 5.165139e-16 6.513361e-12
40 28 2.946e-3 6.392456e-16 1.180507e-11
45 30 3.746e-3 7.350890e-16 2.389886e-11
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$ N $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
30 26 1.632e-3 5.193328e-16 7.355250e-12
35 27 2.241e-3 5.165139e-16 6.513361e-12
40 28 2.946e-3 6.392456e-16 1.180507e-11
45 30 3.746e-3 7.350890e-16 2.389886e-11
Table 3.  Numerical results for two dimensional space while varying $ \nu $
$ \nu $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
0.001 300 29.460541e-4 1.227802e-13 3.695397e-05
0.01 29 29.460541e-4 5.676882e-16 5.966039e-11
0.1 29 29.460544e-4 5.676882e-16 5.966039e-11
0.5 28 29.460549e-4 7.650130e-16 2.529592e-11
1 28 29.460551e-4 6.392456e-16 1.180507e-11
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$ \nu $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
0.001 300 29.460541e-4 1.227802e-13 3.695397e-05
0.01 29 29.460541e-4 5.676882e-16 5.966039e-11
0.1 29 29.460544e-4 5.676882e-16 5.966039e-11
0.5 28 29.460549e-4 7.650130e-16 2.529592e-11
1 28 29.460551e-4 6.392456e-16 1.180507e-11
Table 4.  Numerical results for two dimensional space while varying $ \delta $
$ \delta $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
$ h^{3} $ 27 29.460541e-4 8.135853e-16 8.627484e-12
$ h^{3.5} $ 27 29.460543e-4 8.270294e-16 8.809409e-12
$ h^{4} $ 28 29.460551e-4 6.392456e-16 1.180507e-11
$ h^{4.5} $ 33 29.460566e-4 9.358833e-16 1.016003e-10
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$ \delta $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
$ h^{3} $ 27 29.460541e-4 8.135853e-16 8.627484e-12
$ h^{3.5} $ 27 29.460543e-4 8.270294e-16 8.809409e-12
$ h^{4} $ 28 29.460551e-4 6.392456e-16 1.180507e-11
$ h^{4.5} $ 33 29.460566e-4 9.358833e-16 1.016003e-10
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