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doi: 10.3934/mcrf.2021054
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Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

Department of Mathematics, TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, Germany

* Corresponding author: Xenia Kerkhoff

Received  March 2021 Revised  July 2021 Early access October 2021

We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.

Citation: Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021054
References:
[1]

T. Akman and B. Karasözen, Variational time discretization methods for optimal control problems governed by diffusion-convection-reaction equations, Comput. Appl. Math., 272 (2014), 41-56.  doi: 10.1016/j.cam.2014.05.002.  Google Scholar

[2]

N. AllahverdiA. Pozo and E. Zuazua, Numerical aspects of large-time optimal control of Burgers equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1371-1401.  doi: 10.1051/m2an/2015076.  Google Scholar

[3]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[4]

G. Chen and S. Collis, Optimal control for Burgers flow using the discontinuous Galerkin method, AIAA Region IV Student Paper Conference, 2003. Google Scholar

[5]

A. ChertockA. Kurganov and T. Wu, Operator splitting based central-upwind schemes for the shallow water equations with moving bottom topography, Commun. Math. Sci., 18 (2020), 2149-2168.  doi: 10.4310/CMS.2020.v18.n8.a3.  Google Scholar

[6]

B. Cockburn, An introduction to the discontinuous Galerkin Method for convection-dominated problems, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Lecture Notes in Math., 1697 (1998), 151-268.  doi: 10.1007/BFb0096353.  Google Scholar

[7]

J. C. de los Reyes and K. Kunisch, A comparison of algorithms for control constrained optimal control of the Burgers equation, Calcolo, 41 (2004), 203-225.  doi: 10.1007/s10092-004-0092-7.  Google Scholar

[8]

V. Dolejší and M. Feistauer, Discontinuous Galerkin Methods, Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics, 48. Springer, Cham, 2015. doi: 10.1007/978-3-319-19267-3.  Google Scholar

[9]

K. T. Elgindy and B. Karasözen, Distributed optimal control of viscous Burgers' equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method, Optimal Control Appl. Methods, 41 (2020), 253-277.  doi: 10.1002/oca.2541.  Google Scholar

[10]

B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), 321-351.  doi: 10.1090/S0025-5718-1981-0606500-X.  Google Scholar

[11]

L. Failer, Optimal Control of Time-Dependent Nonlinear Fluid-Structure Interaction, PhD thesis, Technical University of Munich, Munich, 2017. Google Scholar

[12]

M. FeistauerV. KučeraK. Najzar and J. Prokopová, Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math., 117 (2011), 251-288.  doi: 10.1007/s00211-010-0348-x.  Google Scholar

[13]

G. GassnerA. Winters and D. Kopriva, A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations, Appl. Math. Comput., 272 (2016), 291-308.  doi: 10.1016/j.amc.2015.07.014.  Google Scholar

[14]

J. Greenberg and A. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16.  doi: 10.1137/0733001.  Google Scholar

[15]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54. Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar

[16]

A. Hiltebrand and S. May, Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier-Stokes equations, Commun. Math. Sci., 16 (2018), 2095-2124.  doi: 10.4310/CMS.2018.v16.n8.a3.  Google Scholar

[17]

A. Hiltebrand and S. Mishra, Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography, Netw. Heterog. Media, 11 (2016), 145-162.  doi: 10.3934/nhm.2016.11.145.  Google Scholar

[18]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152. Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[19]

L. I. IgnatA. Pozo and E. Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comp., 84 (2015), 1633-1662.  doi: 10.1090/S0025-5718-2014-02915-3.  Google Scholar

[20]

K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.  doi: 10.1023/A:1021732508059.  Google Scholar

[21] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[22]

D. Leykekhman, Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems, J. Sci. Comput., 53 (2012), 483-511.  doi: 10.1007/s10915-012-9582-y.  Google Scholar

[23]

S. May, Spacetime discontinuous Galerkin methods for solving convection-diffusion systems, ESAIM Math. Model. Numer. Anal., 51 (2017), 1755-1781.  doi: 10.1051/m2an/2017001.  Google Scholar

[24]

