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doi: 10.3934/jimo.2022030
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Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

2. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China

3. 

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 102488, China

*Corresponding author: Bing Sun

Received  September 2021 Revised  December 2021 Early access March 2022

Fund Project: The second author is supported by NSFC grant 11471036

In this paper, we are concerned with the Galerkin spectral approximation of an optimal control problem governed by the Stokes equation with $ L^2 $-norm constraint on the control variable. By means of the derived optimality conditions for both the original control system and its spectral approximation one, we establish a priori error estimates and then obtain a posteriori error estimator. A numerical example is, subsequently, executed to illustrate the effectiveness of method and the high performance of estimators. Furthermore, we conjecture that the similar conclusions should hold for optimal control of the Navier-Stokes equation. It is then confirmed by another numerical example.

Citation: Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $ L^2 $-norm control constraint. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022030
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), Vol. 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4$^{nd}$ edition, Springer Monographs in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-2247-7.

[3]

M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim., 35 (1997), 1524-1543.  doi: 10.1137/S036301299529330X.

[4]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8.

[5]

E. CasasM. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.

[6]

E. CasasM. Mateos and A. Rösch, Error estimates for semilinear parabolic control problems in the absence of Tikhonov term, SIAM J. Control Optim., 57 (2019), 2515-2550.  doi: 10.1137/18M117220X.

[7]

Y. Chen and F. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput., 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[8]

Y. Chen and F. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[9]

Y. ChenF. HuangN. Yi and W. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), 1625-1648.  doi: 10.1137/080726057.

[10]

F. H. Clarkel, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[11]

J. C. de los Reyes and K. Kunisch, A semi-smooth Newton method for regularized state-constrained optimal control of the Navier-Stokes equations, Computing, 78 (2006), 287-309.  doi: 10.1007/s00607-006-0183-1.

[12]

K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361.

[13]

R. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47.  doi: 10.1016/0022-247X(73)90022-X.

[14]

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numér, 13 (1979), 313-328.  doi: 10.1051/m2an/1979130403131.

[15]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[16]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 26, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977.

[17]

B.-Y. Guo, Spectral Methods and Their Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812816641.

[18]

Y. He, Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations, J. Math. Anal. Appl., 423 (2015), 1129-1149.  doi: 10.1016/j.jmaa.2014.10.037.

[19]

M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[20]

F. HuangZ. Zheng and Y. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[21]

K. KohlsA. Rösch and K. G. Siebert, A posteriori error analysis of optimal control problems with control constraints, SIAM J. Control. Optim., 52 (2014), 1832-1861.  doi: 10.1137/130909251.

[22]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York, 1971.

[23]

W. LiuW. Gong and N. Yan, A new finite element approximation of a state-constrained optimal control problem, J. Comput. Math., 27 (2009), 97-114. 

[24]

W. Liu and N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.

[25]

H. Liu and N. Yan, Global superconvergence for optimal control problems governed by Stokes equations, Int. J. Numer. Anal. Model., 3 (2006), 283-302. 

[26]

W. LiuD. YangL. Yuan and C. Ma, Finite element approximations of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48 (2010), 1163-1185.  doi: 10.1137/080737095.

[27]

K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1982), 69-95.  doi: 10.1007/BF01447752.

[28]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[29]

H. Niu and D. Yang, Finite element analysis of optimal control problem governed by Stokes equations with $L^2$-norm state-constriants, J. Comput. Math., 29 (2011), 589-604.  doi: 10.4208/jcm.1103-m3514.

[30]

H. NiuL. Yuan and D. Yang, Adaptive finite element method for an optimal control problem of Stokes flow with $L^2$-norm state constraint, Internat. J. Numer. Methods Fluids, 69 (2012), 534-549.  doi: 10.1002/fld.2572.

[31]

T. Rees and A. J. Wathen, Preconditioning iterative methods for the optimal control of the Stokes equations, SIAM J. Sci. Comput., 33 (2011), 2903-2926.  doi: 10.1137/100798491.

[32]

J. Shen, On fast direct Poisson solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys., 116 (1995), 184-188.  doi: 10.1006/jcph.1995.1017.

