doi: 10.3934/dcdss.2022060
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Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

Dedicated to Professor Georg Hetzer on the occasion of his 75th birthday

Received  July 2021 Revised  January 2022 Early access March 2022

Fund Project: This work is partially supported by the US National Science Foundation under grant number DMS-1912626

This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.

Citation: Thi-Thao-Phuong Hoang. Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022060
References:
[1]

T. ArbogastL. C. CowsarM. F. Wheeler and I. Yotov, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37 (2000), 1295-1315.  doi: 10.1137/S0036142996308447.

[2]

T. ArbogastG. PenchevaM. F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6 (2007), 319-346.  doi: 10.1137/060662587.

[3]

E. Audusse, P. Dreyfus and B. Merlet, Optimized Schwarz waveform relaxation for the primitive equations of the ocean, SIAM J. Sci. Comput., 32 (20100, 2908–2936. doi: 10.1137/090770059.

[4]

D. BennequinM. J. GanderL. Gouarin and L. Halpern, Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions, Numerische Mathematik, 134 (2016), 513-567.  doi: 10.1007/s00211-015-0784-8.

[5]

D. BennequinM. J. Gander and L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp., 78 (2009), 185-223.  doi: 10.1090/S0025-5718-08-02145-5.

[6]

E. BlayoL. Debreu and F. Lemarié, Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients. Part 1: The constant coefficients case, Electron. Trans. Numer. Anal., 40 (2013), 170-186. 

[7]

E. BlayoL. Halpern and C. Japhet, Optimized Schwarz waveform relaxation algorithms with nonconforming time discretization for coupling convection-diffusion problems with discontinuous coefficients, in Domain Decomposition Methods in Science and Engineering XVI, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 55 (2007), 267-274.  doi: 10.1007/978-3-540-34469-8_31.

[8]

W. M. Boon, D. Glaser, R. Helmig and I. Yotov, Flux-mortar mixed finite element methods on non-matching grids, arXiv: 2008.09372.

[9]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Elements Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[10]

F. BrunnerF. A. Radu and P. Knabner, Analysis of an upwind-mixed hybrid finite element method for transport problems, SIAM J. Numer. Anal., 52 (2014), 83-102.  doi: 10.1137/130908191.

[11]

C. Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations, SIAM J. Numer. Anal., 35 (1998), 1709-1724.  doi: 10.1137/S0036142993259421.

[12]

R. E. Ewing, R. D. Lazarov, T. F. Russell and P. S. Vassilevski, Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, PA, (1990), 98–114.

[13]

M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal., 44 (2006), 699-731.  doi: 10.1137/S0036142903425409.

[14]

M. J. Gander, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1999), 125-145.  doi: 10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4.

[15]

M. J. Gander, Schwarz methods over the course of time, Electron. Trans. Numer. Anal., 31 (2008), 228-255. 

[16]

M. J. Gander and L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing, Math. Comp., 74 (2005), 153-176.  doi: 10.1090/S0025-5718-04-01635-7.

[17]

M. J. Gander and L. Halpern, Optimized Schwarz waveform relaxation for advection reaction diffusion problems, SIAM J. Numer. Anal., 45 (2007), 666-697.  doi: 10.1137/050642137.

[18]

M. J. GanderL. Halpern and M. Kern, A Schwarz waveform relaxation method for advection-diffusion-reaction problems with discontinuous coefficients and non-matching grids, Domain Decomposition Methods in Science and Engineering XVI, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 55 (2007), 283-290.  doi: 10.1007/978-3-540-34469-8_33.

[19]

M. J. Gander, L. Halpern and F. Nataf, Optimal convergence for overlapping and nonoverlapping Schwarz waveform relaxation, Eleventh International Conference on Domain Decomposition Methods (London, 1998), DDM.org, Augsburg, (1999), 27–36.

[20]

M. J. GanderL. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), 1643-1681.  doi: 10.1137/S003614290139559X.

[21]

M. J. GanderC. JaphetY. Maday and F. Nataf, A new cement to glue nonconforming grids with Robin interface conditions: The finite element case, Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 40 (2005), 259-266.  doi: 10.1007/3-540-26825-1_24.

[22]

M. J. Gander and C. Rohde, Overlapping Schwarz waveform relaxation for convection dominated nonlinear conservation laws, SIAM J. Sci. Comp., 27 (2005), 415-439.  doi: 10.1137/030601090.