S. NoelleY. Xing and C.-W. Shu, High-order well-balanced schemes, Numerical methods for Balance Laws, Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta, 24 (2009), 1-66.   Google Scholar

[25]

J. Pfefferer, Numerical Analysis for Elliptic Neumann Boundary Control Problems on Polygonal Domains, PhD thesis, Bundeswehr University Munich, Munich, 2014. Google Scholar

[26]

W. J. Rider and R. B. Lowrie, The use of classical Lax-Friedrichs Riemann solvers with discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids, 40 (2002), 479-486.  doi: 10.1002/fld.334.  Google Scholar

[27]

S. Volkwein, Mesh-Independence of an Augmented Lagrangian-SQP Method in Hilbert Spaces and Control Problems for the Burgers Equation, PhD thesis, Technical University of Berlin, Berlin, 1997. Google Scholar

[28]

S. Volkwein, Distributed control problems for the Burgers equation, Comput. Optim. Appl., 18 (2001), 115-140.  doi: 10.1023/A:1008770404256.  Google Scholar

[29]

L. WilcoxG. StadlerT. Bui-Thanh and O. Ghattas, Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, J. Sci. Comput., 63 (2015), 138-162.  doi: 10.1007/s10915-014-9890-5.  Google Scholar

[30]

F. Yilmaz and B. Karasözen, Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics, J. Comput. Appl. Math., 235 (2011), 4839-4850.  doi: 10.1016/j.cam.2011.01.002.  Google Scholar

show all references

References:
[1]

T. Akman and B. Karasözen, Variational time discretization methods for optimal control problems governed by diffusion-convection-reaction equations, Comput. Appl. Math., 272 (2014), 41-56.  doi: 10.1016/j.cam.2014.05.002.  Google Scholar

[2]

N. AllahverdiA. Pozo and E. Zuazua, Numerical aspects of large-time optimal control of Burgers equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1371-1401.  doi: 10.1051/m2an/2015076.  Google Scholar

[3]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[4]

G. Chen and S. Collis, Optimal control for Burgers flow using the discontinuous Galerkin method, AIAA Region IV Student Paper Conference, 2003. Google Scholar

[5]

A. ChertockA. Kurganov and T. Wu, Operator splitting based central-upwind schemes for the shallow water equations with moving bottom topography, Commun. Math. Sci., 18 (2020), 2149-2168.  doi: 10.4310/CMS.2020.v18.n8.a3.  Google Scholar

[6]

B. Cockburn, An introduction to the discontinuous Galerkin Method for convection-dominated problems, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Lecture Notes in Math., 1697 (1998), 151-268.  doi: 10.1007/BFb0096353.  Google Scholar

[7]

J. C. de los Reyes and K. Kunisch, A comparison of algorithms for control constrained optimal control of the Burgers equation, Calcolo, 41 (2004), 203-225.  doi: 10.1007/s10092-004-0092-7.  Google Scholar

[8]

V. Dolejší and M. Feistauer, Discontinuous Galerkin Methods, Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics, 48. Springer, Cham, 2015. doi: 10.1007/978-3-319-19267-3.  Google Scholar

[9]

K. T. Elgindy and B. Karasözen, Distributed optimal control of viscous Burgers' equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method, Optimal Control Appl. Methods, 41 (2020), 253-277.  doi: 10.1002/oca.2541.  Google Scholar

[10]

B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), 321-351.  doi: 10.1090/S0025-5718-1981-0606500-X.  Google Scholar

[11]

L. Failer, Optimal Control of Time-Dependent Nonlinear Fluid-Structure Interaction, PhD thesis, Technical University of Munich, Munich, 2017. Google Scholar

[12]

M. FeistauerV. KučeraK. Najzar and J. Prokopová, Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math., 117 (2011), 251-288.  doi: 10.1007/s00211-010-0348-x.  Google Scholar

[13]

G. GassnerA. Winters and D. Kopriva, A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations, Appl. Math. Comput., 272 (2016), 291-308.  doi: 10.1016/j.amc.2015.07.014.  Google Scholar

[14]

J. Greenberg and A. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16.  doi: 10.1137/0733001.  Google Scholar

[15]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54. Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar

[16]

