American Institute of Mathematical Sciences

Advanced
doi: 10.3934/era.2020102

Decoupling PDE computation with intrinsic or inertial Robin interface condition

Lian Zhang 1, , Mingchao Cai 2,, and  Mo Mu 1,

1. 

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hongkong, China
2. 

Department of Mathematics, Morgan State University, 1700 E Cold Spring Ln, MD 21251, USA

* Corresponding author: Mingchao Cai

Received  May 2020 Revised  August 2020 Published  September 2020

Fund Project: The first and the third authors' research is supported in part by the Hong Kong RGC Competitive Earmarked Research Grant HKUST16301218 and NSFC (91530319, 11772281). The second author's research is supported in part by NIH-BUILD grant through UL1GM118973, NIH-RCMI grant through U54MD013376, and the National Science Foundation Awards (1700328, 1831950)

We study decoupled numerical methods for multi-domain, multi-physics applications. By investigating various stages of numerical approximation and decoupling and tracking how the information is transmitted across the interface for a typical multi-modeling model problem, we derive an approximate intrinsic or inertial type Robin condition for its semi-discrete model. This new interface condition is justified both mathematically and physically in contrast to the classical Robin interface condition conventionally introduced for decoupling multi-modeling problems. Based on the intrinsic or inertial Robin condition, an equivalent semi-discrete model is introduced, which provides a general framework for devising effective decoupled numerical methods. Numerical experiments also confirm the effectiveness of this new decoupling approach.

Keywords: Multi-domain, multi-physics models, coupled PDEs, interface coupling conditions, decoupling, intrinsic or inertial type Robin condition, decoupled numerical methods.
Mathematics Subject Classification: 65N30, 65M55, 74F10.
Citation: Lian Zhang, Mingchao Cai, Mo Mu. Decoupling PDE computation with intrinsic or inertial Robin interface condition. Electronic Research Archive, doi: 10.3934/era.2020102
References:
[1]

L. Badea, M. Discacciati and A. Quarteroni, Numerical analysis of the Navier–Stokes/Darcy coupling, Numer. Math., 115 (2010), 195-227.  doi: 10.1007/s00211-009-0279-6.  Google Scholar

[2]

S. Badia, F. Nobile and C. Vergara, Fluid–structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051.  doi: 10.1016/j.jcp.2008.04.006.  Google Scholar

[3]

A. T. Barker and X.-C. Cai, Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling, J. Comput. Phys., 229 (2010), 642-659.  doi: 10.1016/j.jcp.2009.10.001.  Google Scholar

[4]

M. Bukac and B. Muha, Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction, SIAM J. Numer. Anal., 54 (2016), 3032-3061.  doi: 10.1137/16M1055396.  Google Scholar

[5]

M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), 3325-3338.  doi: 10.1137/080721868.  Google Scholar

[6]

M. Cai, M. Mu and J. Xu, Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., 233 (2009), 346-355.  doi: 10.1016/j.cam.2009.07.029.  Google Scholar

[7]

S. Canic, B. Muha and M. Bukac, Stability of the kinematically coupled $\beta$-scheme for fluid-structure interaction problems in hemodynamics, Int. J. Numer. Anal. Model., 12 (2015), 54-80.   Google Scholar

[8]

Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8 (2010), 1-25.  doi: 10.4310/CMS.2010.v8.n1.a2.  Google Scholar

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P. Causin, J.-F. Gerbeau and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506-4527.  doi: 10.1016/j.cma.2004.12.005.  Google Scholar

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Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.  doi: 10.1007/s002110050336.  Google Scholar

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P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos and A. Quarteroni, Fluid–structure interaction simulation of aortic blood flow, Comput. & Fluids, 43 (2011), 46-57.  doi: 10.1016/j.compfluid.2010.11.032.  Google Scholar

[12]

M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22 (2009), 315-426.  doi: 10.5209/rev_REMA.2009.v22.n2.16263.  Google Scholar

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M. Discacciati, A. Quarteroni and A. Valli, Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268.  doi: 10.1137/06065091X.  Google Scholar

[14]

M. A. Fernández, J. Mullaert and M. Vidrascu, Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures, Comput. Methods Appl. Mech. Engrg., 267 (2013), 566-593.  doi: 10.1016/j.cma.2013.09.020.  Google Scholar

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C. Förster, W. A. Wall and E. Ramm, Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1278-1293.  doi: 10.1016/j.cma.2006.09.002.  Google Scholar

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[18]

