March  2018, 8(1): 247-276. doi: 10.3934/mcrf.2018011

Optimal control of a non-smooth semilinear elliptic equation

1. 

TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany

2. 

University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Str. 9, 45127 Essen, Germany

* Corresponding author: C. Meyer

Received  April 2017 Revised  October 2017 Published  January 2018

Fund Project: C. Clason was supported by the DFG under grant CL 487/2-1, and C. Christof and C. Meyer were supported by the DFG under grant ME 3281/7-1, both within the priority programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand sub-differential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

Citation: Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research notes in mathematics, Pitman, Boston-London-Melbourne, 1984.  Google Scholar

[2]

M. Bergounioux, Optimal control problems governed by abstract elliptic variational inequalities with state constraints, SIAM Journal on Control and Optimization, 36 (1998), 273-289.   Google Scholar

[3]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994.  Google Scholar

[4]

M. Červinka, Hierarchical Structures in Equilibrium Problems, PhD thesis, Charles University Prague, Faculty of Mathematics and Physics, 2008, URL https://is.cuni.cz/webapps/zzp/detail/69076/ . Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, 2nd edition, SIAM, 1990.  Google Scholar

[6]

C. Clason and T. Valkonen, Stability of saddle points via explicit coderivatives of pointwise subdifferentials, Set-Valued and Variational Analysis, 25 (2017), 69-112.   Google Scholar

[7]

B. D. Craven and B. M. Glover, An approach to vector subdifferentials, Optimization, 38 (1996), 237-251.   Google Scholar

[8]

J. C. De los Reyes and C. Meyer, Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind, J. Optim. Theory. Appl., 168 (2016), 375-409.   Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

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F. Harder and G. Wachsmuth, Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem, Technical Report SPP1962-029, Priority Program 1962, German Research Foundation, 2017, URL https://spp1962.wias-berlin.de/preprints/029.pdf . Google Scholar

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R. HenrionB. S. Mordukhovich and N. M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM Journal on Optimization, 20 (2010), 2199-2227.   Google Scholar

[12]

R. HerzogC. Meyer and G. Wachsmuth, B-and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.   Google Scholar

[13]

M. Hintermüller, An active-set equality constrained Newton solver with feasibility restoration for inverse coefficient problems in elliptic variational inequalities, Inverse Problems, 24 (2008), 034017, 23pp. Google Scholar

[14]

M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C-and strong stationarity and a path-following algorithm, SIAM Journal on Optimization, 20 (2009), 868-902.   Google Scholar

[15]

M. HintermüllerB. S. Mordukhovich and T. Surowiec, Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints, Mathematical Programming, Ser. A, 146 (2014), 555-582.   Google Scholar

[16]

M. Hintermüller and T. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming, Ser. A, 160 (2016), 271-305.   Google Scholar

[17]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Applied Mathematics and Optimization, 41 (2000), 343-364.   Google Scholar

[18]

F. KikuchiK. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Japan Journal of Applied Mathematics, 1 (1984), 369-403.   Google Scholar

[19]

D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, vol. 60 of Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 2002.  Google Scholar

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P. Mehlitz and G. Wachsmuth, The limiting normal cone to pointwise defined sets in Lebesgue spaces, Set-Valued and Variational Analysis, online first (2016), 1-19.   Google Scholar

[21]

P. Mehlitz and G. Wachsmuth, Weak and strong stationarity in generalized bilevel programming and bilevel optimal control, Optimization, 65 (2016), 907-935.   Google Scholar

[22]

C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM Journal on Control and Optimization, 55 (2017), 2206-2234.   Google Scholar

[23]

F. Mignot, Contrôle dans les inéquations variationelles elliptiques, Journal of Functional Analysis, 22 (1976), 130-185.   Google Scholar

[24]

F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM Journal on Control and Optimization, 22 (1984), 466-476.   Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I, vol. 330 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2006, Basic theory.  Google Scholar

[26]

J. OutrataJ. Jarušek and J. Stará, On optimality conditions in control of elliptic variational inequalities, Set-Valued and Variational Analysis, 19 (2011), 23-42.   Google Scholar

[27]

J. V. Outrata and W. Römisch, On optimality conditions for some nonsmooth optimization problems over Lp spaces, Journal of Optimization Theory and Applications, 126 (2005), 411-438.   Google Scholar

[28]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, vol. 28 of Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 1998.  Google Scholar

[29]

J. -P. Penot, Calculus Without Derivatives, vol. 266 of Graduate Texts in Mathematics, Springer, New York, 2013.  Google Scholar

[30]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, Zentrum Mathematik, Technische Universität München, 2015, URL [http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:91-diss-20150420-1241413-1-4 . Google Scholar

[31]

J. Rappaz, Approximation of a nondifferentiable nonlinear problem related to MHD equilibria, Numer. Math., 45 (1984), 117-133.   Google Scholar

[32]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, vol. 317 of Comprehensive Studies in Mathematics, Springer, Berlin, 2004. Google Scholar

[33]

A. Schiela and D. Wachsmuth, Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints, ESAIM: M2AN, 47 (2013), 771-787.   Google Scholar