[23]

M. J. Gander and A. M. Stuart, Space-time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comp., 19 (1998), 2014-2031.  doi: 10.1137/S1064827596305337.

[24]

E. Giladi and H. B. Keller, Space-time domain decomposition for parabolic problems, Numer. Math., 93 (2002), 279-313.  doi: 10.1007/s002110100345.

[25]

F. HaeberleinL. Halpern and A. Michel, Newton-Schwarz optimised waveform relaxation Krylov accelerators for nonlinear reactive transport, Domain Decomposition Methods in Science and Engineering XX, Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 91 (2013), 387-394.  doi: 10.1007/978-3-642-35275-1_45.

[26]

L. Halpern, Schwarz waveform relaxation algorithms, Domain Decomposition Methods in Science and Engineering XVII, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 60 (2008), 57-68.  doi: 10.1007/978-3-540-75199-1_5.

[27]

L. Halpern, C. Japhet and P. Omnes, Nonconforming in time domain decomposition method for porous media applications, Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010, Lisbon, Portugal.

[28]

L. HalpernC. Japhet and J. Szeftel, Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping for heterogeneous problems, SIAM J. Numer. Anal., 50 (2012), 2588-2611.  doi: 10.1137/120865033.

[29]

R. D. Haynes and R. D. Russell, A Schwarz waveform moving mesh method, SIAM J. Sci. Comp., 29 (2007), 656-673.  doi: 10.1137/050631549.

[30]

T. T. P. Hoang, Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations, (2021), submitted.

[31]

T. T. P. HoangJ. JaffréC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition methods for diffusion problems in mixed formulations, SIAM J. Numer. Anal., 51 (2013), 3532-3559.  doi: 10.1137/130914401.

[32]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Ventcel conditions with mixed formulations for flow in porous media, Domain Decomposition Methods in Science and Engineering XXII, Lect. Notes Comput. Sci. Eng., Springer, Cham, 104 (2016), 531-540. 

[33]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition for reduced fracture models in mixed formulation, SIAM J. Numer. Anal., 54 (2016), 288-316.  doi: 10.1137/15M1009651.

[34]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition for advection-diffusion problems in mixed formulations, Math. Comput. Simulat., 137 (2017), 366-389.  doi: 10.1016/j.matcom.2016.11.002.

[35]

T. T. P. Hoang and H. Lee, A global-in-time domain decomposition method for the coupled nonlinear stokes and darcy flows, J. Sci. Comput., 87 (2021), 22 pp. doi: 10.1007/s10915-021-01422-1.

[36]

C. Japhet, Optimized Krylov-Ventcell method. Application to convection-diffusion problems, Proceedings of the 9th International Conference on Domain Decomposition Methods, Bergen, Norway, (1998), 382–389.

[37]

E. LelarasmeeA. E. Ruehli and A. L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. CAD IC Syst., 1 (1982), 131-145.  doi: 10.1109/TCAD.1982.1270004.

[38]

V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math., 52 (2005), 401-428.  doi: 10.1016/j.apnum.2004.08.022.

[39]

F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models Methods Appl. Sci., 5 (1995), 67-93.  doi: 10.1142/S021820259500005X.

[40]

F. A. RaduN. SuciuJ. HoffmannA. VogelO. KolditzC.-H. Park and S. Attinger, Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study, Adv. Water Resources, 34 (2011), 47-61.  doi: 10.1016/j.advwatres.2010.09.012.

[41]

J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 2 (1991), 523-639. 

[42]

H. A. Schwarz, Ueber einen Grenzübergang durch alternirendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15 (1870), 272-286. 

[43]

A. D. Ventcel', On boundary conditions for multidimensional diffusion processes, Theory Probab. Appl., 4 (1959), 164-177.  doi: 10.1137/1104014.

show all references

References:
[1]

T. ArbogastL. C. CowsarM. F. Wheeler and I. Yotov, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37 (2000), 1295-1315.  doi: 10.1137/S0036142996308447.

[2]

T. ArbogastG. PenchevaM. F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6 (2007), 319-346.  doi: 10.1137/060662587.

[3]

E. Audusse, P. Dreyfus and B. Merlet, Optimized Schwarz waveform relaxation for the primitive equations of the ocean, SIAM J. Sci. Comput., 32 (20100, 2908–2936. doi: 10.1137/090770059.

[4]

D. BennequinM. J. GanderL. Gouarin and L. Halpern, Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions, Numerische Mathematik, 134 (2016), 513-567.  doi: 10.1007/s00211-015-0784-8.

