doi: 10.3934/era.2021045
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An adjoint-based a posteriori analysis of numerical approximation of Richards equation

Department of Mathematics and Statistics, University of Wyoming, Laramie, WY 82071, USA

* Corresponding author: Victor Ginting

Received  November 2020 Revised  April 2021 Early access June 2021

This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.

Citation: Victor Ginting. An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, doi: 10.3934/era.2021045
References:
[1]

W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-7605-6.  Google Scholar

[2]

V. BaronY. Coudière and P. Sochala, Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation, Appl. Numer. Math., 112 (2017), 104-125.  doi: 10.1016/j.apnum.2016.10.005.  Google Scholar

[3]

M. Bause and P. Knabner, Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods, Advances in Water Resources, 27 (2004), 565-581.  doi: 10.1016/j.advwatres.2004.03.005.  Google Scholar

[4]

C. BernardiL. El Alaoui and Z. Mghazli, A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media, IMA J. Numer. Anal., 34 (2014), 1002-1036.  doi: 10.1093/imanum/drt014.  Google Scholar

[5]

M. A. CeliaE. T. Bouloutas and R. L. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26 (1990), 1483-1496.  doi: 10.1029/WR026i007p01483.  Google Scholar

[6]

P. Chatzipantelidis, Finite volume methods for elliptic PDE's: A new approach, M2AN Math. Model. Numer. Anal., 36 (2002), 307-324.  doi: 10.1051/m2an:2002014.  Google Scholar

[7]

P. ChatzipantelidisV. Ginting and R. D. Lazarov, A finite volume element method for a non-linear elliptic problem, Numer. Linear Algebra Appl., 12 (2005), 515-546.  doi: 10.1002/nla.439.  Google Scholar

[8]

Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, vol. 2 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718942.  Google Scholar

[9]

S.-H. Chou and Q. Li, Error estimates in $L^2, H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp., 69 (2000), 103-120.  doi: 10.1090/S0025-5718-99-01192-8.  Google Scholar

[10]

B. CummingT. Moroney and I. Turner, A mass-conservative control volume-finite element method for solving Richards'equation in heterogeneous porous media, BIT, 51 (2011), 845-864.  doi: 10.1007/s10543-011-0335-3.  Google Scholar

[11] K. ErikssonD. EstepP. Hansbo and C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, 1996.   Google Scholar
[12] K. ErikssonD. EstepP. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995.  doi: 10.1017/S0962492900002531.  Google Scholar
[13] K. ErikssonD. EstepP. Hansbo and C. Johnson, Introduction to computational methods for differential equations, in Theory and Numerics of Ordinary and Partial Differential Equations (Leicester, 1994), Adv. Numer. Anal., IV, Oxford Univ. Press, New York, 1995.   Google Scholar
[14]

D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), 1-48.  doi: 10.1137/0732001.  Google Scholar

[15]

D. J. Estep, M. G. Larson and R. D. Williams, Estimating the error of numerical solutions of systems of reaction-diffusion equations, Mem. Amer. Math. Soc., 146 (2000), no. 696. doi: 10.1090/memo/0696.  Google Scholar

[16]

R. EymardM. Gutnic and D. Hilhorst, The finite volume method for Richards equation, Comput. Geosci., 3 (1999), 259-294.  doi: 10.1023/A:1011547513583.  Google Scholar

[17]

M. W. Farthing and F. L. Ogden, Numerical solution of Richards' equation: A review of advances and challenges, Soil Science Society of America Journal, 81 (2017), 1257-1269.  doi: 10.2136/sssaj2017.02.0058.  Google Scholar

[18]

W. R. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Science, 85 (1958), 228-232.  doi: 10.1097/00010694-195804000-00006.  Google Scholar

[19]

M. B. Giles and E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), 145-236.  doi: 10.1017/S096249290200003X.  Google Scholar

[20]

P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal., 15 (1978), 912-928.  doi: 10.1137/0715059.  Google Scholar

[21]

G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems, vol. 295 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1995, Translated from the 1992 Russian edition by Guennadi Kontarev and revised by the author. doi: 10.1007/978-94-017-0621-6.  Google Scholar

[22]

F. Marinelli and D. S. Durnford, Semianalytical solution to Richards' equation for layered porous media, Journal of Irrigation and Drainage Engineering, 124 (1998), 290-299.  doi: 10.1061/(ASCE)0733-9437(1998)124:6(290).  Google Scholar

[23]

L. A. Richards, Capillary conduction of liquids through porous mediums, Physics, 1 (1931), 318-333.  doi: 10.1063/1.1745010.  Google Scholar

[24]

R. Srivastava and T.-C. Jim Yeh, Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Water Resour. Res., 27 (1991), 753-762.   Google Scholar

[25]

A. W. WarrickA. Islas and D. O. Lomen, An analytical solution to Richards' equation for time-varying infiltration, Water Resour. Res., 27 (1991), 763-766.  doi: 10.1029/91WR00310.  Google Scholar

show all references

References:
[1]

W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-7605-6.  Google Scholar

[2]

V. BaronY. Coudière and P. Sochala, Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation, Appl. Numer. Math., 112 (2017), 104-125.  doi: 10.1016/j.apnum.2016.10.005.  Google Scholar

[3]

M. Bause and P. Knabner, Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods, Advances in Water Resources, 27 (2004), 565-581.  doi: 10.1016/j.advwatres.2004.03.005.  Google Scholar

