American Institute of Mathematical Sciences

Advanced
October  2020, 25(10): 3807-3830. doi: 10.3934/dcdsb.2020229

Higher-order time-stepping schemes for fluid-structure interaction problems

Daniele Boffi 1,2,, , Lucia Gastaldi 3, and  Sebastian Wolf 4,

1. 

Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
2. 

Dipartimento di Matematica "F. Casorati", University of Pavia, Pavia, Italy
3. 

DICATAM, University of Brescia, Brescia, Italy
4. 

Technische Universität München (TUM), München, Germany

* Corresponding author: Daniele Boffi

Received  June 2019 Revised  March 2020 Published  October 2020 Early access  July 2020

We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results.

Keywords: Finite elements, Fluid-structure interactions, Fictitious domain, higher order time stepping, stability estimates.
Mathematics Subject Classification: Primary: 65M60, 65M85; Secondary: 65M12, 74F10.
Citation: Daniele Boffi, Lucia Gastaldi, Sebastian Wolf. Higher-order time-stepping schemes for fluid-structure interaction problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3807-3830. doi: 10.3934/dcdsb.2020229
References:
[1]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Local mass conservation of Stokes finite elements, J. Sci. Comput., 52 (2012), 383-400.  doi: 10.1007/s10915-011-9549-4.  Google Scholar

[2]

D. Boffi, N. Cavallini and L. Gastaldi, Finite element approach to immersed boundary method with different fluid and solid densities, Math. Models Methods Appl. Sci, 21 (2011), 2523-2550.  doi: 10.1142/S0218202511005829.  Google Scholar

[3]

D. Boffi, N. Cavallini and L. Gastaldi, The finite element immersed boundary method with distributed lagrange multiplier, SIAM J. Numer. Anal., 53 (2015), 2584-2604.  doi: 10.1137/140978399.  Google Scholar

[4]

D. Boffi and L. Gastaldi, Discrete models for fluid-structure interactions: The Finite Element Immersed Boundary Method, Discrete Contin. Dyn. Syst., Ser. S, 9 (2016), 89-107.  doi: 10.3934/dcdss.2016.9.89.  Google Scholar

[5]

D. Boffi and L. Gastaldi, A fictious domain approach with distributed lagrange multipliers for fluid-structure interactions, Numer. Math., 135 (2017), 711-732.  doi: 10.1007/s00211-016-0814-1.  Google Scholar

[6]

D. Boffi, L. Gastaldi and L. Heltai, Numerical stability of the finite element immersed boundary method, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1479-1505.  doi: 10.1142/S0218202507002352.  Google Scholar

[7]

D. Boffi, L. Gastaldi and L. Heltai, A distributed Lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications (eds. B. D., P. L., R. G., S. S. and V. C.), vol. 16 of SEMA SIMAI Springer Series, Springer, 2018, 1–21, URL https://arXiv.org/abs/1712.02545.  Google Scholar

[8]

D. Boffi, L. Gastaldi, L. Heltai and C. S. Peskin, On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng., 197 (2008), 2210-2231.  doi: 10.1016/j.cma.2007.09.015.  Google Scholar

[9]

M. Boulakia and S. Guerrero, On the interaction problem between a compressible fluid and a Saint–Venant Kirchoff elastic structure, Adv. Differential Equations, 22 (2017), 1-48.   Google Scholar

[10]

M. Boulakia, S. Guerrero and T. Takahashi, Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure, Nonlinearity, 32 (2019), 3548-3592.  doi: 10.1088/1361-6544/ab128c.  Google Scholar

[11]

W. Chen, M. Gunzburger, D. Sun and X. Wang, Efficient and long-time accurate second-order methods for Stokes-Darcy Systems, SIAM J. Numer. Anal., 51 (2013), 2563-2584.  doi: 10.1137/120897705.  Google Scholar

[12]

C. Coutand and S. Shloller, Motion of an elastic solid inside an incompressible fluid-structure interaction, Arch. Ration. Mech. Anal., 176 (2005), 25-102.  doi: 10.1007/s00205-004-0340-7.  Google Scholar

[13]

C. Coutand and S. Shloller, The interaction between quasilinear elastidynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[14]

P. Deuflhard and F. Bornemann, Numerische Mathematik 2, revised edition, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2008, Gewöhnliche Differentialgleichungen. [Ordinary differential equations].  Google Scholar

[15]

S. Dong, BDF-like methods for nonlinear dynamic analysis, J. Comput. Phys., 229 (2010), 3019-3045.  doi: 10.1016/j.jcp.2009.12.028.  Google Scholar

[16]

