Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

Bikram Bir; Deepjyoti Goswami; Amiya K. Pani

doi:10.1515/cmam-2022-0012

Published by De Gruyter February 12, 2022

Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

Bikram Bir , Deepjyoti Goswami and Amiya K. Pani

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2022-0012

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Abstract

In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 . New regularity results which are valid uniformly in time as t → ∞ and in the penalty parameter ε as ε → 0 are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 as well in 𝐇 1 -norms and for the penalized pressure in L 2 -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

Keywords: Oldroyd Fluid of Order One; Penalty Method; Uniform-in-Time Bound; Optimal Error Estimates; Non-smooth Initial Data

MSC 2010: 65M60; 65M15; 35Q35

Funding statement: The first author would like to express his gratitude to the Department of Science and Technology (DST), Government of India, for the financial support (DST/INSPIRE Fellowship/IF170401).

References

[1] B. Bir and D. Goswami, On a three step two-grid finite element method for the Oldroyd model of order one, Z. Angew. Math. Mech. 101 (2021), Article ID e202000373. 10.1002/zamm.202000373Search in Google Scholar

[2] B. Bir, D. Goswami and A. K. Pani, Backward euler method for the equations of motion arising in Oldroyd fluids of order one with non-smooth initial data, IMA J. Numer. Anal. (2021), 10.1093/imanum/drab072. 10.1093/imanum/drab072Search in Google Scholar

[3] B. Brefort, J.-M. Ghidaglia and R. Temam, Attractors for the penalized Navier–Stokes equations, SIAM J. Math. Anal. 19 (1988), no. 1, 1–21. 10.1137/0519001Search in Google Scholar

[4] J. R. Cannon and Y. P. Lin, A priori L 2 error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal. 27 (1990), no. 3, 595–607. 10.1137/0727036Search in Google Scholar

[5] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1–23. 10.1201/b16924-2Search in Google Scholar

[6] U. Ghia, K. N. Ghia and C. T. Shin, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), 387–411. 10.1016/0021-9991(82)90058-4Search in Google Scholar

[7] D. Goswami and A. K. Pani, A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one, Int. J. Numer. Anal. Model. 8 (2011), no. 2, 324–352. Search in Google Scholar

[8] Y. He, Optimal error estimate of the penalty finite element method for the time-dependent Navier–Stokes equations, Math. Comp. 74 (2005), no. 251, 1201–1216. 10.1090/S0025-5718-05-01751-5Search in Google Scholar

[9] Y. He, Y. Lin, S. S. P. Shen, W. Sun and R. Tait, Finite element approximation for the viscoelastic fluid motion problem, J. Comput. Appl. Math. 155 (2003), no. 2, 201–222. 10.1016/S0377-0427(02)00864-6Search in Google Scholar

[10] Y. He, Y. Lin, S. S. P. Shen and R. Tait, On the convergence of viscoelastic fluid flows to a steady state, Adv. Differential Equations 7 (2002), no. 6, 717–742. 10.57262/ade/1356651735Search in Google Scholar

[11] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3–4, 251–265. 10.1515/jnum-2012-0013Search in Google Scholar

[12] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311. 10.1137/0719018Search in Google Scholar

[13] J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. 10.1137/0727022Search in Google Scholar

[14] A. A. Kotsiolis and A. P. Oskolkov, Initial-boundary value problems for equations of weakly compressible Jeffreys–Oldroyd fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 208 (1993), no. 7, 200–218, 223. Search in Google Scholar

[15] Y. P. Lin, V. Thomée and L. B. Wahlbin, Ritz–Volterra projections to finite-element spaces and applications to integrodifferential and related equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 1047–1070. 10.1137/0728056Search in Google Scholar

[16] C. Liu and Z. Si, An incremental pressure correction finite element method for the time-dependent Oldroyd flows, Appl. Math. Comput. 351 (2019), 99–115. 10.1016/j.amc.2019.01.026Search in Google Scholar