[4]

C. BernardiL. El Alaoui and Z. Mghazli, A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media, IMA J. Numer. Anal., 34 (2014), 1002-1036.  doi: 10.1093/imanum/drt014.  Google Scholar

[5]

M. A. CeliaE. T. Bouloutas and R. L. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26 (1990), 1483-1496.  doi: 10.1029/WR026i007p01483.  Google Scholar

[6]

P. Chatzipantelidis, Finite volume methods for elliptic PDE's: A new approach, M2AN Math. Model. Numer. Anal., 36 (2002), 307-324.  doi: 10.1051/m2an:2002014.  Google Scholar

[7]

P. ChatzipantelidisV. Ginting and R. D. Lazarov, A finite volume element method for a non-linear elliptic problem, Numer. Linear Algebra Appl., 12 (2005), 515-546.  doi: 10.1002/nla.439.  Google Scholar

[8]

Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, vol. 2 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718942.  Google Scholar

[9]

S.-H. Chou and Q. Li, Error estimates in $L^2, H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp., 69 (2000), 103-120.  doi: 10.1090/S0025-5718-99-01192-8.  Google Scholar

[10]

B. CummingT. Moroney and I. Turner, A mass-conservative control volume-finite element method for solving Richards'equation in heterogeneous porous media, BIT, 51 (2011), 845-864.  doi: 10.1007/s10543-011-0335-3.  Google Scholar

[11] K. ErikssonD. EstepP. Hansbo and C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, 1996.   Google Scholar
[12] K. ErikssonD. EstepP. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995.  doi: 10.1017/S0962492900002531.  Google Scholar
[13] K. ErikssonD. EstepP. Hansbo and C. Johnson, Introduction to computational methods for differential equations, in Theory and Numerics of Ordinary and Partial Differential Equations (Leicester, 1994), Adv. Numer. Anal., IV, Oxford Univ. Press, New York, 1995.   Google Scholar
[14]

D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), 1-48.  doi: 10.1137/0732001.  Google Scholar

[15]

D. J. Estep, M. G. Larson and R. D. Williams, Estimating the error of numerical solutions of systems of reaction-diffusion equations, Mem. Amer. Math. Soc., 146 (2000), no. 696. doi: 10.1090/memo/0696.  Google Scholar

[16]

R. EymardM. Gutnic and D. Hilhorst, The finite volume method for Richards equation, Comput. Geosci., 3 (1999), 259-294.  doi: 10.1023/A:1011547513583.  Google Scholar

[17]

M. W. Farthing and F. L. Ogden, Numerical solution of Richards' equation: A review of advances and challenges, Soil Science Society of America Journal, 81 (2017), 1257-1269.  doi: 10.2136/sssaj2017.02.0058.  Google Scholar

[18]

W. R. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Science, 85 (1958), 228-232.  doi: 10.1097/00010694-195804000-00006.  Google Scholar

[19]

M. B. Giles and E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), 145-236.  doi: 10.1017/S096249290200003X.  Google Scholar

[20]

P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal., 15 (1978), 912-928.  doi: 10.1137/0715059.  Google Scholar

[21]

G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems, vol. 295 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1995, Translated from the 1992 Russian edition by Guennadi Kontarev and revised by the author. doi: 10.1007/978-94-017-0621-6.  Google Scholar

[22]

F. Marinelli and D. S. Durnford, Semianalytical solution to Richards' equation for layered porous media, Journal of Irrigation and Drainage Engineering, 124 (1998), 290-299.  doi: 10.1061/(ASCE)0733-9437(1998)124:6(290).  Google Scholar

[23]

L. A. Richards, Capillary conduction of liquids through porous mediums, Physics, 1 (1931), 318-333.  doi: 10.1063/1.1745010.  Google Scholar

[24]

R. Srivastava and T.-C. Jim Yeh, Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Water Resour. Res., 27 (1991), 753-762.   Google Scholar

[25]

A. W. WarrickA. Islas and D. O. Lomen, An analytical solution to Richards' equation for time-varying infiltration, Water Resour. Res., 27 (1991), 763-766.  doi: 10.1029/91WR00310.  Google Scholar