[5]

D. BennequinM. J. Gander and L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp., 78 (2009), 185-223.  doi: 10.1090/S0025-5718-08-02145-5.

[6]

E. BlayoL. Debreu and F. Lemarié, Toward an optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients. Part 1: The constant coefficients case, Electron. Trans. Numer. Anal., 40 (2013), 170-186. 

[7]

E. BlayoL. Halpern and C. Japhet, Optimized Schwarz waveform relaxation algorithms with nonconforming time discretization for coupling convection-diffusion problems with discontinuous coefficients, in Domain Decomposition Methods in Science and Engineering XVI, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 55 (2007), 267-274.  doi: 10.1007/978-3-540-34469-8_31.

[8]

W. M. Boon, D. Glaser, R. Helmig and I. Yotov, Flux-mortar mixed finite element methods on non-matching grids, arXiv: 2008.09372.

[9]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Elements Methods, Springer Series in Computational Mathematics, 15. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[10]

F. BrunnerF. A. Radu and P. Knabner, Analysis of an upwind-mixed hybrid finite element method for transport problems, SIAM J. Numer. Anal., 52 (2014), 83-102.  doi: 10.1137/130908191.

[11]

C. Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations, SIAM J. Numer. Anal., 35 (1998), 1709-1724.  doi: 10.1137/S0036142993259421.

[12]

R. E. Ewing, R. D. Lazarov, T. F. Russell and P. S. Vassilevski, Local refinement via domain decomposition techniques for mixed finite element methods with rectangular Raviart-Thomas elements, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, PA, (1990), 98–114.

[13]

M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal., 44 (2006), 699-731.  doi: 10.1137/S0036142903425409.

[14]

M. J. Gander, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1999), 125-145.  doi: 10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4.

[15]

M. J. Gander, Schwarz methods over the course of time, Electron. Trans. Numer. Anal., 31 (2008), 228-255. 

[16]

M. J. Gander and L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing, Math. Comp., 74 (2005), 153-176.  doi: 10.1090/S0025-5718-04-01635-7.

[17]

M. J. Gander and L. Halpern, Optimized Schwarz waveform relaxation for advection reaction diffusion problems, SIAM J. Numer. Anal., 45 (2007), 666-697.  doi: 10.1137/050642137.

[18]

M. J. GanderL. Halpern and M. Kern, A Schwarz waveform relaxation method for advection-diffusion-reaction problems with discontinuous coefficients and non-matching grids, Domain Decomposition Methods in Science and Engineering XVI, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 55 (2007), 283-290.  doi: 10.1007/978-3-540-34469-8_33.

[19]

M. J. Gander, L. Halpern and F. Nataf, Optimal convergence for overlapping and nonoverlapping Schwarz waveform relaxation, Eleventh International Conference on Domain Decomposition Methods (London, 1998), DDM.org, Augsburg, (1999), 27–36.

[20]

M. J. GanderL. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., 41 (2003), 1643-1681.  doi: 10.1137/S003614290139559X.

[21]

M. J. GanderC. JaphetY. Maday and F. Nataf, A new cement to glue nonconforming grids with Robin interface conditions: The finite element case, Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 40 (2005), 259-266.  doi: 10.1007/3-540-26825-1_24.

[22]

M. J. Gander and C. Rohde, Overlapping Schwarz waveform relaxation for convection dominated nonlinear conservation laws, SIAM J. Sci. Comp., 27 (2005), 415-439.  doi: 10.1137/030601090.

[23]

M. J. Gander and A. M. Stuart, Space-time continuous analysis of waveform relaxation for the heat equation, SIAM J. Sci. Comp., 19 (1998), 2014-2031.  doi: 10.1137/S1064827596305337.

[24]

E. Giladi and H. B. Keller, Space-time domain decomposition for parabolic problems, Numer. Math., 93 (2002), 279-313.  doi: 10.1007/s002110100345.

[25]

F. HaeberleinL. Halpern and A. Michel, Newton-Schwarz optimised waveform relaxation Krylov accelerators for nonlinear reactive transport, Domain Decomposition Methods in Science and Engineering XX, Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 91 (2013), 387-394.  doi: 10.1007/978-3-642-35275-1_45.

[26]

L. Halpern, Schwarz waveform relaxation algorithms, Domain Decomposition Methods in Science and Engineering XVII, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 60 (2008), 57-68.  doi: 10.1007/978-3-540-75199-1_5.