B. E. Griffith, On the volume conservation of the immersed boundary method, Commun. Comput. Phys., 12 (2012), 401-432.  doi: 10.4208/cicp.120111.300911s.  Google Scholar

[17]

B. E. Griffith and X. Luo, Hybrid finite difference/finite element immersed boundary method, Int. J. Numer. Meth. Biomed. Engng., 33 (2017), e2888, 31pp. doi: 10.1002/cnm.2888.  Google Scholar

[18]

L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Eng., 229/232 (2012), 110-127.  doi: 10.1016/j.cma.2012.04.001.  Google Scholar

[19]

J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary navier–stokes problem. part iv: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[20]

O. R. Isik, G. Yuksel and B. Demir, Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation, Numer. Methods Partial Differ. Equations, 34 (2017), 2060-2078.  doi: 10.1002/num.22276.  Google Scholar

[21]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer, 2016. doi: 10.1007/978-3-319-45750-5.  Google Scholar

[22]

Y. Okamoto, K. Fujiwara and Y. Ishihara, Effectiveness of higher order time integration in time-domain finite-element analysis, IEEE Transactions on Magnetics, 46 (2010), 3321-3324.  doi: 10.1109/TMAG.2010.2044771.  Google Scholar

[23]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar

[24]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier–Sstokes equations and the Lamé system, J. Mat. Pura Appl., 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[25]

S. Roy, L. Heltai and F. Costanzo, Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69 (2015), 1167–1188. doi: 10.1016/j.camwa.2015.03.012.  Google Scholar

[26]

X. Wang and L. T. Zhang, Interpolation functions in the immersed boundary and finite element methods, Comput. Mech., 45 (2009), 321. doi: 10.1007/s00466-009-0449-5.  Google Scholar

show all references

References:
[1]

D. Boffi, N. Cavallini, F. Gardini and L. Gastaldi, Local mass conservation of Stokes finite elements, J. Sci. Comput., 52 (2012), 383-400.  doi: 10.1007/s10915-011-9549-4.  Google Scholar

[2]

D. Boffi, N. Cavallini and L. Gastaldi, Finite element approach to immersed boundary method with different fluid and solid densities, Math. Models Methods Appl. Sci, 21 (2011), 2523-2550.  doi: 10.1142/S0218202511005829.  Google Scholar

[3]

D. Boffi, N. Cavallini and L. Gastaldi, The finite element immersed boundary method with distributed lagrange multiplier, SIAM J. Numer. Anal., 53 (2015), 2584-2604.  doi: 10.1137/140978399.  Google Scholar

[4]

D. Boffi and L. Gastaldi, Discrete models for fluid-structure interactions: The Finite Element Immersed Boundary Method, Discrete Contin. Dyn. Syst., Ser. S, 9 (2016), 89-107.  doi: 10.3934/dcdss.2016.9.89.  Google Scholar

[5]

D. Boffi and L. Gastaldi, A fictious domain approach with distributed lagrange multipliers for fluid-structure interactions, Numer. Math., 135 (2017), 711-732.  doi: 10.1007/s00211-016-0814-1.  Google Scholar

[6]

D. Boffi, L. Gastaldi and L. Heltai, Numerical stability of the finite element immersed boundary method, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1479-1505.  doi: 10.1142/S0218202507002352.  Google Scholar

[7]

D. Boffi, L. Gastaldi and L. Heltai, A distributed Lagrange formulation of the finite element immersed boundary method for fluids interacting with compressible solids, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications (eds. B. D., P. L., R. G., S. S. and V. C.), vol. 16 of SEMA SIMAI Springer Series, Springer, 2018, 1–21, URL https://arXiv.org/abs/1712.02545.  Google Scholar

[8]

D. Boffi, L. Gastaldi, L. Heltai and C. S. Peskin, On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng., 197 (2008), 2210-2231.  doi: 10.1016/j.cma.2007.09.015.  Google Scholar

[9]

M. Boulakia and S. Guerrero, On the interaction problem between a compressible fluid and a Saint–Venant Kirchoff elastic structure, Adv. Differential Equations, 22 (2017), 1-48.   Google Scholar

[10]

M. Boulakia, S. Guerrero and T. Takahashi, Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure, Nonlinearity, 32 (2019), 3548-3592.  doi: 10.1088/1361-6544/ab128c.  Google Scholar

[11]

W. Chen, M. Gunzburger, D. Sun and X. Wang, Efficient and long-time accurate second-order methods for Stokes-Darcy Systems, SIAM J. Numer. Anal., 51 (2013), 2563-2584.  doi: 10.1137/120897705.  Google Scholar