Figure 4.1.  Ex. 1: $ u(z,t) $ (top) and $ \vartheta(u(z,t)) $ (bottom)
Figure 4.2.  Ex. 2: $ u(z,t) $ (top) and $ \vartheta(u(z,t)) $ (bottom)
Figure 4.3.  Ex. 1: $ \varphi(z,t) $ for $ Q $ in (4.9) (top) and for $ Q $ in (4.10) (bottom). Each of them is obtained from numerical approximation of (3.8), with $ \widetilde{u} \in \mathcal{W}^0_h $, $ h = (b-a)/96 $, and $ k = T/8 $
Figure 4.4.  Ex. 2: $ \varphi(z,t) $ for $ Q $ in (4.9) (left) and for $ Q $ in (4.10) (right). Each of them is obtained from numerical approximation of (3.8), with $ \widetilde{u} \in \mathcal{W}^0_h $, $ h = (b-a)/50 $, and $ k = T/8 $
Table 4.1.  Data for all examples with closed form solution
Ex. 1 Ex. 2
$ (a,b) $ $ (0,60) \text{cm} $ $ (0,2) \text{m} $
$ T $ $ 1000 \text{s} $ $ 0.5 \text{hr} $
$ \mathcal{B}_a u $ $ u(a,t) $ $ \kappa(u) \partial_z(u-z) \big|_{(a,t)} $
$ \mathcal{B}_b u $ $ u(b,t) $ $ u(b) $
$ g_a $ $ -65 \text{cm} $ $ -0.15 \text{m/hr} $
$ g_b $ $ 0 \text{cm} $ $ 0 \text{m} $
$ u_0 $ (4.6) & (4.7) (4.6) & (4.8)
$ \alpha $ $ 0.01 \text{cm}^{-1} $ $ 4 \text{m}^{-1} $
$ \kappa_s $ $ 0.001 \text{cm/s} $ $ 0.1 \text{m/hr} $
$ \theta_s $ $ 0.3 $ $ 0.6 $
$ \theta_r $ $ 0.08 $ $ 0.02 $
Ex. 1 Ex. 2
$ (a,b) $ $ (0,60) \text{cm} $ $ (0,2) \text{m} $
$ T $ $ 1000 \text{s} $ $ 0.5 \text{hr} $
$ \mathcal{B}_a u $ $ u(a,t) $ $ \kappa(u) \partial_z(u-z) \big|_{(a,t)} $
$ \mathcal{B}_b u $ $ u(b,t) $ $ u(b) $
$ g_a $ $ -65 \text{cm} $ $ -0.15 \text{m/hr} $
$ g_b $ $ 0 \text{cm} $ $ 0 \text{m} $
$ u_0 $ (4.6) & (4.7) (4.6) & (4.8)
$ \alpha $ $ 0.01 \text{cm}^{-1} $ $ 4 \text{m}^{-1} $
$ \kappa_s $ $ 0.001 \text{cm/s} $ $ 0.1 \text{m/hr} $
$ \theta_s $ $ 0.3 $ $ 0.6 $
$ \theta_r $ $ 0.08 $ $ 0.02 $
Table 4.2.  Ex. 1: Error of $ \vartheta(\widetilde{u}(T)) $ quantified in $ L^2(a,b) $-norm
$ M $ $ N $ FVEM-dG0 FVEM-dG1
12 1 58.0082e-03 14.5267e-03
24 1 58.1606e-03 15.1200e-03
48 1 58.1991e-03 15.2688e-03
96 1 58.2087e-03 15.3060e-03
12 2 32.5818e-03 0.6939e-03
24 2 32.5477e-03 0.8597e-03
48 2 32.5396e-03 0.9086e-03
96 2 32.5376e-03 0.9211e-03
12 4 17.3786e-03 0.2823e-03
24 4 17.2600e-03 0.0916e-03
48 4 17.2310e-03 0.1222e-03
96 4 17.2238e-03 0.1344e-03
12 8 9.0728e-03 0.3586e-03
24 8 8.9160e-03 0.0793e-03
48 8 8.8779e-03 0.0154e-03
96 8 8.8685e-03 0.0153e-03
$ M $ $ N $ FVEM-dG0 FVEM-dG1
12 1 58.0082e-03 14.5267e-03
24 1 58.1606e-03 15.1200e-03
48 1 58.1991e-03 15.2688e-03
96 1 58.2087e-03 15.3060e-03
12 2 32.5818e-03 0.6939e-03
24 2 32.5477e-03 0.8597e-03
48 2 32.5396e-03 0.9086e-03
96 2 32.5376e-03 0.9211e-03
12 4 17.3786e-03 0.2823e-03
24 4 17.2600e-03 0.0916e-03
48 4 17.2310e-03 0.1222e-03
96 4 17.2238e-03 0.1344e-03
12 8 9.0728e-03 0.3586e-03
24 8 8.9160e-03 0.0793e-03
48 8 8.8779e-03 0.0154e-03
96 8 8.8685e-03 0.0153e-03
Table 4.3.  Ex. 2: Error of $ \vartheta(\widetilde{u}(T)) $ quantified in $ L^2(a,b) $-norm
$ M $ $ N $ FVEM-dG0 FVEM-dG1
5 4 4.9527e-02 5.0631e-02
10 4 2.0846e-02 1.8016e-02
20 4 1.0534e-02 0.4128e-02
40 4 0.8508e-02 0.0992e-02
5 8 4.9824e-02 5.0641e-02
10 8 1.8868e-02 1.7991e-02
20 8 0.6907e-02 0.4088e-03
40 8 0.4671e-02 0.0965e-02
5 16 5.0159e-02 5.0645e-02
10 16 1.8208e-02 1.7988e-02
20 16 0.5197e-02 0.4079e-02
40 16 0.2637e-02 0.0959e-02
5 32 5.0382e-02 5.0647e-02