[33]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Mathematics Monograph Series, Vol. 3, Science Press Beijing, Beijing, 2006.

[34]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[35]

I. Silberman, Planetary waves in the atmosphere, J. Atmospheric Sci., 11 (1954), 27-34. 

[36]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, in Mathematics Applied to Engineering, Modelling, and Social Issues, Stud. Syst. Decis. Control (eds. F. T. Smith, H. Dutta and J. N. Mordeson), Vol. 200, Springer, Cham, 2019,363–420.

[37]

J. Zhou and D. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[38]

J. ZhouJ. Zhang and X. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput. Math. Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), Vol. 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, 4$^{nd}$ edition, Springer Monographs in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-2247-7.

[3]

M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim., 35 (1997), 1524-1543.  doi: 10.1137/S036301299529330X.

[4]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8.

[5]

E. CasasM. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.

[6]

E. CasasM. Mateos and A. Rösch, Error estimates for semilinear parabolic control problems in the absence of Tikhonov term, SIAM J. Control Optim., 57 (2019), 2515-2550.  doi: 10.1137/18M117220X.

[7]

Y. Chen and F. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput., 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[8]

Y. Chen and F. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[9]

Y. ChenF. HuangN. Yi and W. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations, SIAM J. Numer. Anal., 49 (2011), 1625-1648.  doi: 10.1137/080726057.

[10]

F. H. Clarkel, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[11]

J. C. de los Reyes and K. Kunisch, A semi-smooth Newton method for regularized state-constrained optimal control of the Navier-Stokes equations, Computing, 78 (2006), 287-309.  doi: 10.1007/s00607-006-0183-1.

[12]

K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361.

[13]

R. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28-47.  doi: 10.1016/0022-247X(73)90022-X.

[14]

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numér, 13 (1979), 313-328.  doi: 10.1051/m2an/1979130403131.

[15]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[16]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 26, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977.

[17]

B.-Y. Guo, Spectral Methods and Their Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812816641.

[18]

Y. He, Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations, J. Math. Anal. Appl., 423 (2015), 1129-1149.  doi: 10.1016/j.jmaa.2014.10.037.

[19]

M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[20]

F. HuangZ. Zheng and Y. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[21]

K. KohlsA. Rösch and K. G. Siebert, A posteriori error analysis of optimal control problems with control constraints, SIAM J. Control. Optim., 52 (2014), 1832-1861.  doi: 10.1137/130909251.

[22]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York, 1971.

[23]

W. LiuW. Gong and N. Yan, A new finite element approximation of a state-constrained optimal control problem, J. Comput. Math., 27 (2009), 97-114. 

[24]

W. Liu and N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.

[25]

H. Liu and N. Yan, Global superconvergence for optimal control problems governed by Stokes equations, Int. J. Numer. Anal. Model., 3 (2006), 283-302. 

[26]

W. LiuD. YangL. Yuan and C. Ma, Finite element approximations of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48 (2010), 1163-1185.  doi: 10.1137/080737095.

[27]

K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1982), 69-95.  doi: 10.1007/BF01447752.

[28]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[29]

H. Niu and D. Yang, Finite element analysis of optimal control problem governed by Stokes equations with $L^2$-norm state-constriants, J. Comput. Math., 29 (2011), 589-604.  doi: 10.4208/jcm.1103-m3514.

[30]

H. NiuL. Yuan and D. Yang, Adaptive finite element method for an optimal control problem of Stokes flow with $L^2$-norm state constraint, Internat. J. Numer. Methods Fluids, 69 (2012), 534-549.  doi: 10.1002/fld.2572.

[31]

T. Rees and A. J. Wathen, Preconditioning iterative methods for the optimal control of the Stokes equations, SIAM J. Sci. Comput., 33 (2011), 2903-2926.  doi: 10.1137/100798491.

[32]

J. Shen, On fast direct Poisson solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys., 116 (1995), 184-188.  doi: 10.1006/jcph.1995.1017.

[33]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Mathematics Monograph Series, Vol. 3, Science Press Beijing, Beijing, 2006.