W. Leng, C.-S. Zhang, P. Sun, B. Gao and J. Xu, Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications, J. Comput. Sci., 30 (2019), 79-89.  doi: 10.1016/j.jocs.2018.11.010.  Google Scholar

[19]

M. Mu and J. Xu, A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801-1813.  doi: 10.1137/050637820.  Google Scholar

[20]

M. Mu and X. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comp., 79 (2010), 707-731.  doi: 10.1090/S0025-5718-09-02302-3.  Google Scholar

[21]

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999.  Google Scholar

[22]

M. Razzaq, H. Damanik, J. Hron, A. Ouazzi and S. Turek, FEM multigrid techniques for fluid-structure interaction with application to hemodynamics, Appl. Numer. Math., 62 (2012), 1156-1170.  doi: 10.1016/j.apnum.2010.12.010.  Google Scholar

[23]

B. Rivière, Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. Sci. Comput., 22/23 (2005), 479-500.  doi: 10.1007/s10915-004-4147-3.  Google Scholar

[24]

B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal., 42 (2005), 1959-1977.  doi: 10.1137/S0036142903427640.  Google Scholar

[25]

R. K. Sinha and B. Deka, Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.  doi: 10.1137/040605357.  Google Scholar

[26]

P. Sun and C. Wang, Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients, Appl. Numer. Math., 152 (2020), 199-220.  doi: 10.1016/j.apnum.2019.12.009.  Google Scholar

[27]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-33122-0.  Google Scholar

[28]

S. Turek and J. Hron, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, in Fluid Structure Interaction, Lect. Notes Comput. Sci. Eng., 53, Springer, Berlin, 2006,371–385. doi: 10.1007/3-540-34596-5_15.  Google Scholar

[29]

S. Turek, J. Hron, M. Razzaq, H. Wobker and M. Schäfer, Numerical benchmarking of fluid-structure interaction: A comparison of different discretization and solution approaches, In Fluid Structure Interaction II, Lect. Notes Comput. Sci. Eng., 73, Springer, Heidelberg, 2010,413–424. doi: 10.1007/978-3-642-14206-2_15.  Google Scholar

[30]

T. Wick, Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. II Library, Arch. Numer. Software, 1 (2013), 1-19.  doi: 10.11588/ans.2013.1.10305.  Google Scholar

show all references

References:
[1]

L. Badea, M. Discacciati and A. Quarteroni, Numerical analysis of the Navier–Stokes/Darcy coupling, Numer. Math., 115 (2010), 195-227.  doi: 10.1007/s00211-009-0279-6.  Google Scholar

[2]

S. Badia, F. Nobile and C. Vergara, Fluid–structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227 (2008), 7027-7051.  doi: 10.1016/j.jcp.2008.04.006.  Google Scholar

[3]

A. T. Barker and X.-C. Cai, Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling, J. Comput. Phys., 229 (2010), 642-659.  doi: 10.1016/j.jcp.2009.10.001.  Google Scholar

[4]

M. Bukac and B. Muha, Stability and convergence analysis of the extensions of the kinematically coupled scheme for the fluid-structure interaction, SIAM J. Numer. Anal., 54 (2016), 3032-3061.  doi: 10.1137/16M1055396.  Google Scholar

[5]

M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), 3325-3338.  doi: 10.1137/080721868.  Google Scholar

[6]

M. Cai, M. Mu and J. Xu, Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. Comput. Appl. Math., 233 (2009), 346-355.  doi: 10.1016/j.cam.2009.07.029.  Google Scholar

[7]

S. Canic, B. Muha and M. Bukac, Stability of the kinematically coupled $\beta$-scheme for fluid-structure interaction problems in hemodynamics, Int. J. Numer. Anal. Model., 12 (2015), 54-80.   Google Scholar

[8]

Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8 (2010), 1-25.  doi: 10.4310/CMS.2010.v8.n1.a2.  Google Scholar

[9]

P. Causin, J.-F. Gerbeau and F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4506-4527.  doi: 10.1016/j.cma.2004.12.005.  Google Scholar

[10]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.  doi: 10.1007/s002110050336.  Google Scholar

[11]

P. Crosetto, P. Reymond, S. Deparis, D. Kontaxakis, N. Stergiopulos and A. Quarteroni, Fluid–structure interaction simulation of aortic blood flow, Comput. & Fluids, 43 (2011), 46-57.  doi: 10.1016/j.compfluid.2010.11.032.  Google Scholar

[12]