10 32 1.8027e-02 1.7988e-02
20 32 0.4504e-02 0.4077e-02
40 32 0.1646e-02 0.0958e-02
$ M $ $ N $ FVEM-dG0 FVEM-dG1
5 4 4.9527e-02 5.0631e-02
10 4 2.0846e-02 1.8016e-02
20 4 1.0534e-02 0.4128e-02
40 4 0.8508e-02 0.0992e-02
5 8 4.9824e-02 5.0641e-02
10 8 1.8868e-02 1.7991e-02
20 8 0.6907e-02 0.4088e-03
40 8 0.4671e-02 0.0965e-02
5 16 5.0159e-02 5.0645e-02
10 16 1.8208e-02 1.7988e-02
20 16 0.5197e-02 0.4079e-02
40 16 0.2637e-02 0.0959e-02
5 32 5.0382e-02 5.0647e-02
10 32 1.8027e-02 1.7988e-02
20 32 0.4504e-02 0.4077e-02
40 32 0.1646e-02 0.0958e-02
Table 4.4.  True Value of Quantities of Interest
Ex. 1 Ex. 2
$ Q $ in (4.9) 0.231831624739998887 0.129975678959476710
$ Q $ in (4.10) 0.221196137487056291 0.111225678959476525
Ex. 1 Ex. 2
$ Q $ in (4.9) 0.231831624739998887 0.129975678959476710
$ Q $ in (4.10) 0.221196137487056291 0.111225678959476525
Table 4.5.  Ex. 1: Performance of the Error Estimator in Theorem 3.1 for $ Q $ in (4.9)
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
12 1 -5.5601e-05 6.6298e-03 2.6571e-04 2.2485e-05 6.8623e-03 1.149
24 1 -1.3843e-05 6.6974e-03 1.3440e-04 5.3567e-06 6.8233e-03 1.139
48 1 -3.4568e-06 6.7142e-03 6.7378e-05 1.3220e-06 6.7794e-03 1.131
96 1 -8.6396e-07 6.7184e-03 3.3706e-05 3.2943e-07 6.7515e-03 1.126
12 2 -4.4365e-05 3.8268e-03 7.1941e-05 3.6142e-05 3.8906e-03 1.120
24 2 -1.1046e-05 3.8548e-03 3.9509e-05 8.5556e-06 3.8918e-03 1.120
48 2 -2.7579e-06 3.8617e-03 2.0874e-05 2.1051e-06 3.8820e-03 1.120
96 2 -6.8914e-07 3.8635e-03 1.0730e-05 5.2410e-07 3.8740e-03 1.115
12 4 -3.9779e-05 2.0010e-03 2.5003e-05 4.3906e-05 2.0310e-03 1.067
24 4 -9.9212e-06 2.0128e-03 1.2165e-06 1.0291e-05 2.0254e-03 1.069
48 4 -2.4774e-06 2.0158e-03 6.9258e-06 2.5084e-06 2.0227e-03 1.069
96 4 -6.1911e-07 2.0165e-03 3.7657e-06 6.2266e-07 2.0203e-03 1.068
12 8 -3.8042e-05 1.0175e-03 1.9493e-05 4.6704e-05 1.0457e-03 1.037
24 8 -9.4764e-06 1.0227e-03 5.0975e-06 1.1390e-05 1.0297e-03 1.036
48 8 -2.3676e-06 1.0240e-03 2.3602e-06 2.7559e-06 1.0268e-03 1.036
96 8 -5.9172e-07 1.0244e-03 1.3885e-06 6.8063e-07 1.0258e-03 1.036
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
12 1 -5.5601e-05 6.6298e-03 2.6571e-04 2.2485e-05 6.8623e-03 1.149
24 1 -1.3843e-05 6.6974e-03 1.3440e-04 5.3567e-06 6.8233e-03 1.139
48 1 -3.4568e-06 6.7142e-03 6.7378e-05 1.3220e-06 6.7794e-03 1.131
96 1 -8.6396e-07 6.7184e-03 3.3706e-05 3.2943e-07 6.7515e-03 1.126
12 2 -4.4365e-05 3.8268e-03 7.1941e-05 3.6142e-05 3.8906e-03 1.120
24 2 -1.1046e-05 3.8548e-03 3.9509e-05 8.5556e-06 3.8918e-03 1.120
48 2 -2.7579e-06 3.8617e-03 2.0874e-05 2.1051e-06 3.8820e-03 1.120
96 2 -6.8914e-07 3.8635e-03 1.0730e-05 5.2410e-07 3.8740e-03 1.115
12 4 -3.9779e-05 2.0010e-03 2.5003e-05 4.3906e-05 2.0310e-03 1.067
24 4 -9.9212e-06 2.0128e-03 1.2165e-06 1.0291e-05 2.0254e-03 1.069
48 4 -2.4774e-06 2.0158e-03 6.9258e-06 2.5084e-06 2.0227e-03 1.069
96 4 -6.1911e-07 2.0165e-03 3.7657e-06 6.2266e-07 2.0203e-03 1.068
12 8 -3.8042e-05 1.0175e-03 1.9493e-05 4.6704e-05 1.0457e-03 1.037
24 8 -9.4764e-06 1.0227e-03 5.0975e-06 1.1390e-05 1.0297e-03 1.036
48 8 -2.3676e-06 1.0240e-03 2.3602e-06 2.7559e-06 1.0268e-03 1.036
96 8 -5.9172e-07 1.0244e-03 1.3885e-06 6.8063e-07 1.0258e-03 1.036
Table 4.6.  Ex. 1: Performance of the Error Estimator in Theorem 3.1 for $ Q $ in (4.10)