A. Hiltebrand and S. May, Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier-Stokes equations, Commun. Math. Sci., 16 (2018), 2095-2124.  doi: 10.4310/CMS.2018.v16.n8.a3.  Google Scholar

[17]

A. Hiltebrand and S. Mishra, Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography, Netw. Heterog. Media, 11 (2016), 145-162.  doi: 10.3934/nhm.2016.11.145.  Google Scholar

[18]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, 152. Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[19]

L. I. IgnatA. Pozo and E. Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comp., 84 (2015), 1633-1662.  doi: 10.1090/S0025-5718-2014-02915-3.  Google Scholar

[20]

K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.  doi: 10.1023/A:1021732508059.  Google Scholar

[21] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar
[22]

D. Leykekhman, Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems, J. Sci. Comput., 53 (2012), 483-511.  doi: 10.1007/s10915-012-9582-y.  Google Scholar

[23]

S. May, Spacetime discontinuous Galerkin methods for solving convection-diffusion systems, ESAIM Math. Model. Numer. Anal., 51 (2017), 1755-1781.  doi: 10.1051/m2an/2017001.  Google Scholar

[24]

S. NoelleY. Xing and C.-W. Shu, High-order well-balanced schemes, Numerical methods for Balance Laws, Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta, 24 (2009), 1-66.   Google Scholar

[25]

J. Pfefferer, Numerical Analysis for Elliptic Neumann Boundary Control Problems on Polygonal Domains, PhD thesis, Bundeswehr University Munich, Munich, 2014. Google Scholar

[26]

W. J. Rider and R. B. Lowrie, The use of classical Lax-Friedrichs Riemann solvers with discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids, 40 (2002), 479-486.  doi: 10.1002/fld.334.  Google Scholar

[27]

S. Volkwein, Mesh-Independence of an Augmented Lagrangian-SQP Method in Hilbert Spaces and Control Problems for the Burgers Equation, PhD thesis, Technical University of Berlin, Berlin, 1997. Google Scholar

[28]

S. Volkwein, Distributed control problems for the Burgers equation, Comput. Optim. Appl., 18 (2001), 115-140.  doi: 10.1023/A:1008770404256.  Google Scholar

[29]

L. WilcoxG. StadlerT. Bui-Thanh and O. Ghattas, Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, J. Sci. Comput., 63 (2015), 138-162.  doi: 10.1007/s10915-014-9890-5.  Google Scholar

[30]

F. Yilmaz and B. Karasözen, Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics, J. Comput. Appl. Math., 235 (2011), 4839-4850.  doi: 10.1016/j.cam.2011.01.002.  Google Scholar