[34]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[35]

I. Silberman, Planetary waves in the atmosphere, J. Atmospheric Sci., 11 (1954), 27-34. 

[36]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, in Mathematics Applied to Engineering, Modelling, and Social Issues, Stud. Syst. Decis. Control (eds. F. T. Smith, H. Dutta and J. N. Mordeson), Vol. 200, Springer, Cham, 2019,363–420.

[37]

J. Zhou and D. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[38]

J. ZhouJ. Zhang and X. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput. Math. Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

Figure 1.  Error estimates for the Stokes problem
Figure 2.  The point-wise error estimates of velocity when $ N = 20 $ for the Stokes problem
Figure 3.  The point-wise error estimates of control and pressure when $ N = 20 $ for the Stokes problem
Figure 4.  Error estimates for the Navier-Stokes problem
Figure 5.  The point-wise error estimates of velocity when $ N = 12 $ for the Navier-Stokes problem
Figure 6.  The point-wise error estimates of control and pressure when $ N = 12 $ for the Navier-Stokes problem
Table 1.  Error results and indicators $ \eta $ and $ \zeta $ for the Stokes problem
$ N $ 4 8 12 16 20
$ \left\|\mathit{\boldsymbol{u}}-\mathit{\boldsymbol{u}}_N \right\|_{0,\Omega} $ 2.1205e-1 1.0128e-3 5.5297e-7 9.7982e-11 6.9339e-15
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{1,\Omega} $ 3.4839e-1 2.9315e-3 2.4041e-6 4.1206e-10 2.8449e-14
$ \left\|r-r_N \right\|_{0,\Omega} $ 2.5714e-1 2.7753e-3 2.3353e-6 3.9898e-10 3.0040e-14
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{1,\Omega} $ 1.3120e+0 1.1464e-2 9.5451e-6 1.6466e-9 1.1391e-13
$ \left\|r^\ast-r^\ast_N \right\|_{0,\Omega} $ 1.0238e+0 1.1186e-2 9.4062e-6 1.6070e-9 1.2138e-13
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{0,\Omega} $ 5.6836e-2 2.5945e-4 1.3929e-7 2.4527e-11 1.2809e-15
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{0,\Omega} $ 2.1187e-1 1.0128e-3 5.5297e-7 9.7981e-11 4.8888e-15
$ \Sigma $ 3.1534e+0 2.9370e-2 2.4244e-5 4.1626e-9 3.0071e-13
$ \Sigma_1 $ 4.8075e-1 2.2850e-3 1.2452e-6 2.2049e-10 1.3104e-14
$ \eta $ 3.7276e+0 3.0111e-2 2.2922e-5 3.6442e-9 2.4850e-13
$ \zeta $ 1.2136e+0 6.7677e-3 4.2722e-6 6.0057e-10 3.8516e-14
$ {\rm Order}(\mathit{\boldsymbol{u}}) $ $ \setminus $ 13.1803 26.1153 38.7117 52.4135
$ {\rm Order}(\mathit{\boldsymbol{y}}) $ $ \setminus $ 13.2918 26.1739 38.7398 54.0802
$ N $ 4 8 12 16 20
$ \left\|\mathit{\boldsymbol{u}}-\mathit{\boldsymbol{u}}_N \right\|_{0,\Omega} $ 2.1205e-1 1.0128e-3 5.5297e-7 9.7982e-11 6.9339e-15
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{1,\Omega} $ 3.4839e-1 2.9315e-3 2.4041e-6 4.1206e-10 2.8449e-14
$ \left\|r-r_N \right\|_{0,\Omega} $ 2.5714e-1 2.7753e-3 2.3353e-6 3.9898e-10 3.0040e-14