[12]

C. Coutand and S. Shloller, Motion of an elastic solid inside an incompressible fluid-structure interaction, Arch. Ration. Mech. Anal., 176 (2005), 25-102.  doi: 10.1007/s00205-004-0340-7.  Google Scholar

[13]

C. Coutand and S. Shloller, The interaction between quasilinear elastidynamics and the Navier–Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 303-352.  doi: 10.1007/s00205-005-0385-2.  Google Scholar

[14]

P. Deuflhard and F. Bornemann, Numerische Mathematik 2, revised edition, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 2008, Gewöhnliche Differentialgleichungen. [Ordinary differential equations].  Google Scholar

[15]

S. Dong, BDF-like methods for nonlinear dynamic analysis, J. Comput. Phys., 229 (2010), 3019-3045.  doi: 10.1016/j.jcp.2009.12.028.  Google Scholar

[16]

B. E. Griffith, On the volume conservation of the immersed boundary method, Commun. Comput. Phys., 12 (2012), 401-432.  doi: 10.4208/cicp.120111.300911s.  Google Scholar

[17]

B. E. Griffith and X. Luo, Hybrid finite difference/finite element immersed boundary method, Int. J. Numer. Meth. Biomed. Engng., 33 (2017), e2888, 31pp. doi: 10.1002/cnm.2888.  Google Scholar

[18]

L. Heltai and F. Costanzo, Variational implementation of immersed finite element methods, Comput. Methods Appl. Mech. Eng., 229/232 (2012), 110-127.  doi: 10.1016/j.cma.2012.04.001.  Google Scholar

[19]

J. Heywood and R. Rannacher, Finite-element approximation of the nonstationary navier–stokes problem. part iv: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[20]

O. R. Isik, G. Yuksel and B. Demir, Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation, Numer. Methods Partial Differ. Equations, 34 (2017), 2060-2078.  doi: 10.1002/num.22276.  Google Scholar

[21]

V. John, Finite Element Methods for Incompressible Flow Problems, Springer, 2016. doi: 10.1007/978-3-319-45750-5.  Google Scholar

[22]

Y. Okamoto, K. Fujiwara and Y. Ishihara, Effectiveness of higher order time integration in time-domain finite-element analysis, IEEE Transactions on Magnetics, 46 (2010), 3321-3324.  doi: 10.1109/TMAG.2010.2044771.  Google Scholar

[23]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar

[24]

J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling the Navier–Sstokes equations and the Lamé system, J. Mat. Pura Appl., 102 (2014), 546-596.  doi: 10.1016/j.matpur.2013.12.004.  Google Scholar

[25]

S. Roy, L. Heltai and F. Costanzo, Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69 (2015), 1167–1188. doi: 10.1016/j.camwa.2015.03.012.  Google Scholar

[26]

X. Wang and L. T. Zhang, Interpolation functions in the immersed boundary and finite element methods, Comput. Mech., 45 (2009), 321. doi: 10.1007/s00466-009-0449-5.  Google Scholar

