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)
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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 |
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.
Keywords:
Optimal control,
discontinuous Galerkin method,
optimize-then-discretize,
viscous Burgers equation,
conservative formulation.
Mathematics Subject Classification:
Primary: 49M41; Secondary: 65M60.
Citation:
Xenia Kerkhoff, Sandra May.
Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021054
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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. |
| [2] |
N. Allahverdi, A. 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. |
| [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. |
| [4] |
G. Chen and S. Collis, Optimal control for Burgers flow using the discontinuous Galerkin method, AIAA Region IV Student Paper Conference, 2003. |
| [5] |
A. Chertock, A. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
L. Failer, Optimal Control of Time-Dependent Nonlinear Fluid-Structure Interaction, PhD thesis, Technical University of Munich, Munich, 2017. |
| [12] |
M. Feistauer, V. Kučera, K. 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. |
| [13] |
G. Gassner, A. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
L. I. Ignat, A. 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. |
| [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. |
| [21] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253.
|
| [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. |
| [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. |
| [24] |
S. Noelle, Y. 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.
|
| [25] |
J. Pfefferer, Numerical Analysis for Elliptic Neumann Boundary Control Problems on Polygonal Domains, PhD thesis, Bundeswehr University Munich, Munich, 2014. |
| [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. |
| [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. |
| [28] |
S. Volkwein,
Distributed control problems for the Burgers equation, Comput. Optim. Appl., 18 (2001), 115-140.
doi: 10.1023/A:1008770404256. |
| [29] |
L. Wilcox, G. Stadler, T. 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. |
| [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. |
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. |
| [2] |
N. Allahverdi, A. 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. |
| [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. |
| [4] |
G. Chen and S. Collis, Optimal control for Burgers flow using the discontinuous Galerkin method, AIAA Region IV Student Paper Conference, 2003. |
| [5] |
A. Chertock, A. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
L. Failer, Optimal Control of Time-Dependent Nonlinear Fluid-Structure Interaction, PhD thesis, Technical University of Munich, Munich, 2017. |
| [12] |
M. Feistauer, V. Kučera, K. 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. |
| [13] |
G. Gassner, A. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
L. I. Ignat, A. 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. |
| [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. |
| [21] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253.
|
| [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. |
| [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. |
| [24] |
S. Noelle, Y. 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.
|
| [25] |
J. Pfefferer, Numerical Analysis for Elliptic Neumann Boundary Control Problems on Polygonal Domains, PhD thesis, Bundeswehr University Munich, Munich, 2014. |
| [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. |
| [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. |
| [28] |
S. Volkwein,
Distributed control problems for the Burgers equation, Comput. Optim. Appl., 18 (2001), 115-140.
doi: 10.1023/A:1008770404256. |
| [29] |
L. Wilcox, G. Stadler, T. 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. |
| [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. |
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} $
| error |
order | error |
order | error |
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 |
| error |
order | error |
order | error |
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} $
| error |
order | error |
order | error |
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 |
| error |
order | error |
order | error |
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 |
order | iter. | gradient | value |
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 |
order | iter. | gradient | value |
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 | error |
order | iter. | gradient | error |
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 |
order | iter. | gradient | error |
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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