[17] M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Stoch. Anal. Appl. 38 (2020), no. 1, 1–61. 10.1080/07362994.2019.1646138Search in Google Scholar

[18] M. T. Mohan, Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one, Stochastic Process. Appl. 130 (2020), no. 8, 4513–4562. 10.1016/j.spa.2020.01.007Search in Google Scholar

[19] J. G. Oldroyd, Non-Newtonian flow of liquids and solids, Rheology: Theory and Applications, Vol. I, Academic Press, New York (1956), 653–682. 10.1016/B978-0-12-395694-1.50022-1Search in Google Scholar

[20] A. P. Oskolkov, The penalty method for the equations of viscoelastic media, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 224 (1995), no. 13, 267–278, 340–341. 10.1007/BF02364990Search in Google Scholar

[21] A. K. Pani and J. Y. Yuan, Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model, IMA J. Numer. Anal. 25 (2005), no. 4, 750–782. 10.1093/imanum/dri016Search in Google Scholar

[22] A. K. Pani, J. Y. Yuan and P. D. Damázio, On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one, SIAM J. Numer. Anal. 44 (2006), no. 2, 804–825. 10.1137/S0036142903428967Search in Google Scholar

[23] J. Shen, On error estimates of the penalty method for unsteady Navier–Stokes equations, SIAM J. Numer. Anal. 32 (1995), no. 2, 386–403. 10.1137/0732016Search in Google Scholar

[24] R. Temam, Une méthode d’approximation de la solution des équations de Navier–Stokes, Bull. Soc. Math. France 96 (1968), 115–152. 10.24033/bsmf.1662Search in Google Scholar

[25] K. Wang, Y. He and X. Feng, On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization, Appl. Math. Model. 34 (2010), no. 12, 4089–4105. 10.1016/j.apm.2010.04.008Search in Google Scholar

[26] K. Wang, Y. He and X. Feng, On error estimates of the fully discrete penalty method for the viscoelastic flow problem, Int. J. Comput. Math. 88 (2011), no. 10, 2199–2220. 10.1080/00207160.2010.534781Search in Google Scholar

[27] K. Wang, Y. He and Y. Shang, Fully discrete finite element method for the viscoelastic fluid motion equations, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), no. 3, 665–684. 10.3934/dcdsb.2010.13.665Search in Google Scholar

[28] K. Wang, Y. Lin and Y. He, Asymptotic analysis of the equations of motion for viscoelastic Oldroyd fluid, Discrete Contin. Dyn. Syst. 32 (2012), no. 2, 657–677. 10.3934/dcds.2012.32.657Search in Google Scholar

[29] K. Wang, Y. Shang and H. Wei, A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations, Comput. Math. Appl. 62 (2011), no. 4, 1814–1827. 10.1016/j.camwa.2011.06.025Search in Google Scholar

[30] K. Wang, Y. Shang and R. Zhao, Optimal error estimates of the penalty method for the linearized viscoelastic flows, Int. J. Comput. Math. 87 (2010), no. 14, 3236–3253. 10.1080/00207160902980500Search in Google Scholar

[31] Y. Yang, Y. Lei and Z. Si, Unconditional stability and error estimates of the modified characteristics FEM for the time-dependent viscoelastic Oldroyd flows, Adv. Appl. Math. Mech. 13 (2021), no. 2, 311–332. 10.4208/aamm.OA-2018-0169Search in Google Scholar

[32] T. Zhang and Y. Qian, Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data, Numer. Methods Partial Differential Equations 34 (2018), no. 6, 2180–2216. 10.1002/num.22283Search in Google Scholar

[33] T. Zhang, Y. Qian, T. Jiang and J. Yuan, Stability and convergence of the higher projection method for the time-dependent viscoelastic flow problem, J. Comput. Appl. Math. 338 (2018), 1–21. 10.1016/j.cam.2017.12.045Search in Google Scholar

Received: 2022-01-09

Accepted: 2022-01-09

Published Online: 2022-02-12

Published in Print: 2022-04-01

Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

Abstract

References

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