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September  2021, 29(4): 2755-2770. doi: 10.3934/era.2021012

An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows

Hsueh-Chen Lee 1,, and  Hyesuk Lee 2,

1. 

Center for General Education, Wenzao Ursuline University of Languages, 900 Mintsu 1st Road Kaohsiung, Taiwan
2. 

School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC 29634-0975, USA

* Corresponding author: Hsueh-Chen Lee

†The first author of this article would like to dedicate this article to her advisor Professor Tsu-Fen Chen (1956-2020) who was the main inspiration and guidance in her research in this area

Received  July 2020 Revised  December 2020 Published  September 2021 Early access  February 2021

Fund Project: The first author is supported by the Taiwan MOST grant 107-2115-M-160-001-MY2. The second author is grateful for the financial support provided in part by the US NSF grant DMS-1818842

We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.

Keywords: Least-squares, viscoelastic fluid, adaptive mesh refinement, a posteriori error estimator, transverse slot.
Mathematics Subject Classification: Primary: 65N30, 65N50, Secondary: 76A10.
Citation: Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows. Electronic Research Archive, 2021, 29 (4) : 2755-2770. doi: 10.3934/era.2021012
References:
[1]

J. H. Adler, T. A. Manteuffel, S. F. McCormick, J. W. Nolting, J. W. Ruge and L. Tang, Efficiency based adaptive local refinement for first-order system least-squares formulations, SIAM J. Sci. Comput., 33 (2011), 1-24.  doi: 10.1137/100786897.  Google Scholar

[2]

M. A. Alves, P. J. Oliveira and F. T. Pinho, Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions, J. Non-Newtonian Fluid Mech., 110 (2003), 45-75.  doi: 10.1016/S0377-0257(02)00191-X.  Google Scholar

[3]

R. B. Bird, R. C. Armstrong and O. Hassager, Fluid Mechanics, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, 2nd ed. New York, John Wiley and Sons, 1987. Google Scholar

[4]

Z. Cai and C. R. Westphal, An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type, J. Non-Newtonian Fluid Mech., 159 (2009), 72-80.  doi: 10.1016/j.jnnfm.2009.02.004.  Google Scholar

[5]

T. F. Chen, C. L. Cox, H. C. Lee and K. L. Tung, Least-squares finite element methods for generalized Newtonian and viscoelastic flows, Appl. Numer. Math., 60 (2010), 1024-1040.  doi: 10.1016/j.apnum.2010.07.006.  Google Scholar

[6]

T. F. Chen, H. Lee and C. C. Liu, Numerical approximation of the Oldroyd-B model by the weighted least-squares/discontinuous Galerkin method, Numer. Methods Partial Differ. Equations, 29 (2013), 531-548.  doi: 10.1002/num.21719.  Google Scholar

[7]

L. L. Ferrás, M. L. Morgado, M. Rebeloc, Gareth H. McKinley and A. M. Afonso, A generalised Phan-Thien-Tanner model, J. Non-Newtonian Fluid Mech., 269 (2019), 88-89.  doi: 10.1016/j.jnnfm.2019.06.001.  Google Scholar

[8]

J. Ku and E. J. Park, A posteriori error estimators for the first-order least-squares finite element method, J. Comput. Appl. Math., 235 (2010), 293-300.  doi: 10.1016/j.cam.2010.06.004.  Google Scholar

[9]

H. C. Lee, A nonlinear weighted least-squares finite element method for the Oldroyd-B viscoelastic flow, Appl. Math. Comput., 219 (2012), 421-434.  doi: 10.1016/j.amc.2012.06.036.  Google Scholar

[10]

H. C. Lee, An adaptively refined least-squares finite element method for generalized Newtonian fluid flows using the Carreau model, SIAM J. Sci. Comput., 36 (2014), 193-218.  doi: 10.1137/130912682.  Google Scholar

[11]

H. C. Lee, Adaptive weights for mass conservation in a least-squares finite element method, Int. J. Comput. Math., 95 (2018), 20-35.  doi: 10.1080/00207160.2017.1397639.  Google Scholar

[12]

H. C. Lee and T. F. Chen, A nonlinear weighted least-squares finite element method for the Stokes equations, Comput. Math. Appl., 59 (2010), 215-224.  doi: 10.1016/j.camwa.2009.08.033.  Google Scholar

[13]

H. C. Lee and T. F. Chen, Adaptive least-squares finite element approximations to Stokes equations, J. Comput. Appl. Math., 280 (2015), 396-412.  doi: 10.1016/j.cam.2014.11.041.  Google Scholar

[14]

H. C. Lee and H. Lee, Numerical simulations of viscoelastic fluid flows past a transverse slot using least-squares finite element methods, J. Sci. Comput., 79 (2019), 369-388.  doi: 10.1007/s10915-018-0856-x.  Google Scholar

[15]