Figure 1.  Test 1: Results for gradient test. The $ x- $axis denotes $ \frac{1}{\rho} $ with $ \rho $ being the step length in the difference quotient, the $ y- $axis denotes the error of the gradient as given by (24)
Figure 2.  Test 2: Computed solution $ (q^h,u^h) $ for $ p = 2 $ and $ N = 160 $. Left column: control $ q^h $, right column: state $ u^h $. The second row shows the solutions $ q^h(\cdot,t) $ and $ u^h(\cdot,t) $ at time instances $ t = 0.25 $, $ t = 0.5 $, and $ t = 0.75 $ as a function of the space coordinate $ x $
Table 1.  Test 1: Errors and orders of convergence for $ \varepsilon = 10^{-3} $
$ p $ $ N $ error $ u $ order error $ q $ order error $ z $ order iter. gradient
OD-wo
1 40 2.35e-04 1.93e-03 2.31e-04 25 4.66e-05
1 80 5.43e-05 2.11 5.09e-04 1.92 7.36e-05 1.65 28 1.97e-05
1 160 1.28e-05 2.08 1.44e-04 1.82 2.48e-05 1.57 31 7.11e-06
1 320 3.19e-06 2.01 4.05e-05 1.83 8.22e-06 1.59 36 2.81e-06
2 20 2.31e-05 1.67e-04 1.67e-05 49 2.18e-07
2 40 2.95e-06 2.97 2.05e-05 3.03 2.06e-06 3.02 52 3.00e-08
2 80 3.41e-07 3.11 2.61e-06 2.98 2.61e-07 2.98 54 9.51e-09
2 160 4.65e-08 2.87 3.93e-07 2.73 3.47e-08 2.91 54 9.51e-09
3 10 4.15e-06 4.97e-05 4.96e-06 54 9.51e-09
3 20 1.81e-07 4.52 3.02e-06 4.04 3.00e-07 4.05 54 9.51e-09
3 40 2.97e-08 2.61 2.86e-07 3.40 1.70e-08 4.14 54 9.51e-09
3 80 2.62e-08 0.18 2.07e-07 0.47 3.33e-09 2.35 54 9.51e-09
OD-w
1 40 2.31e-04 1.87e-03 1.87e-04 54 9.51e-09
1 80 5.05e-05 2.19 4.63e-04 2.02 4.63e-05 2.02 54 9.51e-09
1 160 9.91e-06 2.35 1.15e-04 2.01 1.15e-05 2.01 54 9.51e-09
1 320 1.72e-06 2.52 2.87e-05 2.01 2.87e-06 2.01 54 9.51e-09
2 20 2.26e-05 1.67e-04 3.07 1.67e-05 54 9.51e-09
2 40 2.88e-06 2.98 2.01e-05 3.06 2.01e-06 3.06 54 9.51e-09
2 80 3.23e-07 3.15 2.45e-06 3.04 2.44e-07 3.04 54 9.51e-09
2 160 4.35e-08 2.89 3.62e-07 2.76 3.01e-08 3.02 54 9.51e-09
3 10 4.16e-06 4.96e-05 4.96e-06 54 9.51e-09
3 20 1.73e-07 4.59 2.99e-06 4.05 2.98e-07 4.05 54 9.51e-09
3 40 2.66e-08 2.70 2.65e-07 3.49 1.72e-08 4.11 54 9.51e-09
3 80 2.59e-08 0.04 2.05e-07 0.37 3.52e-09 2.29 54 9.51e-09
$ p $ $ N $ error $ u $ order error $ q $ order error $ z $ order iter. gradient
OD-wo
1 40 2.35e-04 1.93e-03 2.31e-04 25 4.66e-05
1 80 5.43e-05 2.11 5.09e-04 1.92 7.36e-05 1.65 28 1.97e-05
1 160 1.28e-05 2.08 1.44e-04 1.82 2.48e-05 1.57 31 7.11e-06
1 320 3.19e-06 2.01 4.05e-05 1.83 8.22e-06 1.59 36 2.81e-06
2 20 2.31e-05 1.67e-04 1.67e-05 49 2.18e-07
2 40 2.95e-06 2.97 2.05e-05 3.03 2.06e-06 3.02 52 3.00e-08
2 80 3.41e-07 3.11 2.61e-06 2.98 2.61e-07 2.98 54 9.51e-09
2 160 4.65e-08 2.87 3.93e-07 2.73 3.47e-08 2.91 54 9.51e-09
3 10 4.15e-06 4.97e-05 4.96e-06 54 9.51e-09