M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22 (2009), 315-426.  doi: 10.5209/rev_REMA.2009.v22.n2.16263.  Google Scholar

[13]

M. Discacciati, A. Quarteroni and A. Valli, Robin–Robin domain decomposition methods for the Stokes–Darcy coupling, SIAM J. Numer. Anal., 45 (2007), 1246-1268.  doi: 10.1137/06065091X.  Google Scholar

[14]

M. A. Fernández, J. Mullaert and M. Vidrascu, Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures, Comput. Methods Appl. Mech. Engrg., 267 (2013), 566-593.  doi: 10.1016/j.cma.2013.09.020.  Google Scholar

[15]

M. A. Fernández, J. Mullaert and M. Vidrascu, Generalized Robin–Neumann explicit coupling schemes for incompressible fluid-structure interaction: Stability analysis and numerics, Internat. J. Numer. Methods Engrg., 101 (2015), 199-229.  doi: 10.1002/nme.4785.  Google Scholar

[16]

C. Förster, W. A. Wall and E. Ramm, Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1278-1293.  doi: 10.1016/j.cma.2006.09.002.  Google Scholar

[17]

R. Lan, P. Sun and M. Mu, Mixed finite element analysis for an elliptic/mixed elliptic interface problem with jump coefficients, Procedia Comput. Sci., 108 (2017), 1913-1922.  doi: 10.1016/j.procs.2017.05.001.  Google Scholar

[18]

W. Leng, C.-S. Zhang, P. Sun, B. Gao and J. Xu, Numerical simulation of an immersed rotating structure in fluid for hemodynamic applications, J. Comput. Sci., 30 (2019), 79-89.  doi: 10.1016/j.jocs.2018.11.010.  Google Scholar

[19]

M. Mu and J. Xu, A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 45 (2007), 1801-1813.  doi: 10.1137/050637820.  Google Scholar

[20]

M. Mu and X. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comp., 79 (2010), 707-731.  doi: 10.1090/S0025-5718-09-02302-3.  Google Scholar

[21]

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999.  Google Scholar

[22]

M. Razzaq, H. Damanik, J. Hron, A. Ouazzi and S. Turek, FEM multigrid techniques for fluid-structure interaction with application to hemodynamics, Appl. Numer. Math., 62 (2012), 1156-1170.  doi: 10.1016/j.apnum.2010.12.010.  Google Scholar

[23]

B. Rivière, Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems, J. Sci. Comput., 22/23 (2005), 479-500.  doi: 10.1007/s10915-004-4147-3.  Google Scholar

[24]

B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal., 42 (2005), 1959-1977.  doi: 10.1137/S0036142903427640.  Google Scholar

[25]

R. K. Sinha and B. Deka, Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.  doi: 10.1137/040605357.  Google Scholar

[26]

P. Sun and C. Wang, Distributed Lagrange multiplier/fictitious domain finite element method for Stokes/parabolic interface problems with jump coefficients, Appl. Numer. Math., 152 (2020), 199-220.  doi: 10.1016/j.apnum.2019.12.009.  Google Scholar

[27]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-33122-0.  Google Scholar

[28]

S. Turek and J. Hron, Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, in Fluid Structure Interaction, Lect. Notes Comput. Sci. Eng., 53, Springer, Berlin, 2006,371–385. doi: 10.1007/3-540-34596-5_15.  Google Scholar

[29]

S. Turek, J. Hron, M. Razzaq, H. Wobker and M. Schäfer, Numerical benchmarking of fluid-structure interaction: A comparison of different discretization and solution approaches, In Fluid Structure Interaction II, Lect. Notes Comput. Sci. Eng., 73, Springer, Heidelberg, 2010,413–424. doi: 10.1007/978-3-642-14206-2_15.  Google Scholar

[30]

T. Wick, Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal. II Library, Arch. Numer. Software, 1 (2013), 1-19.  doi: 10.11588/ans.2013.1.10305.  Google Scholar