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
12 1 -7.6641e-05 -5.9702e-03 6.5348e-06 2.7273e-05 -6.0131e-03 1.290
24 1 -1.9082e-05 -5.9818e-03 1.6059e-06 6.7751e-06 -5.9925e-03 1.290
48 1 -4.7653e-06 -5.9847e-03 3.9971e-07 1.6907e-06 -5.9874e-03 1.290
96 1 -1.1910e-06 -5.9854e-03 9.9817e-08 4.2248e-07 -5.9861e-03 1.290
12 2 -7.4772e-05 -3.0291e-03 1.6147e-05 2.5436e-05 -3.0623e-03 1.123
24 2 -1.8616e-05 -3.0342e-03 3.9483e-06 6.3250e-06 -3.0426e-03 1.123
48 2 -4.6488e-06 -3.0355e-03 9.8092e-07 1.5784e-06 -3.0376e-03 1.123
96 2 -1.1619e-06 -3.0358e-03 2.4483e-07 3.9441e-07 -3.0363e-03 1.123
12 4 -7.4397e-05 -1.5584e-03 2.3449e-05 2.4137e-05 -1.5852e-03 1.057
24 4 -1.8523e-05 -1.5609e-03 5.7189e-06 6.0235e-06 -1.5677e-03 1.057
48 4 -4.6257e-06 -1.5615e-03 1.4158e-06 1.5036e-06 -1.5632e-03 1.057
96 4 -1.1561e-06 -1.5617e-03 3.5295e-07 3.7571e-07 -1.5621e-03 1.057
12 8 -7.4326e-05 -7.9524e-04 2.7824e-05 2.3365e-05 -8.1837e-04 1.027
24 8 -1.8504e-05 -7.9647e-04 6.8634e-06 5.8679e-06 -8.0225e-04 1.027
48 8 -4.6210e-06 -7.9678e-04 1.6962e-06 1.4661e-06 -7.9824e-04 1.027
96 8 -1.1549e-06 -7.9686e-04 4.2208e-07 3.6638e-07 -7.9722e-04 1.027
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
12 1 -7.6641e-05 -5.9702e-03 6.5348e-06 2.7273e-05 -6.0131e-03 1.290
24 1 -1.9082e-05 -5.9818e-03 1.6059e-06 6.7751e-06 -5.9925e-03 1.290
48 1 -4.7653e-06 -5.9847e-03 3.9971e-07 1.6907e-06 -5.9874e-03 1.290
96 1 -1.1910e-06 -5.9854e-03 9.9817e-08 4.2248e-07 -5.9861e-03 1.290
12 2 -7.4772e-05 -3.0291e-03 1.6147e-05 2.5436e-05 -3.0623e-03 1.123
24 2 -1.8616e-05 -3.0342e-03 3.9483e-06 6.3250e-06 -3.0426e-03 1.123
48 2 -4.6488e-06 -3.0355e-03 9.8092e-07 1.5784e-06 -3.0376e-03 1.123
96 2 -1.1619e-06 -3.0358e-03 2.4483e-07 3.9441e-07 -3.0363e-03 1.123
12 4 -7.4397e-05 -1.5584e-03 2.3449e-05 2.4137e-05 -1.5852e-03 1.057
24 4 -1.8523e-05 -1.5609e-03 5.7189e-06 6.0235e-06 -1.5677e-03 1.057
48 4 -4.6257e-06 -1.5615e-03 1.4158e-06 1.5036e-06 -1.5632e-03 1.057
96 4 -1.1561e-06 -1.5617e-03 3.5295e-07 3.7571e-07 -1.5621e-03 1.057
12 8 -7.4326e-05 -7.9524e-04 2.7824e-05 2.3365e-05 -8.1837e-04 1.027
24 8 -1.8504e-05 -7.9647e-04 6.8634e-06 5.8679e-06 -8.0225e-04 1.027
48 8 -4.6210e-06 -7.9678e-04 1.6962e-06 1.4661e-06 -7.9824e-04 1.027
96 8 -1.1549e-06 -7.9686e-04 4.2208e-07 3.6638e-07 -7.9722e-04 1.027
Table 4.7.  Ex. 2: Performance of the Error Estimator in Theorem 3.1 for $ Q $ in (4.9)
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
5 4 -3.6102e-05 -6.2164e-08 -3.6456e-07 1.0290e-02 1.0253e-02 0.977
10 4 -1.3302e-05 3.3879e-09 -4.6917e-09 3.9111e-03 3.8978e-03 0.993
20 4 -5.7876e-06 9.9041e-10 -2.2824e-10 1.0090e-03 1.0032e-03 0.998
40 4 -2.7053e-06 5.6955e-10 -3.8568e-11 2.4965e-04 2.4695e-04 0.999
5 8 -3.6107e-05 -4.3218e-08 -3.5113e-07 1.1133e-02 1.1096e-02 0.974
10 8 -1.3302e-05 1.8737e-10 -4.7768e-10 4.3243e-03 4.3110e-03 0.993
20 8 -5.7876e-06 1.9063e-11 -6.0320e-12 1.0842e-03 1.0784e-03 0.998
40 8 -2.7053e-06 5.5338e-12 -4.5001e-13 2.6644e-04 2.6373e-02 0.999
5 16 -3.6121e-05 6.8028e-08 -4.5382e-07 1.1595e-02 1.1559e-02 0.973
10 16 -1.3302e-05 1.6801e-11 -8.8583e-11 4.5811e-03 4.5678e-03 0.992
20 16 -5.7876e-06 5.2323e-13 -3.2271e-13 1.1290e-03 1.1232e-03 0.998