J. L. Liu, Exact a posteriori error analysis of the least squares finite element method, Appl. Math. Comput., 116 (2000), 297-305.  doi: 10.1016/S0096-3003(99)00153-8.  Google Scholar

[16]

N. P. Thien, A nonlinear network viscoelstic model, J. Rheol., 22 (1978), 259-283.  doi: 10.1122/1.549481.  Google Scholar

[17]

N. P. Thien and R. I. Tanner, A new constitutive equation derived from network theory, Journal of Non-Newtonian Fluid Mechanics, 2 (1977), 353-365.  doi: 10.1016/0377-0257(77)80021-9.  Google Scholar

show all references

References:
[1]

J. H. Adler, T. A. Manteuffel, S. F. McCormick, J. W. Nolting, J. W. Ruge and L. Tang, Efficiency based adaptive local refinement for first-order system least-squares formulations, SIAM J. Sci. Comput., 33 (2011), 1-24.  doi: 10.1137/100786897.  Google Scholar

[2]

M. A. Alves, P. J. Oliveira and F. T. Pinho, Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions, J. Non-Newtonian Fluid Mech., 110 (2003), 45-75.  doi: 10.1016/S0377-0257(02)00191-X.  Google Scholar

[3]

R. B. Bird, R. C. Armstrong and O. Hassager, Fluid Mechanics, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, 2nd ed. New York, John Wiley and Sons, 1987. Google Scholar

[4]

Z. Cai and C. R. Westphal, An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type, J. Non-Newtonian Fluid Mech., 159 (2009), 72-80.  doi: 10.1016/j.jnnfm.2009.02.004.  Google Scholar

[5]

T. F. Chen, C. L. Cox, H. C. Lee and K. L. Tung, Least-squares finite element methods for generalized Newtonian and viscoelastic flows, Appl. Numer. Math., 60 (2010), 1024-1040.  doi: 10.1016/j.apnum.2010.07.006.  Google Scholar

[6]

T. F. Chen, H. Lee and C. C. Liu, Numerical approximation of the Oldroyd-B model by the weighted least-squares/discontinuous Galerkin method, Numer. Methods Partial Differ. Equations, 29 (2013), 531-548.  doi: 10.1002/num.21719.  Google Scholar

[7]

L. L. Ferrás, M. L. Morgado, M. Rebeloc, Gareth H. McKinley and A. M. Afonso, A generalised Phan-Thien-Tanner model, J. Non-Newtonian Fluid Mech., 269 (2019), 88-89.  doi: 10.1016/j.jnnfm.2019.06.001.  Google Scholar

[8]

J. Ku and E. J. Park, A posteriori error estimators for the first-order least-squares finite element method, J. Comput. Appl. Math., 235 (2010), 293-300.  doi: 10.1016/j.cam.2010.06.004.  Google Scholar

[9]

H. C. Lee, A nonlinear weighted least-squares finite element method for the Oldroyd-B viscoelastic flow, Appl. Math. Comput., 219 (2012), 421-434.  doi: 10.1016/j.amc.2012.06.036.  Google Scholar

[10]

H. C. Lee, An adaptively refined least-squares finite element method for generalized Newtonian fluid flows using the Carreau model, SIAM J. Sci. Comput., 36 (2014), 193-218.  doi: 10.1137/130912682.  Google Scholar

[11]

H. C. Lee, Adaptive weights for mass conservation in a least-squares finite element method, Int. J. Comput. Math., 95 (2018), 20-35.  doi: 10.1080/00207160.2017.1397639.  Google Scholar

[12]

H. C. Lee and T. F. Chen, A nonlinear weighted least-squares finite element method for the Stokes equations, Comput. Math. Appl., 59 (2010), 215-224.  doi: 10.1016/j.camwa.2009.08.033.  Google Scholar

[13]

H. C. Lee and T. F. Chen, Adaptive least-squares finite element approximations to Stokes equations, J. Comput. Appl. Math., 280 (2015), 396-412.  doi: 10.1016/j.cam.2014.11.041.  Google Scholar

[14]

H. C. Lee and H. Lee, Numerical simulations of viscoelastic fluid flows past a transverse slot using least-squares finite element methods, J. Sci. Comput., 79 (2019), 369-388.  doi: 10.1007/s10915-018-0856-x.  Google Scholar

[15]

J. L. Liu, Exact a posteriori error analysis of the least squares finite element method, Appl. Math. Comput., 116 (2000), 297-305.  doi: 10.1016/S0096-3003(99)00153-8.  Google Scholar

[16]

N. P. Thien, A nonlinear network viscoelstic model, J. Rheol., 22 (1978), 259-283.  doi: 10.1122/1.549481.  Google Scholar

[17]

N. P. Thien and R. I. Tanner, A new constitutive equation derived from network theory, Journal of Non-Newtonian Fluid Mechanics, 2 (1977), 353-365.  doi: 10.1016/0377-0257(77)80021-9.  Google Scholar