Figure 1.  Geometrical configuration of the FSI problem
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Figure 2.  Sparsity pattern for a matrix arising from Equation (24)
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Figure 3.  Meshes for the fluid and the structure
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Figure 4.  The deformed annulus 1
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Figure 5.  Volume preservation over time
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Figure 6.  Volume preservation over time for coarser parameters
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Figure 7.  Volume preservation over time for IFEM method
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Table 1.  Mesh parameters
DOFs $ \mathbf{u}_h $ DOFs $ p_h $ DOFs $ \mathbf{X}_h $ DOFs $ \lambda_h $
coarse mesh (M = $ 8 $) $ 578 $ $ 209 $ $ 306 $ $ 306 $
fine mesh (M = $ 16 $) $ 2,178 $ $ 801 $ $ 1,122 $ $ 1,122 $
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DOFs $ \mathbf{u}_h $ DOFs $ p_h $ DOFs $ \mathbf{X}_h $ DOFs $ \lambda_h $
coarse mesh (M = $ 8 $) $ 578 $ $ 209 $ $ 306 $ $ 306 $
fine mesh (M = $ 16 $) $ 2,178 $ $ 801 $ $ 1,122 $ $ 1,122 $
Table 2.  Convergence results for the fully implicit scheme on the coarse mesh
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 7.63\cdot 10^{-2} $ $ 2.97\cdot 10^{-2} $ $ 2.37\cdot 10^{-1} $ $ 2.42\cdot 10^{-1} $
$ 0.025 $ $ 4.11\cdot 10^{-2} $ $ 0.89 $ $ 4.90\cdot 10^{-3} $ $ 2.60 $ $ 6.24\cdot 10^{-2} $ $ 1.92 $ $ 6.02\cdot 10^{-2} $ $ 2.00 $
$ 0.0125 $ $ 2.13\cdot 10^{-2} $ $ 0.95 $ $ 1.13\cdot 10^{-3} $ $ 2.11 $ $ 1.21\cdot 10^{-2} $ $ 2.36 $ $ 1.10\cdot 10^{-2} $ $ 2.45 $
$ 0.00625 $ $ 1.08\cdot 10^{-2} $ $ 0.97 $ $ 2.86\cdot 10^{-4} $ $ 1.98 $ $ 2.03\cdot 10^{-3} $ $ 2.58 $ $ 9.95\cdot 10^{-4} $ $ 3.47 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.60\cdot 10^{-3} $ $ 4.39\cdot 10^{-4} $ $ 1.43\cdot 10^{-3} $ $ 2.95\cdot 10^{-4} $
$ 0.025 $ $ 8.40\cdot 10^{-4} $ $ 0.93 $ $ 9.75\cdot 10^{-5} $ $ 2.17 $ $ 7.90\cdot 10^{-4} $ $ 0.86 $ $ 6.89\cdot 10^{-5} $ $ 2.10 $
$ 0.0125 $ $ 4.29\cdot 10^{-4} $ $ 0.97 $ $ 2.53\cdot 10^{-5} $ $ 1.95 $ $ 4.06\cdot 10^{-4} $ $ 0.96 $ $ 7.53\cdot 10^{-6} $ $ 3.19 $
$ 0.00625 $ $ 2.17\cdot 10^{-4} $ $ 0.98 $ $ 6.41\cdot 10^{-6} $ $ 1.98 $ $ 2.04\cdot 10^{-4} $ $ 0.99 $ $ 2.05\cdot 10^{-6} $ $ 1.88 $
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Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 7.63\cdot 10^{-2} $ $ 2.97\cdot 10^{-2} $ $ 2.37\cdot 10^{-1} $ $ 2.42\cdot 10^{-1} $
$ 0.025 $ $ 4.11\cdot 10^{-2} $ $ 0.89 $ $ 4.90\cdot 10^{-3} $ $ 2.60 $ $ 6.24\cdot 10^{-2} $ $ 1.92 $ $ 6.02\cdot 10^{-2} $ $ 2.00 $
$ 0.0125 $ $ 2.13\cdot 10^{-2} $ $ 0.95 $ $ 1.13\cdot 10^{-3} $ $ 2.11 $ $ 1.21\cdot 10^{-2} $ $ 2.36 $ $ 1.10\cdot 10^{-2} $ $ 2.45 $
$ 0.00625 $ $ 1.08\cdot 10^{-2} $ $ 0.97 $ $ 2.86\cdot 10^{-4} $ $ 1.98 $ $ 2.03\cdot 10^{-3} $ $ 2.58 $ $ 9.95\cdot 10^{-4} $ $ 3.47 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.60\cdot 10^{-3} $ $ 4.39\cdot 10^{-4} $ $ 1.43\cdot 10^{-3} $ $ 2.95\cdot 10^{-4} $