3 20 1.81e-07 4.52 3.02e-06 4.04 3.00e-07 4.05 54 9.51e-09
3 40 2.97e-08 2.61 2.86e-07 3.40 1.70e-08 4.14 54 9.51e-09
3 80 2.62e-08 0.18 2.07e-07 0.47 3.33e-09 2.35 54 9.51e-09
OD-w
1 40 2.31e-04 1.87e-03 1.87e-04 54 9.51e-09
1 80 5.05e-05 2.19 4.63e-04 2.02 4.63e-05 2.02 54 9.51e-09
1 160 9.91e-06 2.35 1.15e-04 2.01 1.15e-05 2.01 54 9.51e-09
1 320 1.72e-06 2.52 2.87e-05 2.01 2.87e-06 2.01 54 9.51e-09
2 20 2.26e-05 1.67e-04 3.07 1.67e-05 54 9.51e-09
2 40 2.88e-06 2.98 2.01e-05 3.06 2.01e-06 3.06 54 9.51e-09
2 80 3.23e-07 3.15 2.45e-06 3.04 2.44e-07 3.04 54 9.51e-09
2 160 4.35e-08 2.89 3.62e-07 2.76 3.01e-08 3.02 54 9.51e-09
3 10 4.16e-06 4.96e-05 4.96e-06 54 9.51e-09
3 20 1.73e-07 4.59 2.99e-06 4.05 2.98e-07 4.05 54 9.51e-09
3 40 2.66e-08 2.70 2.65e-07 3.49 1.72e-08 4.11 54 9.51e-09
3 80 2.59e-08 0.04 2.05e-07 0.37 3.52e-09 2.29 54 9.51e-09
Table 2.  Test 1: Errors and orders of convergence for $ \varepsilon = 10^{-5} $
$ p $ $ N $ error $ u $ order error $ q $ order error $ z $ order iter. gradient
OD-wo
1 40 2.79e-04 1.97e-03 2.47e-04 24 5.33e-05
1 80 7.51e-05 1.89 5.51e-04 1.84 9.20e-05 1.42 27 2.81e-05
1 160 2.28e-05 1.72 1.93e-04 1.51 4.08e-05 1.18 29 1.26e-05
1 320 8.15e-06 1.49 7.90e-05 1.29 1.99e-05 1.04 33 7.11e-06
2 20 2.52e-05 1.72e-04 1.72e-05 44 2.25e-07
2 40 3.98e-06 2.66 2.18e-05 2.98 2.18e-06 2.97 51 4.63e-08
2 80 6.28e-07 2.67 3.09e-06 2.82 3.10e-07 2.82 54 9.51e-09
2 160 1.01e-07 2.64 5.60e-07 2.46 5.42e-08 2.51 54 9.51e-09
3 10 4.63e-06 4.87e-05 4.87e-06 54 9.51e-09
3 20 2.76e-07 4.07 3.12e-06 3.96 3.08e-07 3.98 54 9.51e-09
3 40 3.58e-08 2.94 3.19e-07 3.29 1.87e-08 4.04 54 9.51e-09
3 80 2.68e-08 0.42 2.12e-07 0.59 2.94e-09 2.67 54 9.51e-09
OD-w
1 40 2.74e-04 1.89e-03 1.89e-04 54 9.51e-09
1 80 6.99e-05 1.97 4.68e-04 2.01 4.68e-05 2.01 54 9.51e-09
1 160 1.76e-05 1.99 1.17e-04 2.01 1.17e-05 2.01 54 9.51e-09
1 320 4.39e-06 2.00 2.91e-05 2.00 2.91e-06 2.00 54 9.51e-09
2 20 2.46e-05 1.72e-04 1.72e-05 54 9.51e-09
2 40 3.84e-06 2.68 2.12e-05 3.02 2.12e-06 3.02 54 9.51e-09
2 80 5.86e-07 2.71 2.65e-06 3.00 2.65e-07 3.00 54 9.51e-09
2 160 8.82e-08 2.73 3.89e-07 2.77 3.33e-08 2.99 54 9.51e-09
3 10 4.65e-06 4.85e-05 4.85e-06 54 9.51e-09
3 20 2.65e-07 4.13 3.05e-06 3.99 3.04e-07 4.00 54 9.51e-09
3 40 2.96e-08 3.16 2.75e-07 3.47 1.86e-08 4.03 54 9.51e-09
3 80 2.59e-08 0.19 2.05e-07 0.42 3.53e-09 2.40 54 9.51e-09
$ p $ $ N $ error $ u $ order error $ q $ order error $ z $ order iter. gradient
OD-wo
1 40 2.79e-04 1.97e-03 2.47e-04 24 5.33e-05
1 80 7.51e-05 1.89 5.51e-04 1.84 9.20e-05 1.42 27 2.81e-05
1 160 2.28e-05 1.72 1.93e-04 1.51 4.08e-05 1.18 29 1.26e-05
1 320 8.15e-06 1.49 7.90e-05 1.29 1.99e-05 1.04 33 7.11e-06