Figure 1.  Reduction of the functional norm $ \delta g^{1/2}_{m} $ on three meshes for various weights $ W_2 = 10^{m} $ ($ 2 \le m \le 6 $)
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Figure 2.  $ L^{2} $ errors of $ {\boldsymbol \tau} $, $ {\boldsymbol \sigma} $, $ {p} $, $ H^{1} $ error in $ {\bf u} $, and $ g^{1/2} $ functional norm of the LS solutions for (a) $ W_2 = 10^{2} $ and (b) $ W_2 = 10^{6} $
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Figure 3.  The slot channel with $ d = h = l = 1 $
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Figure 4.  $ \Delta g^{1/2}_{(k)}/\Delta N_{k} = |g^{1/2}_{(k)}-g^{1/2}_{(k-1)}|/|N_{k}-N_{k-1}| $ versus the number of elements $ N_{k} $ for $ k = $1, 2, and 3, where $ g^{1/2}_{(k)} $ denotes the functional value by $ N_{k} $
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Figure 5.  Mesh G with various number of elements $ N_k $ in refinement steps
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Figure 6.  The LS solutions using $ W_2 = 10^{6} $ on the uniformly refined Mesh U with 6144 elements. Reduction of functional norm $ \delta g^{1/2}_{m} $ and $ u(2.5,0.5) $ for various weights $ W_2 = 10^{m} $ ($ 2 \le m \le 6) $
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Figure 7.  Streamlines of LS solutions on the refined Mesh U with 6144 elements using (a) $ W_2 = 10^{4} $ and (b) $ W_2 = 10^{6} $
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Figure 8.  LS solutions at $ Re = 1 $, $ We = 0.1 $, and $ \epsilon = 0.5 $ using $ W_2 = 10^{6} $. (a) Grid independence results for $ g^{1/2} $. (b) Convergence of $ g^{1/2} $ by the LS and ALS methods
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Figure 9.  Plots of $ g^{1/2}_e $ on (a) the initial Mesh T with 1536 elements, (b) the uniformly refined Mesh U with 24576 elements, and (c) the adaptive Mesh G with 12993 elements
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Figure 10.  Plots of $ g^{1/2}_e $ on (a) the initial Mesh S with 384 elements and (b) the adaptive Mesh H with 18320 elements at the 5-th refinement step
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Figure 11.  Streamlines of the flow over a slot by the ALS method for (a) the exponential PTT ($ a = 1 $), (b) the linear PTT ($ a = 0 $), and (c) the Oldroyd-B ($ \epsilon = 0 $) models at $ We = 1 $
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Table 1.  Viscoelastic fluid models for $ (a,\ We,\ \beta,\ \epsilon) $
Constitutive equation $ a $ $ We $ $ \beta $ $ \epsilon $
Newtonian 0 0 $ \beta $ 0
Upper Convected Maxwell 0 $ We $ 0 0
Oldroyd-B 0 $ We $ $ \beta $ 0
Linear PTT 0 $ We $ $ \beta $ $ \epsilon $
Exponential PTT 1 $ We $ $ \beta $ $ \epsilon $
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Constitutive equation $ a $ $ We $ $ \beta $ $ \epsilon $
Newtonian 0 0 $ \beta $ 0
Upper Convected Maxwell 0 $ We $ 0 0
Oldroyd-B 0 $ We $ $ \beta $ 0
Linear PTT 0 $ We $ $ \beta $ $ \epsilon $
Exponential PTT 1 $ We $ $ \beta $ $ \epsilon $
Table 2.  Meshes considered for $ (We,\ \epsilon,\ Re) $ = $ (0.1,\ 0.5,\ 1) $
Mesh Type Method $ N_k ^{a}$ $ S ^{b}$ $ k ^{c}$
Mesh S Initial uniform grids LS 348 4 –
Mesh T Initial uniform grids LS 1536 4 –
Mesh U Uniform refined grids with Mesh T LS 24576 5 2
Mesh G Adaptive grids by $ {\boldsymbol g^{1/2}} $ with Mesh T ALS 12993 4 3
Mesh H Adaptive grids by $ {\boldsymbol g^{1/2}} $ with Mesh S ALS 18320 4 5
aNk represents the number of elements at the k refinement step.
b S is the number of Newton steps for convergence.
c k is the number of mesh refinements.
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Mesh Type Method $ N_k ^{a}$ $ S ^{b}$ $ k ^{c}$
Mesh S Initial uniform grids LS 348 4 –
Mesh T Initial uniform grids LS 1536 4 –
Mesh U Uniform refined grids with Mesh T LS 24576 5 2
Mesh G Adaptive grids by $ {\boldsymbol g^{1/2}} $ with Mesh T ALS 12993 4 3
Mesh H Adaptive grids by $ {\boldsymbol g^{1/2}} $ with Mesh S ALS 18320 4 5
aNk represents the number of elements at the k refinement step.
b S is the number of Newton steps for convergence.
c k is the number of mesh refinements.
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