40 16 -2.7053e-06 4.0884e-14 -2.5154e-14 2.7647e-04 2.7376e-04 0.999
5 32 -3.6089e-05 -4.4295e-08 4.5726e-08 1.1837e-02 1.1801e-02 0.972
10 32 -1.3302e-05 2.5330e-12 -2.9891e-11 4.7280e-03 4.7144e-03 0.992
20 32 -5.7876e-06 2.6583e-14 -9.4480e-14 1.1545e-03 1.1487e-03 0.998
40 32 -2.7053e-06 -1.0436e-14 -1.8777e-14 2.8211e-04 2.7941e-04 0.999
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
5 4 -3.6102e-05 -6.2164e-08 -3.6456e-07 1.0290e-02 1.0253e-02 0.977
10 4 -1.3302e-05 3.3879e-09 -4.6917e-09 3.9111e-03 3.8978e-03 0.993
20 4 -5.7876e-06 9.9041e-10 -2.2824e-10 1.0090e-03 1.0032e-03 0.998
40 4 -2.7053e-06 5.6955e-10 -3.8568e-11 2.4965e-04 2.4695e-04 0.999
5 8 -3.6107e-05 -4.3218e-08 -3.5113e-07 1.1133e-02 1.1096e-02 0.974
10 8 -1.3302e-05 1.8737e-10 -4.7768e-10 4.3243e-03 4.3110e-03 0.993
20 8 -5.7876e-06 1.9063e-11 -6.0320e-12 1.0842e-03 1.0784e-03 0.998
40 8 -2.7053e-06 5.5338e-12 -4.5001e-13 2.6644e-04 2.6373e-02 0.999
5 16 -3.6121e-05 6.8028e-08 -4.5382e-07 1.1595e-02 1.1559e-02 0.973
10 16 -1.3302e-05 1.6801e-11 -8.8583e-11 4.5811e-03 4.5678e-03 0.992
20 16 -5.7876e-06 5.2323e-13 -3.2271e-13 1.1290e-03 1.1232e-03 0.998
40 16 -2.7053e-06 4.0884e-14 -2.5154e-14 2.7647e-04 2.7376e-04 0.999
5 32 -3.6089e-05 -4.4295e-08 4.5726e-08 1.1837e-02 1.1801e-02 0.972
10 32 -1.3302e-05 2.5330e-12 -2.9891e-11 4.7280e-03 4.7144e-03 0.992
20 32 -5.7876e-06 2.6583e-14 -9.4480e-14 1.1545e-03 1.1487e-03 0.998
40 32 -2.7053e-06 -1.0436e-14 -1.8777e-14 2.8211e-04 2.7941e-04 0.999
Table 4.8.  Ex. 2: Performance of the Error Estimator in Theorem 3.1 for $ Q $ in (4.10)
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
5 4 -3.6102e-05 -4.6874e-03 -7.6435e-08 7.2999e-03 2.5763e-03 0.921
10 4 -1.3302e-05 -4.6875e-03 -8.4587e-10 3.3265e-03 -1.3743e-03 1.026
20 4 -5.7876e-06 -4.6875e-03 -3.4359e-11 9.8963e-04 -3.7036e-03 1.001
40 4 -2.7054e-06 -4.6875e-03 -4.2724e-12 2.5707e-04 -4.4331e-03 1.000
5 8 -3.6102e-05 -2.3437e-03 -4.6414e-08 7.2458e-03 4.8659e-03 0.950
10 8 -1.3302e-05 -2.3437e-03 -6.5517e-11 3.5887e-03 1.2316e-03 0.961
20 8 -5.7876e-06 -2.3437e-03 -7.0707e-13 1.0899e-03 -1.2596e-03 1.004
40 8 -2.7054e-06 -2.3437e-03 -4.9054e-14 2.8339e-04 -2.0631e-03 1.000
5 16 -3.6105e-05 -1.1719e-03 -7.9249e-08 7.1892e-03 5.9812e-03 0.956
10 16 -1.3302e-05 -1.1719e-03 -1.0129e-11 3.7626e-03 2.5774e-03 0.977
20 16 -5.7876e-06 -1.1719e-03 -6.6506e-14 1.1637e-03 -1.3957e-05 1.823
40 16 -2.7054e-06 -1.1719e-03 -1.1789e-14 3.0287e-04 -8.7171e-04 1.001
5 32 -3.6103e-05 -5.8593e-04 -2.1637e-08 7.1541e-03 6.5320e-03 0.959
10 32 -1.3302e-06 -5.8594e-04 -3.1392e-12 3.8643e-03 3.2650e-03 0.979
20 32 -5.7876e-06 -5.8594e-04 -4.6973e-14 1.2132e-03 6.2149e-04 0.988
40 32 -2.7054e-06 -5.8594e-04 -1.1491e-14 3.1628e-04 -2.7236e-04 1.002
$ M $ $ N $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $ Err. Est. Eff.
5 4 -3.6102e-05 -4.6874e-03 -7.6435e-08 7.2999e-03 2.5763e-03 0.921
10 4 -1.3302e-05 -4.6875e-03 -8.4587e-10 3.3265e-03 -1.3743e-03 1.026
20 4 -5.7876e-06 -4.6875e-03 -3.4359e-11 9.8963e-04 -3.7036e-03 1.001
40 4 -2.7054e-06 -4.6875e-03 -4.2724e-12 2.5707e-04 -4.4331e-03 1.000
5 8 -3.6102e-05 -2.3437e-03 -4.6414e-08 7.2458e-03 4.8659e-03 0.950
10 8 -1.3302e-05 -2.3437e-03 -6.5517e-11 3.5887e-03 1.2316e-03 0.961