[27]

L. Halpern, C. Japhet and P. Omnes, Nonconforming in time domain decomposition method for porous media applications, Proceedings of the 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010, Lisbon, Portugal.

[28]

L. HalpernC. Japhet and J. Szeftel, Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping for heterogeneous problems, SIAM J. Numer. Anal., 50 (2012), 2588-2611.  doi: 10.1137/120865033.

[29]

R. D. Haynes and R. D. Russell, A Schwarz waveform moving mesh method, SIAM J. Sci. Comp., 29 (2007), 656-673.  doi: 10.1137/050631549.

[30]

T. T. P. Hoang, Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations, (2021), submitted.

[31]

T. T. P. HoangJ. JaffréC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition methods for diffusion problems in mixed formulations, SIAM J. Numer. Anal., 51 (2013), 3532-3559.  doi: 10.1137/130914401.

[32]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Ventcel conditions with mixed formulations for flow in porous media, Domain Decomposition Methods in Science and Engineering XXII, Lect. Notes Comput. Sci. Eng., Springer, Cham, 104 (2016), 531-540. 

[33]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition for reduced fracture models in mixed formulation, SIAM J. Numer. Anal., 54 (2016), 288-316.  doi: 10.1137/15M1009651.

[34]

T. T. P. HoangC. JaphetM. Kern and J. E. Roberts, Space-time domain decomposition for advection-diffusion problems in mixed formulations, Math. Comput. Simulat., 137 (2017), 366-389.  doi: 10.1016/j.matcom.2016.11.002.

[35]

T. T. P. Hoang and H. Lee, A global-in-time domain decomposition method for the coupled nonlinear stokes and darcy flows, J. Sci. Comput., 87 (2021), 22 pp. doi: 10.1007/s10915-021-01422-1.

[36]

C. Japhet, Optimized Krylov-Ventcell method. Application to convection-diffusion problems, Proceedings of the 9th International Conference on Domain Decomposition Methods, Bergen, Norway, (1998), 382–389.

[37]

E. LelarasmeeA. E. Ruehli and A. L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. CAD IC Syst., 1 (1982), 131-145.  doi: 10.1109/TCAD.1982.1270004.

[38]

V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math., 52 (2005), 401-428.  doi: 10.1016/j.apnum.2004.08.022.

[39]

F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models Methods Appl. Sci., 5 (1995), 67-93.  doi: 10.1142/S021820259500005X.

[40]

F. A. RaduN. SuciuJ. HoffmannA. VogelO. KolditzC.-H. Park and S. Attinger, Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study, Adv. Water Resources, 34 (2011), 47-61.  doi: 10.1016/j.advwatres.2010.09.012.

[41]

J. E. Roberts and J. M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 2 (1991), 523-639. 

[42]

H. A. Schwarz, Ueber einen Grenzübergang durch alternirendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15 (1870), 272-286. 

[43]

A. D. Ventcel', On boundary conditions for multidimensional diffusion processes, Theory Probab. Appl., 4 (1959), 164-177.  doi: 10.1137/1104014.

