Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Amiya K. Pani; Bikram Bir; Deepjyoti Goswami

arxiv: 2106.16052 · v1 · submitted 2021-06-30 · 🧮 math.NA · cs.NA

Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Bikram Bir , Deepjyoti Goswami , Amiya K. Pani This is my paper

Pith reviewed 2026-05-24 13:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Oldroyd modelbackward Euler methodfinite element discretizationa priori error estimatesviscoelastic fluidsnonsmooth initial datauniform in time

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The pith

The backward Euler finite element method for the Oldroyd model achieves optimal L2 error estimates that are uniform in time for nonsmooth initial data under a uniqueness condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops and analyzes a numerical scheme for the equations describing viscoelastic fluid motion in the Oldroyd model of order one in two dimensions. It uses backward Euler stepping in time together with finite element approximation in space, for cases where the forcing is time-independent or bounded in time. The discrete solutions are shown to stay bounded in the Dirichlet norm for all time. Optimal error bounds in the L2 norm are proved for problems starting from nonsmooth data, and these bounds do not grow with time provided the continuous problem has unique solutions.

Core claim

For the 2D Oldroyd model of viscoelastic fluids of order one, the backward Euler method with finite element discretization in space yields discrete solutions that are bounded uniformly in time in the Dirichlet norm. Optimal a priori error estimates in the L2 norm are derived for nonsmooth initial data, and these estimates remain uniform in time when the uniqueness condition holds for the continuous problem.

What carries the argument

Backward Euler time discretization combined with finite element spatial discretization applied to the Oldroyd viscoelastic fluid equations.

If this is right

The discrete solutions remain bounded in the Dirichlet norm uniformly in time.
Optimal L2-norm error estimates are obtained for nonsmooth initial data.
These error estimates are uniform in time under the uniqueness assumption.
Numerical results confirm the theoretical predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be applied to long-time integration of viscoelastic flows without accumulating errors over time.
Similar uniform estimates might be possible for other time discretizations if they preserve the structure used here.
Extensions to three dimensions would require establishing the uniqueness condition first.

Load-bearing premise

The continuous Oldroyd problem satisfies a uniqueness condition.

What would settle it

A numerical experiment where the L2 error grows without bound as time increases, for a case with nonsmooth initial data where uniqueness is known to hold.

Figures

Figures reproduced from arXiv: 2106.16052 by Amiya K. Pani, Bikram Bir, Deepjyoti Goswami.

**Figure 2.** Figure 2: Velocity and pressure errors based on MINI element for E [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Figure 3: Velocity and pressure errors based on P2-P0 element for [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

**Figure 4.** Figure 4: Velocity and pressure errors based on MINI element for E [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

**Figure 5.** Figure 5: Velocity errors in L 2 - norm with respect to time for Example 7.2. For the example 7.2, the numerical results are shown for final time T = 10, 20, 30, 40 and 50 with µ = 1, γ = 0.1, δ = 1, k = 0.1 and h = 2−i , i = 2, 3, . . . , 6. We represent the errors and the convergence rates for the velocity in L 2 -norm for P2-P0 and MINI-elements in [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Uniform in time errors for P2-P0 element (left) and MINI ele [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

read the original abstract

In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the $2D$ Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal {\it a priori} error estimate in $\textbf{L}^2$-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Uniform-in-time L2 error bound for this Oldroyd FEM scheme holds only under an unproven uniqueness assumption on the continuous problem.