$ 0.025 $ $ 8.40\cdot 10^{-4} $ $ 0.93 $ $ 9.75\cdot 10^{-5} $ $ 2.17 $ $ 7.90\cdot 10^{-4} $ $ 0.86 $ $ 6.89\cdot 10^{-5} $ $ 2.10 $
$ 0.0125 $ $ 4.29\cdot 10^{-4} $ $ 0.97 $ $ 2.53\cdot 10^{-5} $ $ 1.95 $ $ 4.06\cdot 10^{-4} $ $ 0.96 $ $ 7.53\cdot 10^{-6} $ $ 3.19 $
$ 0.00625 $ $ 2.17\cdot 10^{-4} $ $ 0.98 $ $ 6.41\cdot 10^{-6} $ $ 1.98 $ $ 2.04\cdot 10^{-4} $ $ 0.99 $ $ 2.05\cdot 10^{-6} $ $ 1.88 $
Table 3.  Convergence results for the fully implicit scheme on the fine mesh
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 9.05\cdot 10^{-2} $ $ 3.62\cdot 10^{-2} $ $ 2.28\cdot 10^{-1} $ $ 2.26\cdot 10^{-1} $
$ 0.025 $ $ 4.87\cdot 10^{-2} $ $ 0.89 $ $ 5.05\cdot 10^{-3} $ $ 2.84 $ $ 6.23\cdot 10^{-2} $ $ 1.87 $ $ 6.04\cdot 10^{-2} $ $ 1.91 $
$ 0.0125 $ $ 2.54\cdot 10^{-2} $ $ 0.94 $ $ 1.20\cdot 10^{-3} $ $ 2.07 $ $ 2.28\cdot 10^{-2} $ $ 1.45 $ $ 2.07\cdot 10^{-2} $ $ 1.54 $
$ 0.00625 $ $ 1.29\cdot 10^{-2} $ $ 0.98 $ $ 3.53\cdot 10^{-4} $ $ 1.77 $ $ 5.27\cdot 10^{-3} $ $ 2.11 $ $ 4.03\cdot 10^{-3} $ $ 2.36 $
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.98\cdot 10^{-3} $ $ 5.19\cdot 10^{-4} $ $ 1.65\cdot 10^{-3} $ $ 4.04\cdot 10^{-4} $
$ 0.025 $ $ 1.05\cdot 10^{-3} $ $ 0.92 $ $ 9.79\cdot 10^{-5} $ $ 2.41 $ $ 9.27\cdot 10^{-4} $ $ 0.84 $ $ 8.48\cdot 10^{-5} $ $ 2.25 $
$ 0.0125 $ $ 5.31\cdot 10^{-4} $ $ 0.99 $ $ 3.13\cdot 10^{-5} $ $ 1.64 $ $ 4.90\cdot 10^{-4} $ $ 0.92 $ $ 2.47\cdot 10^{-5} $ $ 1.78 $
$ 0.00625 $ $ 2.70\cdot 10^{-4} $ $ 0.98 $ $ 1.35\cdot 10^{-5} $ $ 1.22 $ $ 2.50\cdot 10^{-4} $ $ 0.97 $ $ 3.47\cdot 10^{-6} $ $ 2.83 $
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Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 9.05\cdot 10^{-2} $ $ 3.62\cdot 10^{-2} $ $ 2.28\cdot 10^{-1} $ $ 2.26\cdot 10^{-1} $
$ 0.025 $ $ 4.87\cdot 10^{-2} $ $ 0.89 $ $ 5.05\cdot 10^{-3} $ $ 2.84 $ $ 6.23\cdot 10^{-2} $ $ 1.87 $ $ 6.04\cdot 10^{-2} $ $ 1.91 $
$ 0.0125 $ $ 2.54\cdot 10^{-2} $ $ 0.94 $ $ 1.20\cdot 10^{-3} $ $ 2.07 $ $ 2.28\cdot 10^{-2} $ $ 1.45 $ $ 2.07\cdot 10^{-2} $ $ 1.54 $
$ 0.00625 $ $ 1.29\cdot 10^{-2} $ $ 0.98 $ $ 3.53\cdot 10^{-4} $ $ 1.77 $ $ 5.27\cdot 10^{-3} $ $ 2.11 $ $ 4.03\cdot 10^{-3} $ $ 2.36 $
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.98\cdot 10^{-3} $ $ 5.19\cdot 10^{-4} $ $ 1.65\cdot 10^{-3} $ $ 4.04\cdot 10^{-4} $
$ 0.025 $ $ 1.05\cdot 10^{-3} $ $ 0.92 $ $ 9.79\cdot 10^{-5} $ $ 2.41 $ $ 9.27\cdot 10^{-4} $ $ 0.84 $ $ 8.48\cdot 10^{-5} $ $ 2.25 $
$ 0.0125 $ $ 5.31\cdot 10^{-4} $ $ 0.99 $ $ 3.13\cdot 10^{-5} $ $ 1.64 $ $ 4.90\cdot 10^{-4} $ $ 0.92 $ $ 2.47\cdot 10^{-5} $ $ 1.78 $
$ 0.00625 $ $ 2.70\cdot 10^{-4} $ $ 0.98 $ $ 1.35\cdot 10^{-5} $ $ 1.22 $ $ 2.50\cdot 10^{-4} $ $ 0.97 $ $ 3.47\cdot 10^{-6} $ $ 2.83 $
Table 4.  Maximum iterates of the nonlinear solver on the coarse mesh
$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 5 $ $ 5 $ $ 4 $ $ 7 $
$ 0.025 $ $ 4 $ $ 4 $ $ 3 $ $ 4 $
$ 0.0125 $ $ 3 $ $ 3 $ $ 3 $ $ 3 $
