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A four-field mixed finite element method for Biot's consolidation problems
An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
N. P. Thien,
A nonlinear network viscoelstic model, J. Rheol., 22 (1978), 259-283.
doi: 10.1122/1.549481. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
N. P. Thien,
A nonlinear network viscoelstic model, J. Rheol., 22 (1978), 259-283.
doi: 10.1122/1.549481. |
[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. |











Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
Constitutive equation | ||||
Newtonian | 0 | 0 | 0 | |
Upper Convected Maxwell | 0 | 0 | 0 | |
Oldroyd-B | 0 | 0 | ||
Linear PTT | 0 | |||
Exponential PTT | 1 |
Mesh | Type | Method | |||
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 |
ALS | 12993 | 4 | 3 |
Mesh H | Adaptive grids by |
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. |
Mesh | Type | Method | |||
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 |
ALS | 12993 | 4 | 3 |
Mesh H | Adaptive grids by |
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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