Figure 1.  A decomposition of $ \Omega $ into two nonoverlapping subdomains
Figure 2.  Nonconforming time grids in the subdomains
Figure 3.  [Test case 2] Convergence curves by Jacobi (left) and GMRES (right) for different Péclet numbers: $ L^{2}- $norm errors in the concentration at $ T = 1 $ with optimized two-sided Robin (blue curves), optimized one-sided Ventcel (magenta curves), and optimized weighted Ventcel (red curves) parameters
Figure 4.  [Test case 2: Advection dominance] Level curves for the error in concentration (in logarithmic scales) after 12 iterations of Jacobi (left) and GMRES (right) for various values of $ p $ and $ q $. The red star shows the optimized values computed by numerically minimizing the continuous convergence factor of the OSWR algorithm
Figure 5.  [Test case 2] Errors in the concentration $ c $ (left) and the vector field $ \pmb{\varphi} $ (right) between the reference and multidomain solutions
Table 1.  [Test case 1] Accuracy in space (the convergence rates are shown in square brackets) and numbers of Jacobi and GMRES iterations for different optimized parameters
$h$ $1/20$ $1/40$ $1/80$ $1/160$
$L^{2}$ errors $c$ 0.0641 0.0321 [1.00] 0.0160 [1.00] 0.0080 [1.00]
$\mathit{\boldsymbol{\varphi }}$ 0.0453 0.0227 [1.00] 0.0114 [0.99] 0.0057 [1.00]
Jacobi 2-sided Robin 21 21 23 25
1-sided Ventcel 11 11 12 13
weighted Ventcel 11 11 12 13
GMRES 2-sided Robin 16 16 20 22
1-sided Ventcel 10 11 11 12
weighted Ventcel 8 10 10 11
$h$ $1/20$ $1/40$ $1/80$ $1/160$
$L^{2}$ errors $c$ 0.0641 0.0321 [1.00] 0.0160 [1.00] 0.0080 [1.00]
$\mathit{\boldsymbol{\varphi }}$ 0.0453 0.0227 [1.00] 0.0114 [0.99] 0.0057 [1.00]
Jacobi 2-sided Robin 21 21 23 25
1-sided Ventcel 11 11 12 13
weighted Ventcel 11 11 12 13
GMRES 2-sided Robin 16 16 20 22
1-sided Ventcel 10 11 11 12
weighted Ventcel 8 10 10 11
Table 2.  [Test case 1] Accuracy in time (the convergence rates are shown in square brackets) and numbers of Jacobi and GMRES iterations for different optimized parameters
$ \Delta t_{2} $ $ T/6 $ $ T/12 $ $ T/24 $ $ T/48 $
$ L^{2} $ errors $ c $ 0.1859 0.0708 [1.39] 0.0301 [1.23] 0.0145 [1.05]
$ \pmb{\varphi} $ 0.2008 0.0768 [1.39] 0.0325 [1.24] 0.0150 [1.12]
Jacobi 2-sided Robin 33 33 33 35
1-sided Ventcel 17 17 17 17
weighted Ventcel 17 17 17 17
GMRES 2-sided Robin 18 18 20 24
1-sided Ventcel 11 12 13 14
weighted Ventcel 10 11 12 13
$ \Delta t_{2} $ $ T/6 $ $ T/12 $ $ T/24 $ $ T/48 $
$ L^{2} $ errors $ c $ 0.1859 0.0708 [1.39] 0.0301 [1.23] 0.0145 [1.05]
$ \pmb{\varphi} $ 0.2008 0.0768 [1.39] 0.0325 [1.24] 0.0150 [1.12]
Jacobi 2-sided Robin 33 33 33 35
1-sided Ventcel 17 17 17 17
weighted Ventcel 17 17 17 17
GMRES 2-sided Robin 18 18 20 24
1-sided Ventcel 11 12 13 14
weighted Ventcel 10 11 12 13
Table 3.  [Test case 2] Discontinuous diffusion and advection coefficients
Problems $ d_{1} $ $ \pmb{u}_{1} $ $ {\rm{Pe}}_{G,1} $ $ d_{2} $ $ \pmb{u}_{2} $ $ {\rm{Pe}}_{G,2} $
(a) Diffusion dominance $ 1 $ $ (-0.02, \; -0.5)^{T} $ $ \approx 0.5 $ $ 0.1 $ $ (-0.02, \; -0.05)^{T} $ $ \approx 0.5 $
(b) Mixed regime $ 0.01 $ $ (-0.02, \; -0.5)^{T} $ $ \approx 50 $ $ 0.1 $ $ (-0.02, \; -0.05)^{T} $ $ \approx 0.5 $
(c) Advection dominance $ 0.02 $ $ (0.5, \; 1)^{T} $ $ \approx 56 $ $ 0.002 $ $ (0.5, \; 0.1)^{T} $ $ \approx 255 $
Problems $ d_{1} $ $ \pmb{u}_{1} $ $ {\rm{Pe}}_{G,1} $ $ d_{2} $ $ \pmb{u}_{2} $ $ {\rm{Pe}}_{G,2} $
(a) Diffusion dominance $ 1 $ $ (-0.02, \; -0.5)^{T} $ $ \approx 0.5 $ $ 0.1 $ $ (-0.02, \; -0.05)^{T} $ $ \approx 0.5 $
(b) Mixed regime $ 0.01 $ $ (-0.02, \; -0.5)^{T} $ $ \approx 50 $ $ 0.1 $ $ (-0.02, \; -0.05)^{T} $ $ \approx 0.5 $
(c) Advection dominance $ 0.02 $ $ (0.5, \; 1)^{T} $ $ \approx 56 $ $ 0.002 $ $ (0.5, \; 0.1)^{T} $ $ \approx 255 $
[1]

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