2 20 2.52e-05 1.72e-04 1.72e-05 44 2.25e-07
2 40 3.98e-06 2.66 2.18e-05 2.98 2.18e-06 2.97 51 4.63e-08
2 80 6.28e-07 2.67 3.09e-06 2.82 3.10e-07 2.82 54 9.51e-09
2 160 1.01e-07 2.64 5.60e-07 2.46 5.42e-08 2.51 54 9.51e-09
3 10 4.63e-06 4.87e-05 4.87e-06 54 9.51e-09
3 20 2.76e-07 4.07 3.12e-06 3.96 3.08e-07 3.98 54 9.51e-09
3 40 3.58e-08 2.94 3.19e-07 3.29 1.87e-08 4.04 54 9.51e-09
3 80 2.68e-08 0.42 2.12e-07 0.59 2.94e-09 2.67 54 9.51e-09
OD-w
1 40 2.74e-04 1.89e-03 1.89e-04 54 9.51e-09
1 80 6.99e-05 1.97 4.68e-04 2.01 4.68e-05 2.01 54 9.51e-09
1 160 1.76e-05 1.99 1.17e-04 2.01 1.17e-05 2.01 54 9.51e-09
1 320 4.39e-06 2.00 2.91e-05 2.00 2.91e-06 2.00 54 9.51e-09
2 20 2.46e-05 1.72e-04 1.72e-05 54 9.51e-09
2 40 3.84e-06 2.68 2.12e-05 3.02 2.12e-06 3.02 54 9.51e-09
2 80 5.86e-07 2.71 2.65e-06 3.00 2.65e-07 3.00 54 9.51e-09
2 160 8.82e-08 2.73 3.89e-07 2.77 3.33e-08 2.99 54 9.51e-09
3 10 4.65e-06 4.85e-05 4.85e-06 54 9.51e-09
3 20 2.65e-07 4.13 3.05e-06 3.99 3.04e-07 4.00 54 9.51e-09
3 40 2.96e-08 3.16 2.75e-07 3.47 1.86e-08 4.03 54 9.51e-09
3 80 2.59e-08 0.19 2.05e-07 0.42 3.53e-09 2.40 54 9.51e-09
Table 3.  Test 2: Values and orders of convergence (computed using values from 3 subsequent meshes) for functional $ J $ for $ p = 1 $
OD-w OD-wo
N value $ J $ order iter. gradient value $ J $ order iter. gradient
20 0.10640082692 - 44 8.58e-08 0.10641958284 - 11 2.10e-03
40 0.10818515778 - 44 7.94e-08 0.10819225166 - 14 1.02e-03
80 0.10855578246 2.27 43 8.03e-08 0.10855750188 2.28 18 3.44e-04
160 0.10862035158 2.52 43 8.00e-08 0.10862054051 2.53 22 1.09e-04
OD-w OD-wo
N value $ J $ order iter. gradient value $ J $ order iter. gradient
20 0.10640082692 - 44 8.58e-08 0.10641958284 - 11 2.10e-03
40 0.10818515778 - 44 7.94e-08 0.10819225166 - 14 1.02e-03
80 0.10855578246 2.27 43 8.03e-08 0.10855750188 2.28 18 3.44e-04
160 0.10862035158 2.52 43 8.00e-08 0.10862054051 2.53 22 1.09e-04
Table 4.  Test 2: Errors and orders of convergence for the control $ q $ for $ p = 1 $
OD-w OD-wo
N error $ q $ order iter. gradient error $ q $ order iter. gradient
20 1.67e-02 - 44 8.58e-08 1.77e-02 - 11 2.10e-03
40 4.56e-03 1.87 44 7.94e-08 6.10e-03 1.54 14 1.02e-03
80 1.47e-03 1.64 43 8.03e-08 2.83e-03 1.11 18 3.44e-04
160 4.20e-04 1.80 43 8.00e-08 1.00e-03 1.50 22 1.09e-04
OD-w OD-wo
N error $ q $ order iter. gradient error $ q $ order iter. gradient
20 1.67e-02 - 44 8.58e-08 1.77e-02 - 11 2.10e-03
40 4.56e-03 1.87 44 7.94e-08 6.10e-03 1.54 14 1.02e-03
80 1.47e-03 1.64 43 8.03e-08 2.83e-03 1.11 18 3.44e-04
160 4.20e-04 1.80 43 8.00e-08 1.00e-03 1.50 22 1.09e-04
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