Figure 1.  Computational Domains with Interface
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Figure 2.  A comparison of the exact solution and solutions obtained by the decoupled iRN scheme at $ t = 1 $ with $ \Delta t = h = \frac{1}{8},\frac{1}{16},\frac{1}{32} $
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Table 1.  Errors of $ \|u_{h,N}-u_{ext}(T)\|_{0,\Omega} $ with $ \rho_1 = \rho_2 = 1,\; \beta_1 = \beta_2 = \beta(x, y),\; \Delta t = h^2 $
h Coupled scheme Decoupled iRN scheme
$ \frac{1}{8} $ 3.43107e-2 3.75366e-2
$ \frac{1}{16} $ 9.20988e-3 1.13574e-2
$ \frac{1}{32} $ 2.34374e-3 2.61407e-3
$ \frac{1}{64} $ 5.88545e-4 5.77793e-4
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h Coupled scheme Decoupled iRN scheme
$ \frac{1}{8} $ 3.43107e-2 3.75366e-2
$ \frac{1}{16} $ 9.20988e-3 1.13574e-2
$ \frac{1}{32} $ 2.34374e-3 2.61407e-3
$ \frac{1}{64} $ 5.88545e-4 5.77793e-4
Table 2.  Errors of $ \|u_{h,N}-u_{ext}(T)\|_{0,\Omega} $ with $ \rho_1 = \rho_2 = 1,\; \beta_1 = \beta_2 = \beta(x, y),\; \Delta t = h^2 $
h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.43107e-2 3.53891e-2
$ \frac{1}{16} $ 9.20988e-3 9.46749e-3
$ \frac{1}{32} $ 2.34374e-3 2.40475e-3
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.53436e-2 3.24908e-2 3.80028e-2
$ \frac{1}{16} $ 9.55398e-3 8.81404e-3 1.39637e-2
$ \frac{1}{32} $ 2.25541e-3 2.26911e-3 5.08893e-3
Table Options

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h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.43107e-2 3.53891e-2
$ \frac{1}{16} $ 9.20988e-3 9.46749e-3
$ \frac{1}{32} $ 2.34374e-3 2.40475e-3
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.53436e-2 3.24908e-2 3.80028e-2
$ \frac{1}{16} $ 9.55398e-3 8.81404e-3 1.39637e-2
$ \frac{1}{32} $ 2.25541e-3 2.26911e-3 5.08893e-3
Table 3.  Errors of $ \|u_{h,N}-u_{ext}(T)\|_{0,\Omega} $ with $ \rho_1 = 10,\; \rho_2 = 1,\; \beta_1 = \beta_2 = \beta(x, y),\; \Delta t = h^2 $
h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.36012e-2 $ \infty $
$ \frac{1}{16} $ 9.01136e-3 $ \infty $
$ \frac{1}{32} $ 2.29268e-3 $ \infty $
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.24630e-2 3.18627e-2 3.72954e-2
$ \frac{1}{16} $ 8.80706e-3 8.64307e-3 1.37006e-2
$ \frac{1}{32} $ 2.27580e-3 2.26911e-3 5.00114e-3
Table Options

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h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.36012e-2 $ \infty $
$ \frac{1}{16} $ 9.01136e-3 $ \infty $
$ \frac{1}{32} $ 2.29268e-3 $ \infty $
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.24630e-2 3.18627e-2 3.72954e-2
$ \frac{1}{16} $ 8.80706e-3 8.64307e-3 1.37006e-2
$ \frac{1}{32} $ 2.27580e-3 2.26911e-3 5.00114e-3
Table 4.  Errors of $ \|u_{h,N}-u_{ext}(T)\|_{0,\Omega} $ with $ \rho_1 = 1,\; \rho_2 = 10,\; \beta_1 = \beta_2 = \beta(x, y),\; \Delta t = h^2 $
h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.39101e-2 3.21780e-2
$ \frac{1}{16} $ 9.09826e-3 9.38034e-3
$ \frac{1}{32} $ 2.31506e-3 2.38207e-3
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.23888e-2 3.21780e-2 3.76394e-2
$ \frac{1}{16} $ 8.82536e-3 8.72945e-3 1.38221e-2
$ \frac{1}{32} $ 2.28889e-3 2.24812e-3 5.04232e-3
Table Options

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h Coupled scheme Decoupled DN scheme
$ \frac{1}{8} $ 3.39101e-2 3.21780e-2
$ \frac{1}{16} $ 9.09826e-3 9.38034e-3
$ \frac{1}{32} $ 2.31506e-3 2.38207e-3
h Decoupled iRR scheme Decoupled RR scheme
$ \alpha_1 = 10,\alpha_2 = 5 $ $ \alpha_1 = 1,\alpha_2 = 1 $
$ \frac{1}{8} $ 3.23888e-2 3.21780e-2 3.76394e-2
$ \frac{1}{16} $ 8.82536e-3 8.72945e-3 1.38221e-2
$ \frac{1}{32} $ 2.28889e-3 2.24812e-3 5.04232e-3
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