20 8 -5.7876e-06 -2.3437e-03 -7.0707e-13 1.0899e-03 -1.2596e-03 1.004
40 8 -2.7054e-06 -2.3437e-03 -4.9054e-14 2.8339e-04 -2.0631e-03 1.000
5 16 -3.6105e-05 -1.1719e-03 -7.9249e-08 7.1892e-03 5.9812e-03 0.956
10 16 -1.3302e-05 -1.1719e-03 -1.0129e-11 3.7626e-03 2.5774e-03 0.977
20 16 -5.7876e-06 -1.1719e-03 -6.6506e-14 1.1637e-03 -1.3957e-05 1.823
40 16 -2.7054e-06 -1.1719e-03 -1.1789e-14 3.0287e-04 -8.7171e-04 1.001
5 32 -3.6103e-05 -5.8593e-04 -2.1637e-08 7.1541e-03 6.5320e-03 0.959
10 32 -1.3302e-06 -5.8594e-04 -3.1392e-12 3.8643e-03 3.2650e-03 0.979
20 32 -5.7876e-06 -5.8594e-04 -4.6973e-14 1.2132e-03 6.2149e-04 0.988
40 32 -2.7054e-06 -5.8594e-04 -1.1491e-14 3.1628e-04 -2.7236e-04 1.002
Table 4.9.  Ex. 1: Decomposition of Error according to Theorem 3.2 for $ Q $ in (4.9)
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
12 1 6.9104e-03 7.5953e-06 0
24 1 6.8352e-03 1.8846e-06 6.9389e-18
48 1 6.7824e-03 4.6724e-07 3.4694e-18
96 1 6.7523e-03 1.1620e-07 6.9389e-18
12 2 3.8392e-03 6.1045e-06 8.9582e-05
24 2 3.8789e-03 1.5228e-06 2.2428e-05
48 2 3.8787e-03 3.7986e-07 5.6090e-06
96 2 3.8732e-03 9.4827e-08 1.4024e-06
12 4 1.9444e-03 5.6811e-06 1.1982e-04
24 4 2.0038e-03 1.4190e-06 3.0033e-05
48 4 2.0173e-03 3.5462e-07 7.5129e-06
96 4 2.0189e-03 8.8625e-08 1.8785e-06
12 8 9.4839e-04 5.5176e-06 1.2981e-04
24 8 1.0052e-03 1.3769e-06 3.2575e-05
48 8 1.0206e-03 3.4421e-07 8.1499e-06
96 8 1.0243e-03 8.6050e-08 2.0378e-06
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
12 1 6.9104e-03 7.5953e-06 0
24 1 6.8352e-03 1.8846e-06 6.9389e-18
48 1 6.7824e-03 4.6724e-07 3.4694e-18
96 1 6.7523e-03 1.1620e-07 6.9389e-18
12 2 3.8392e-03 6.1045e-06 8.9582e-05
24 2 3.8789e-03 1.5228e-06 2.2428e-05
48 2 3.8787e-03 3.7986e-07 5.6090e-06
96 2 3.8732e-03 9.4827e-08 1.4024e-06
12 4 1.9444e-03 5.6811e-06 1.1982e-04
24 4 2.0038e-03 1.4190e-06 3.0033e-05
48 4 2.0173e-03 3.5462e-07 7.5129e-06
96 4 2.0189e-03 8.8625e-08 1.8785e-06
12 8 9.4839e-04 5.5176e-06 1.2981e-04
24 8 1.0052e-03 1.3769e-06 3.2575e-05
48 8 1.0206e-03 3.4421e-07 8.1499e-06
96 8 1.0243e-03 8.6050e-08 2.0378e-06
Table 4.10.  Ex. 1: Decomposition of Error according to Theorem 3.2 for $ Q $ in (4.10)
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
12 1 -5.9364e-03 0 0
24 1 -5.9734e-03 0 0
48 1 -5.9826e-03 0 0
96 1 -5.9849e-03 0 0
12 2 -3.0501e-03 2.5710e-06 5.9998e-05
24 2 -3.0397e-03 6.4271e-07 1.5112e-05
48 2 -3.0369e-03 1.6068e-07 3.7850e-06
96 2 -3.0361e-03 4.0169e-08 9.4670e-07
12 4 -1.6024e-03 4.3468e-06 8.7251e-05
24 4 -1.5722e-03 1.0865e-06 2.1970e-05
48 4 -1.5644e-03 2.7161e-07 5.5023e-06
96 4 -1.5624e-03 6.7902e-08 1.3762e-06
12 8 -8.5030e-04 5.5588e-06 1.0069e-04
24 8 -8.1048e-04 1.3899e-06 2.5351e-05
48 8 -8.0031e-04 3.4750e-07 6.3488e-06
96 8 -7.9774e-04 8.6875e-08 1.5879e-06
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
12 1 -5.9364e-03 0 0
24 1 -5.9734e-03 0 0
48 1 -5.9826e-03 0 0
96 1 -5.9849e-03 0 0
12 2 -3.0501e-03 2.5710e-06 5.9998e-05
24 2 -3.0397e-03 6.4271e-07 1.5112e-05
48 2 -3.0369e-03 1.6068e-07 3.7850e-06
96 2 -3.0361e-03 4.0169e-08 9.4670e-07
12 4 -1.6024e-03 4.3468e-06 8.7251e-05
24 4 -1.5722e-03 1.0865e-06 2.1970e-05
48 4 -1.5644e-03 2.7161e-07 5.5023e-06