$ 0.00625 $ $ 3 $ $ 3 $ $ 2 $ $ 3 $
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$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 5 $ $ 5 $ $ 4 $ $ 7 $
$ 0.025 $ $ 4 $ $ 4 $ $ 3 $ $ 4 $
$ 0.0125 $ $ 3 $ $ 3 $ $ 3 $ $ 3 $
$ 0.00625 $ $ 3 $ $ 3 $ $ 2 $ $ 3 $
Table 5.  Maximum iterates of the nonlinear solver on the fine mesh
$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 10 $ $ 5 $ $ 6 $ $ 6 $
$ 0.025 $ $ 6 $ $ 5 $ $ 5 $ $ 4 $
$ 0.0125 $ $ 6 $ $ 4 $ $ 4 $ $ 4 $
$ 0.00625 $ $ 4 $ $ 4 $ $ 3 $ $ 3 $
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$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 10 $ $ 5 $ $ 6 $ $ 6 $
$ 0.025 $ $ 6 $ $ 5 $ $ 5 $ $ 4 $
$ 0.0125 $ $ 6 $ $ 4 $ $ 4 $ $ 4 $
$ 0.00625 $ $ 4 $ $ 4 $ $ 3 $ $ 3 $
Table 6.  Convergence results for the semi-implicit scheme on the coarse mesh
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 7.78\cdot 10^{-2} $ $ 3.05\cdot 10^{-2} $ $ 2.49\cdot 10^{-1} $ $ 2.58\cdot 10^{-1} $
$ 0.025 $ $ 4.17\cdot 10^{-2} $ $ 0.90 $ $ 7.89\cdot 10^{-3} $ $ 1.95 $ $ 6.24\cdot 10^{-2} $ $ 2.00 $ $ 6.74\cdot 10^{-2} $ $ 1.94 $
$ 0.0125 $ $ 2.17\cdot 10^{-2} $ $ 0.95 $ $ 3.14\cdot 10^{-3} $ $ 1.33 $ $ 1.24\cdot 10^{-2} $ $ 2.33 $ $ 2.64\cdot 10^{-2} $ $ 1.35 $
$ 0.00625 $ $ 1.10\cdot 10^{-2} $ $ 0.97 $ $ 1.29\cdot 10^{-3} $ $ 1.29 $ $ 2.25\cdot 10^{-3} $ $ 2.47 $ $ 3.18\cdot 10^{-3} $ $ 3.06 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.67\cdot 10^{-3} $ $ 6.79\cdot 10^{-4} $ $ 1.44\cdot 10^{-3} $ $ 3.52\cdot 10^{-4} $
$ 0.025 $ $ 8.65\cdot 10^{-4} $ $ 0.95 $ $ 2.70\cdot 10^{-4} $ $ 1.33 $ $ 7.91\cdot 10^{-4} $ $ 0.86 $ $ 2.60\cdot 10^{-4} $ $ 0.44 $
$ 0.0125 $ $ 4.36\cdot 10^{-4} $ $ 0.99 $ $ 1.24\cdot 10^{-4} $ $ 1.12 $ $ 4.05\cdot 10^{-4} $ $ 0.97 $ $ 1.53\cdot 10^{-5} $ $ 4.08 $
$ 0.00625 $ $ 2.17\cdot 10^{-4} $ $ 1.01 $ $ 5.71\cdot 10^{-5} $ $ 1.12 $ $ 2.05\cdot 10^{-4} $ $ 0.98 $ $ 9.43\cdot 10^{-6} $ $ 0.70 $
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Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 7.78\cdot 10^{-2} $ $ 3.05\cdot 10^{-2} $ $ 2.49\cdot 10^{-1} $ $ 2.58\cdot 10^{-1} $
$ 0.025 $ $ 4.17\cdot 10^{-2} $ $ 0.90 $ $ 7.89\cdot 10^{-3} $ $ 1.95 $ $ 6.24\cdot 10^{-2} $ $ 2.00 $ $ 6.74\cdot 10^{-2} $ $ 1.94 $
$ 0.0125 $ $ 2.17\cdot 10^{-2} $ $ 0.95 $ $ 3.14\cdot 10^{-3} $ $ 1.33 $ $ 1.24\cdot 10^{-2} $ $ 2.33 $ $ 2.64\cdot 10^{-2} $ $ 1.35 $
$ 0.00625 $ $ 1.10\cdot 10^{-2} $ $ 0.97 $ $ 1.29\cdot 10^{-3} $ $ 1.29 $ $ 2.25\cdot 10^{-3} $ $ 2.47 $ $ 3.18\cdot 10^{-3} $ $ 3.06 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 1.67\cdot 10^{-3} $ $ 6.79\cdot 10^{-4} $ $ 1.44\cdot 10^{-3} $ $ 3.52\cdot 10^{-4} $
$ 0.025 $ $ 8.65\cdot 10^{-4} $ $ 0.95 $ $ 2.70\cdot 10^{-4} $ $ 1.33 $ $ 7.91\cdot 10^{-4} $ $ 0.86 $ $ 2.60\cdot 10^{-4} $ $ 0.44 $
$ 0.0125 $ $ 4.36\cdot 10^{-4} $ $ 0.99 $ $ 1.24\cdot 10^{-4} $ $ 1.12 $ $ 4.05\cdot 10^{-4} $ $ 0.97 $ $ 1.53\cdot 10^{-5} $ $ 4.08 $
$ 0.00625 $ $ 2.17\cdot 10^{-4} $ $ 1.01 $ $ 5.71\cdot 10^{-5} $ $ 1.12 $ $ 2.05\cdot 10^{-4} $ $ 0.98 $ $ 9.43\cdot 10^{-6} $ $ 0.70 $