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{1,\Omega} $ 1.3120e+0 1.1464e-2 9.5451e-6 1.6466e-9 1.1391e-13
$ \left\|r^\ast-r^\ast_N \right\|_{0,\Omega} $ 1.0238e+0 1.1186e-2 9.4062e-6 1.6070e-9 1.2138e-13
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{0,\Omega} $ 5.6836e-2 2.5945e-4 1.3929e-7 2.4527e-11 1.2809e-15
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{0,\Omega} $ 2.1187e-1 1.0128e-3 5.5297e-7 9.7981e-11 4.8888e-15
$ \Sigma $ 3.1534e+0 2.9370e-2 2.4244e-5 4.1626e-9 3.0071e-13
$ \Sigma_1 $ 4.8075e-1 2.2850e-3 1.2452e-6 2.2049e-10 1.3104e-14
$ \eta $ 3.7276e+0 3.0111e-2 2.2922e-5 3.6442e-9 2.4850e-13
$ \zeta $ 1.2136e+0 6.7677e-3 4.2722e-6 6.0057e-10 3.8516e-14
$ {\rm Order}(\mathit{\boldsymbol{u}}) $ $ \setminus $ 13.1803 26.1153 38.7117 52.4135
$ {\rm Order}(\mathit{\boldsymbol{y}}) $ $ \setminus $ 13.2918 26.1739 38.7398 54.0802
Table 2.  Error results and indicators $ \eta_1 $ for the Navier-Stokes problem
$ N $ 4 6 8 10 12
$ \left\|\mathit{\boldsymbol{u}}-\mathit{\boldsymbol{u}}_N \right\|_{0,\Omega} $ 2.1205e-1 2.1146e-2 1.0127e-3 2.9748e-5 4.6690e-6
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{1,\Omega} $ 3.4839e-1 4.7157e-2 2.9317e-3 1.0468e-4 8.8421e-6
$ \left\|r-r_N \right\|_{0,\Omega} $ 2.5715e-1 4.4363e-2 2.7753e-3 1.0680e-4 3.6780e-5
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{1,\Omega} $ 3.1197e+0 1.8114e-1 1.1461e-2 4.1248e-4 3.4989e-5
$ \left\|r^\ast-r^\ast_N \right\|_{0,\Omega} $ 1.0238e+0 1.7913e-1 1.1186e-2 4.1640e-4 1.0156e-4
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{0,\Omega} $ 5.6837e-2 5.5325e-3 2.5949e-4 7.6005e-6 1.4564e-6
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{0,\Omega} $ 2.1187e-1 2.1145e-2 1.0127e-3 2.9990e-5 6.0261e-6
$ \Sigma $ 3.1534e+0 4.7293e-1 2.9367e-2 1.0701e-3 1.8684e-4
$ \eta_1 $ 3.7276e+0 5.1293e-1 3.0100e-2 1.0323e-3 6.4454e-5
$ N $ 4 6 8 10 12
$ \left\|\mathit{\boldsymbol{u}}-\mathit{\boldsymbol{u}}_N \right\|_{0,\Omega} $ 2.1205e-1 2.1146e-2 1.0127e-3 2.9748e-5 4.6690e-6
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{1,\Omega} $ 3.4839e-1 4.7157e-2 2.9317e-3 1.0468e-4 8.8421e-6
$ \left\|r-r_N \right\|_{0,\Omega} $ 2.5715e-1 4.4363e-2 2.7753e-3 1.0680e-4 3.6780e-5
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{1,\Omega} $ 3.1197e+0 1.8114e-1 1.1461e-2 4.1248e-4 3.4989e-5
$ \left\|r^\ast-r^\ast_N \right\|_{0,\Omega} $ 1.0238e+0 1.7913e-1 1.1186e-2 4.1640e-4 1.0156e-4
$ \left\|\mathit{\boldsymbol{y}}-\mathit{\boldsymbol{y}}_N \right\|_{0,\Omega} $ 5.6837e-2 5.5325e-3 2.5949e-4 7.6005e-6 1.4564e-6
$ \left\|\mathit{\boldsymbol{y}}^\ast-\mathit{\boldsymbol{y}}^\ast_N \right\|_{0,\Omega} $ 2.1187e-1 2.1145e-2 1.0127e-3 2.9990e-5 6.0261e-6
$ \Sigma $ 3.1534e+0 4.7293e-1 2.9367e-2 1.0701e-3 1.8684e-4
$ \eta_1 $ 3.7276e+0 5.1293e-1 3.0100e-2 1.0323e-3 6.4454e-5
[1]

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