[34]

W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.  Google Scholar

[35]

R. Temam, A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma, Arch. Rational Mech. Anal., 60 (1975/76), 51-73.   Google Scholar

[36]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer, 1990.  Google Scholar

[37]

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, Theory, methods and applications. Translated from the 2005 German original by Jürgen Sprekels.  Google Scholar

[38]

G. Wachsmuth, Strong stationarity for optimal control of the obstacle problem with control constraints, SIAM Journal on Optimization, 24 (2014), 1914-1932.   Google Scholar

[39]

G. Wachsmuth, Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity, Set-Valued and Variational Analysis, 25 (2017), 133-175.   Google Scholar

[40]

G. Wachsmuth, Towards M-stationarity for optimal control of the obstacle problem with control constraints, SIAM Journal on Control and Optimization, 54 (2016), 964-986.   Google Scholar

show all references

References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, vol. 100 of Research notes in mathematics, Pitman, Boston-London-Melbourne, 1984.  Google Scholar

[2]

M. Bergounioux, Optimal control problems governed by abstract elliptic variational inequalities with state constraints, SIAM Journal on Control and Optimization, 36 (1998), 273-289.   Google Scholar

[3]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994.  Google Scholar

[4]

M. Červinka, Hierarchical Structures in Equilibrium Problems, PhD thesis, Charles University Prague, Faculty of Mathematics and Physics, 2008, URL https://is.cuni.cz/webapps/zzp/detail/69076/ . Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, 2nd edition, SIAM, 1990.  Google Scholar

[6]

C. Clason and T. Valkonen, Stability of saddle points via explicit coderivatives of pointwise subdifferentials, Set-Valued and Variational Analysis, 25 (2017), 69-112.   Google Scholar

[7]

B. D. Craven and B. M. Glover, An approach to vector subdifferentials, Optimization, 38 (1996), 237-251.   Google Scholar

[8]

J. C. De los Reyes and C. Meyer, Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind, J. Optim. Theory. Appl., 168 (2016), 375-409.   Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[10]

F. Harder and G. Wachsmuth, Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem, Technical Report SPP1962-029, Priority Program 1962, German Research Foundation, 2017, URL https://spp1962.wias-berlin.de/preprints/029.pdf . Google Scholar

[11]

R. HenrionB. S. Mordukhovich and N. M. Nam, Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM Journal on Optimization, 20 (2010), 2199-2227.   Google Scholar

[12]

R. HerzogC. Meyer and G. Wachsmuth, B-and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.   Google Scholar

[13]

M. Hintermüller, An active-set equality constrained Newton solver with feasibility restoration for inverse coefficient problems in elliptic variational inequalities, Inverse Problems, 24 (2008), 034017, 23pp. Google Scholar

[14]

M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C-and strong stationarity and a path-following algorithm, SIAM Journal on Optimization, 20 (2009), 868-902.   Google Scholar

[15]

M. HintermüllerB. S. Mordukhovich and T. Surowiec, Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints, Mathematical Programming, Ser. A, 146 (2014), 555-582.   Google Scholar

[16]

M. Hintermüller and T. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming, Ser. A, 160 (2016), 271-305.   Google Scholar

[17]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Applied Mathematics and Optimization, 41 (2000), 343-364.   Google Scholar

[18]

F. KikuchiK. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Japan Journal of Applied Mathematics, 1 (1984), 369-403.   Google Scholar

[19]

D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, vol. 60 of Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 2002.  Google Scholar

[20]

P. Mehlitz and G. Wachsmuth, The limiting normal cone to pointwise defined sets in Lebesgue spaces, Set-Valued and Variational Analysis, online first (2016), 1-19.   Google Scholar

[21]

P. Mehlitz and G. Wachsmuth, Weak and strong stationarity in generalized bilevel programming and bilevel optimal control, Optimization, 65 (2016), 907-935.   Google Scholar

[22]

C. Meyer and L. M. Susu, Optimal control of nonsmooth, semilinear parabolic equations, SIAM Journal on Control and Optimization, 55 (2017), 2206-2234.   Google Scholar

[23]

F. Mignot, Contrôle dans les inéquations variationelles elliptiques, Journal of Functional Analysis, 22 (1976), 130-185.   Google Scholar

[24]

F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM Journal on Control and Optimization, 22 (1984), 466-476.   Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I, vol. 330 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2006, Basic theory.  Google Scholar

[26]

J. OutrataJ. Jarušek and J. Stará, On optimality conditions in control of elliptic variational inequalities, Set-Valued and Variational Analysis, 19 (2011), 23-42.   Google Scholar

[27]

J. V. Outrata and W. Römisch, On optimality conditions for some nonsmooth optimization problems over Lp spaces, Journal of Optimization Theory and Applications, 126 (2005), 411-438.   Google Scholar

[28]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, vol. 28 of Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 1998.  Google Scholar

[29]

J. -P. Penot, Calculus Without Derivatives, vol. 266 of Graduate Texts in Mathematics, Springer, New York, 2013.  Google Scholar