Table 7.  Convergence results for the semi-implicit scheme on the fine mesh
Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 9.18\cdot 10^{-2} $ $ 3.89\cdot 10^{-2} $ $ 2.36\cdot 10^{-1} $ $ 2.39\cdot 10^{-1} $
$ 0.025 $ $ 5.05\cdot 10^{-2} $ $ 0.86 $ $ 8.59\cdot 10^{-3} $ $ 2.18 $ $ 7.54\cdot 10^{-2} $ $ 1.64 $ $ 7.06\cdot 10^{-2} $ $ 1.76 $
$ 0.0125 $ $ 2.63\cdot 10^{-2} $ $ 0.94 $ $ 3.32\cdot 10^{-3} $ $ 1.37 $ $ 4.24\cdot 10^{-2} $ $ 0.83 $ $ 2.22\cdot 10^{-2} $ $ 1.67 $
$ 0.00625 $ $ 1.33\cdot 10^{-2} $ $ 0.98 $ $ 1.40\cdot 10^{-3} $ $ 1.24 $ $ 2.19\cdot 10^{-2} $ $ 0.96 $ $ 4.19\cdot 10^{-3} $ $ 2.40 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 2.03\cdot 10^{-3} $ $ 7.86\cdot 10^{-4} $ $ 1.81\cdot 10^{-3} $ $ 6.51\cdot 10^{-4} $
$ 0.025 $ $ 1.06\cdot 10^{-3} $ $ 0.93 $ $ 3.28\cdot 10^{-4} $ $ 1.26 $ $ 9.75\cdot 10^{-4} $ $ 0.89 $ $ 1.31\cdot 10^{-4} $ $ 2.31 $
$ 0.0125 $ $ 5.34\cdot 10^{-4} $ $ 1.00 $ $ 1.44\cdot 10^{-4} $ $ 1.18 $ $ 5.10\cdot 10^{-4} $ $ 0.93 $ $ 4.82\cdot 10^{-5} $ $ 1.44 $
$ 0.00625 $ $ 2.69\cdot 10^{-4} $ $ 0.99 $ $ 6.31\cdot 10^{-5} $ $ 1.19 $ $ 2.55\cdot 10^{-4} $ $ 1.00 $ $ 1.29\cdot 10^{-5} $ $ 1.90 $
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Fluid velocity
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 9.18\cdot 10^{-2} $ $ 3.89\cdot 10^{-2} $ $ 2.36\cdot 10^{-1} $ $ 2.39\cdot 10^{-1} $
$ 0.025 $ $ 5.05\cdot 10^{-2} $ $ 0.86 $ $ 8.59\cdot 10^{-3} $ $ 2.18 $ $ 7.54\cdot 10^{-2} $ $ 1.64 $ $ 7.06\cdot 10^{-2} $ $ 1.76 $
$ 0.0125 $ $ 2.63\cdot 10^{-2} $ $ 0.94 $ $ 3.32\cdot 10^{-3} $ $ 1.37 $ $ 4.24\cdot 10^{-2} $ $ 0.83 $ $ 2.22\cdot 10^{-2} $ $ 1.67 $
$ 0.00625 $ $ 1.33\cdot 10^{-2} $ $ 0.98 $ $ 1.40\cdot 10^{-3} $ $ 1.24 $ $ 2.19\cdot 10^{-2} $ $ 0.96 $ $ 4.19\cdot 10^{-3} $ $ 2.40 $
Structure deformation
$\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ \Delta t $ $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate $ L^2 $ error rate
$ 0.05 $ $ 2.03\cdot 10^{-3} $ $ 7.86\cdot 10^{-4} $ $ 1.81\cdot 10^{-3} $ $ 6.51\cdot 10^{-4} $
$ 0.025 $ $ 1.06\cdot 10^{-3} $ $ 0.93 $ $ 3.28\cdot 10^{-4} $ $ 1.26 $ $ 9.75\cdot 10^{-4} $ $ 0.89 $ $ 1.31\cdot 10^{-4} $ $ 2.31 $
$ 0.0125 $ $ 5.34\cdot 10^{-4} $ $ 1.00 $ $ 1.44\cdot 10^{-4} $ $ 1.18 $ $ 5.10\cdot 10^{-4} $ $ 0.93 $ $ 4.82\cdot 10^{-5} $ $ 1.44 $
$ 0.00625 $ $ 2.69\cdot 10^{-4} $ $ 0.99 $ $ 6.31\cdot 10^{-5} $ $ 1.19 $ $ 2.55\cdot 10^{-4} $ $ 1.00 $ $ 1.29\cdot 10^{-5} $ $ 1.90 $
Table 8.  Maximum residual in the semi-implicit scheme on the coarse mesh
$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 3.83\cdot 10^{-3} $ $ 3.83\cdot 10^{-3} $ $ 9.87\cdot 10^{-3} $ $ 9.64\cdot 10^{-2} $
$ 0.025 $ $ 2.09\cdot 10^{-3} $ $ 2.30\cdot 10^{-3} $ $ 1.24\cdot 10^{-3} $ $ 2.17\cdot 10^{-2} $
$ 0.0125 $ $ 7.41\cdot 10^{-4} $ $ 8.26\cdot 10^{-4} $ $ 3.62\cdot 10^{-4} $ $ 7.90\cdot 10^{-3} $
$ 0.00625 $ $ 2.23\cdot 10^{-4} $ $ 2.45\cdot 10^{-4} $ $ 1.08\cdot 10^{-4} $ $ 9.55\cdot 10^{-4} $
Table Options