96 4 -1.5624e-03 6.7902e-08 1.3762e-06
12 8 -8.5030e-04 5.5588e-06 1.0069e-04
24 8 -8.1048e-04 1.3899e-06 2.5351e-05
48 8 -8.0031e-04 3.4750e-07 6.3488e-06
96 8 -7.9774e-04 8.6875e-08 1.5879e-06
Table 4.11.  Ex. 2: Decomposition of Error according to Theorem 3.2 for $ Q $ in (4.9)
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
5 4 1.0289e-02 -7.6351e-08 3.0166e-07
10 4 3.9111e-03 -5.9250e-10 4.6158e-09
20 4 1.0090e-03 -1.7676e-11 1.7694e-10
40 4 2.4965e-04 -1.9490e-12 2.1193e-11
5 8 1.1134e-02 7.3138e-08 -1.3466e-06
10 8 4.3243e-03 -6.0938e-11 1.1118e-09
20 8 1.0842e-03 -4.8331e-13 1.1840e-11
40 8 2.6644e-04 -2.7270e-14 6.6398e-13
5 16 1.1601e-02 8.7291e-08 -5.8431e-06
10 16 4.5811e-03 -1.1412e-11 3.7774e-10
20 16 1.1290e-03 -3.7491e-14 1.0044e-12
40 16 2.7647e-04 -4.7254e-15 1.7396e-14
5 32 1.1833e-02 2.1590e-08 3.9613e-06
10 32 4.7277e-03 -3.8970e-12 1.8867e-10
20 32 1.1545e-03 -1.9623e-14 1.5073e-13
40 32 2.8211e-04 -4.4201e-15 -2.7756e-17
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
5 4 1.0289e-02 -7.6351e-08 3.0166e-07
10 4 3.9111e-03 -5.9250e-10 4.6158e-09
20 4 1.0090e-03 -1.7676e-11 1.7694e-10
40 4 2.4965e-04 -1.9490e-12 2.1193e-11
5 8 1.1134e-02 7.3138e-08 -1.3466e-06
10 8 4.3243e-03 -6.0938e-11 1.1118e-09
20 8 1.0842e-03 -4.8331e-13 1.1840e-11
40 8 2.6644e-04 -2.7270e-14 6.6398e-13
5 16 1.1601e-02 8.7291e-08 -5.8431e-06
10 16 4.5811e-03 -1.1412e-11 3.7774e-10
20 16 1.1290e-03 -3.7491e-14 1.0044e-12
40 16 2.7647e-04 -4.7254e-15 1.7396e-14
5 32 1.1833e-02 2.1590e-08 3.9613e-06
10 32 4.7277e-03 -3.8970e-12 1.8867e-10
20 32 1.1545e-03 -1.9623e-14 1.5073e-13
40 32 2.8211e-04 -4.4201e-15 -2.7756e-17
Table 4.12.  Ex. 2: Decomposition of Error according to Theorem 3.2 for $ Q $ in (4.10)
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
5 4 2.6123e-03 -7.7482e-09 1.1954e-07
10 4 -1.3610e-03 -5.0964e-11 1.2934e-09
20 4 -3.6979e-03 -1.4624e-12 4.4327e-11
40 4 -4.4304e-03 -1.6102e-13 5.1326e-12
5 8 4.9021e-03 6.9429e-09 -1.0375e-07
10 8 1.2450e-03 -5.4195e-12 2.1318e-10
20 8 -1.2539e-03 -4.3805e-14 1.8653e-12
40 8 -2.0604e-03 -3.6394e-15 9.6166e-14
5 16 6.0183e-03 1.5114e-08 -1.0262e-06
10 16 2.5907e-03 -1.0270e-12 5.4566e-11
20 16 -8.1695e-06 -9.7700e-15 1.0732e-13
40 16 -8.6900e-04 -2.1094e-15 1.2386e-15
5 32 6.5684e-03 3.4467e-09 -3.0913e-07
10 32 3.2783e-03 -3.6607e-13 2.2585e-11
20 32 6.2727e-04 -8.8020e-15 1.0911e-14
40 32 -2.6966e-04 -2.1545e-15 -4.1980e-16
$ M $ $ N $ $ \mathcal{E}_1 $ $ \mathcal{E}_2 $ $ \mathcal{E}_3 $
5 4 2.6123e-03 -7.7482e-09 1.1954e-07
10 4 -1.3610e-03 -5.0964e-11 1.2934e-09
20 4 -3.6979e-03 -1.4624e-12 4.4327e-11
40 4 -4.4304e-03 -1.6102e-13 5.1326e-12
5 8 4.9021e-03 6.9429e-09 -1.0375e-07
10 8 1.2450e-03 -5.4195e-12 2.1318e-10
20 8 -1.2539e-03 -4.3805e-14 1.8653e-12
40 8 -2.0604e-03 -3.6394e-15 9.6166e-14
5 16 6.0183e-03 1.5114e-08 -1.0262e-06
10 16 2.5907e-03 -1.0270e-12 5.4566e-11
20 16 -8.1695e-06 -9.7700e-15 1.0732e-13
40 16 -8.6900e-04 -2.1094e-15 1.2386e-15
5 32 6.5684e-03 3.4467e-09 -3.0913e-07
10 32 3.2783e-03 -3.6607e-13 2.2585e-11
20 32 6.2727e-04 -8.8020e-15 1.0911e-14
40 32 -2.6966e-04 -2.1545e-15 -4.1980e-16
[1]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014

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