[30]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, Zentrum Mathematik, Technische Universität München, 2015, URL [http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:91-diss-20150420-1241413-1-4 . Google Scholar

[31]

J. Rappaz, Approximation of a nondifferentiable nonlinear problem related to MHD equilibria, Numer. Math., 45 (1984), 117-133.   Google Scholar

[32]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, vol. 317 of Comprehensive Studies in Mathematics, Springer, Berlin, 2004. Google Scholar

[33]

A. Schiela and D. Wachsmuth, Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints, ESAIM: M2AN, 47 (2013), 771-787.   Google Scholar

[34]

W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.  Google Scholar

[35]

R. Temam, A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma, Arch. Rational Mech. Anal., 60 (1975/76), 51-73.   Google Scholar

[36]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Springer, 1990.  Google Scholar

[37]

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, Theory, methods and applications. Translated from the 2005 German original by Jürgen Sprekels.  Google Scholar

[38]

G. Wachsmuth, Strong stationarity for optimal control of the obstacle problem with control constraints, SIAM Journal on Optimization, 24 (2014), 1914-1932.   Google Scholar

[39]

G. Wachsmuth, Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity, Set-Valued and Variational Analysis, 25 (2017), 133-175.   Google Scholar

[40]

G. Wachsmuth, Towards M-stationarity for optimal control of the obstacle problem with control constraints, SIAM Journal on Control and Optimization, 54 (2016), 964-986.   Google Scholar

Table 1.  Numerical results in the first example
$h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\|p_h - p\|_{L^2}$ $\|\chi_h - \chi\|_{L^\infty, h}$ # Newton
$3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.152\text{e}{-}3$ $1.036\text{e}{-}5$ $8.150\text{e}{-}7$ $3$
$1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $2.962\text{e}{-}4$ $2.679\text{e}{-}6$ $8.149\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
$3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.893\text{e}{-}5$ $1.716\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}2 $ - - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}3 $ - - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}5 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $3.178\text{e}{-}6$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $9.178\text{e}{-}6$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}4$ $3.267\text{e}{-}4$ $3.241\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}4$ $2.444\text{e}{-}4$ $2.405\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}4$ $2.449\text{e}{-}6$ $9.204\text{e}{-}9$ $8.149\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}8$ $1\text{e}{-}4$ $1.199\text{e}{-}7$ $9.452\text{e}{-}11$ $8.153\text{e}{-}7$ $3$
$h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\|p_h - p\|_{L^2}$ $\|\chi_h - \chi\|_{L^\infty, h}$ # Newton
$3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.152\text{e}{-}3$ $1.036\text{e}{-}5$ $8.150\text{e}{-}7$ $3$
$1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $2.962\text{e}{-}4$ $2.679\text{e}{-}6$ $8.149\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
$3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.893\text{e}{-}5$ $1.716\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}2 $ - - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}3 $ - - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}5 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $3.178\text{e}{-}6$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $9.178\text{e}{-}6$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}4$ $3.267\text{e}{-}4$ $3.241\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}4$ $2.444\text{e}{-}4$ $2.405\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}4$ $2.449\text{e}{-}6$ $9.204\text{e}{-}9$ $8.149\text{e}{-}7$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}8$ $1\text{e}{-}4$ $1.199\text{e}{-}7$ $9.452\text{e}{-}11$ $8.153\text{e}{-}7$ $3$
Table 2.  Numerical results in the second example
$h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\frac{\|p_h - p\|_{L^2}}{\|p\|_{L^2}}$ # Newton
$3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $8.708\text{e}{-}1$ $1.606\text{e}{-}2$ $4$
$1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $2.281\text{e}{-}1$ $4.541\text{e}{-}3$ $5$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $1.469\text{e}{-}2$ $3.119\text{e}{-}4$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6$ - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}8$ - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}10$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}14$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}12$ $3.007\text{e}{-}3$ $1.747\text{e}{-}3$ $2$
$7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}12$ $1.659\text{e}{-}2$ $1.512\text{e}{-}3$ $2$
$7.752\text{e}{-}3$ $1\text{e}{-}5$ $1\text{e}{-}12$ $1.692\text{e}{-}1$ $8.659\text{e}{-}4$ $5$
$7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}12$ - - no conv.
$h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\frac{\|p_h - p\|_{L^2}}{\|p\|_{L^2}}$ # Newton
$3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $8.708\text{e}{-}1$ $1.606\text{e}{-}2$ $4$
$1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $2.281\text{e}{-}1$ $4.541\text{e}{-}3$ $5$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $1.469\text{e}{-}2$ $3.119\text{e}{-}4$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6$ - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}8$ - - no conv.
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}10$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}14$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
$7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}12$ $3.007\text{e}{-}3$ $1.747\text{e}{-}3$ $2$
$7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}12$ $1.659\text{e}{-}2$ $1.512\text{e}{-}3$ $2$
$7.752\text{e}{-}3$ $1\text{e}{-}5$ $1\text{e}{-}12$ $1.692\text{e}{-}1$ $8.659\text{e}{-}4$ $5$
$7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}12$ - - no conv.
[1]

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