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$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 3.83\cdot 10^{-3} $ $ 3.83\cdot 10^{-3} $ $ 9.87\cdot 10^{-3} $ $ 9.64\cdot 10^{-2} $
$ 0.025 $ $ 2.09\cdot 10^{-3} $ $ 2.30\cdot 10^{-3} $ $ 1.24\cdot 10^{-3} $ $ 2.17\cdot 10^{-2} $
$ 0.0125 $ $ 7.41\cdot 10^{-4} $ $ 8.26\cdot 10^{-4} $ $ 3.62\cdot 10^{-4} $ $ 7.90\cdot 10^{-3} $
$ 0.00625 $ $ 2.23\cdot 10^{-4} $ $ 2.45\cdot 10^{-4} $ $ 1.08\cdot 10^{-4} $ $ 9.55\cdot 10^{-4} $
Table 9.  Maximum residual in the semi-implicit scheme on the fine mesh
$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 3.59\cdot 10^{-2} $ $ 3.59\cdot 10^{-2} $ $ 3.43\cdot 10^{-2} $ $ 3.81\cdot 10^{-2} $
$ 0.025 $ $ 1.03\cdot 10^{-2} $ $ 1.07\cdot 10^{-2} $ $ 8.19\cdot 10^{-3} $ $ 1.02\cdot 10^{-2} $
$ 0.0125 $ $ 5.28\cdot 10^{-3} $ $ 5.87\cdot 10^{-3} $ $ 1.54\cdot 10^{-3} $ $ 2.27\cdot 10^{-3} $
$ 0.00625 $ $ 1.46\cdot 10^{-3} $ $ 1.44\cdot 10^{-3} $ $ 4.33\cdot 10^{-4} $ $ 5.08\cdot 10^{-4} $
Table Options

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$ \Delta t $ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$
$ 0.05 $ $ 3.59\cdot 10^{-2} $ $ 3.59\cdot 10^{-2} $ $ 3.43\cdot 10^{-2} $ $ 3.81\cdot 10^{-2} $
$ 0.025 $ $ 1.03\cdot 10^{-2} $ $ 1.07\cdot 10^{-2} $ $ 8.19\cdot 10^{-3} $ $ 1.02\cdot 10^{-2} $
$ 0.0125 $ $ 5.28\cdot 10^{-3} $ $ 5.87\cdot 10^{-3} $ $ 1.54\cdot 10^{-3} $ $ 2.27\cdot 10^{-3} $
$ 0.00625 $ $ 1.46\cdot 10^{-3} $ $ 1.44\cdot 10^{-3} $ $ 4.33\cdot 10^{-4} $ $ 5.08\cdot 10^{-4} $
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