read the letter

The key point is that the paper obtains time-uniform L2 error estimates for backward Euler plus finite elements on the 2D Oldroyd order-one system with nonsmooth initial data, but only after assuming the continuous problem has a unique solution. That assumption is needed to control the nonlinear terms in the error equation via Gronwall, and the abstract states it explicitly as a requirement. Without it the bound would pick up a time-dependent factor that can grow. They also show the discrete solution stays bounded in the Dirichlet norm uniformly in time when the forcing is time-independent or merely L^infty in time. Numerical tests are included to check the predicted rates. This is a direct extension of standard energy-method arguments for parabolic systems to the extra stress coupling in this viscoelastic model. The focus on nonsmooth data and the uniform-in-time property is the concrete addition over earlier work on similar problems. The analysis appears to follow the usual pattern of splitting the error, using the discrete scheme to cancel terms, and applying the uniqueness hypothesis to absorb the convective and coupling contributions. The numerical section provides basic confirmation that the rates are observed in practice. The soft spot is the uniqueness condition itself. The abstract treats it as an extra hypothesis rather than something proved for the given data class, and the stress-test note indicates the manuscript does not supply a proof or reference establishing it for L^infty forcing and rough initials. In 2D Oldroyd-type systems uniqueness is not automatic for arbitrary data, so the main result remains conditional. If the paper simply invokes the assumption without further justification, that limits how far the uniform bound can be claimed. This paper is for people working on numerical methods for viscoelastic flows who already know the standard backward Euler and FEM toolkit. It is incremental but technically competent on its own terms. It deserves a serious referee who can check the details of the error estimates and the handling of the uniqueness hypothesis. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript analyzes a backward Euler finite-element discretization of the 2D Oldroyd model of order one. It proves that the discrete solution remains bounded uniformly in time in the Dirichlet norm when the forcing is time-independent or merely L^∞ in time. Under an additional uniqueness assumption on the continuous problem, it derives an optimal a priori L² error estimate that is also uniform in time, even for nonsmooth initial data, and supplies numerical experiments to illustrate the theory.

Significance. If the uniqueness condition can be justified for the stated data classes, the uniform-in-time L² error bound would constitute a useful advance in the numerical analysis of viscoelastic flows, extending existing results to nonsmooth data while controlling the growth of the error constant.

major comments (1)

[Abstract] Abstract: The claim of a uniform-in-time optimal L² error estimate is obtained only after invoking an unproven uniqueness condition on the continuous Oldroyd system. No proof or reference is supplied showing that this condition holds for the 2-D model with L^∞-in-time forcing and nonsmooth initial data; without it the Gronwall argument produces a time-dependent factor that may grow, undermining the uniformity assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the role of the uniqueness assumption. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The claim of a uniform-in-time optimal L² error estimate is obtained only after invoking an unproven uniqueness condition on the continuous Oldroyd system. No proof or reference is supplied showing that this condition holds for the 2-D model with L^∞-in-time forcing and nonsmooth initial data; without it the Gronwall argument produces a time-dependent factor that may grow, undermining the uniformity assertion.

Authors: We agree that the uniform-in-time L² error bound is conditional on the uniqueness assumption for the continuous problem. The manuscript already states this explicitly both in the abstract and in the body of the paper. Establishing uniqueness for the 2-D Oldroyd model of order one with merely L^∞-in-time forcing and nonsmooth initial data is a difficult open question in the analysis of viscoelastic flows and lies outside the scope of the present numerical-analysis work. Similar conditional error estimates under a uniqueness hypothesis appear in the literature on the Navier–Stokes equations and related viscoelastic models. Without the assumption the Gronwall factor may indeed grow, which is why the result is stated conditionally. We will revise the abstract to emphasize the conditional character more prominently. revision: yes

Circularity Check

0 steps flagged

No circularity; error estimates derived under explicit external assumption

full rationale

The provided abstract and description show a standard mathematical derivation of a priori L2 error estimates for a backward Euler FEM discretization of the 2D Oldroyd system. The uniform-in-time bound is obtained under an explicitly stated assumption of uniqueness for the continuous problem; this is a hypothesis, not a result derived inside the paper. No self-definitional steps, fitted inputs renamed as predictions, self-citation chains, or ansatzes appear in the text. The analysis is self-contained as a proof under stated conditions, consistent with the default expectation for theoretical papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis depends on standard PDE theory plus one model-specific assumption; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Uniqueness condition for the continuous Oldroyd problem
Explicitly invoked in the abstract to obtain the uniform-in-time error estimate.

pith-pipeline@v0.9.0 · 5651 in / 1156 out tokens · 22288 ms · 2026-05-24T13:21:34.767924+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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Abbaszadeh, M. & Dehghan, M. (2020) Investigation of the Oldroyd model as a generalized incompr essible Navier-Stokes equation via the interpolating stabilized element free Galerkin technique. Appl. Numer. Math. , 150, 274-294. 27 ❍ ❍ ❍ ❍ ❍h k 0.1 0.5 1 1.3 1/10 0.04066058 0.04013758 0.03994627 0.03431507 P2-P0 1/20 0.04060207 0.04007989 0.03988995 0.0342...

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Goswami, D. & Pani, A. K. (2011) A priori error estimates for semidiscrete ﬁnite element app roximations to the equations of motion arising in Oldroyd ﬂuids of order one. Int. J. Numer. Anal. Model. , 8, 324–352

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He, Y., Huang, P. & Feng, X. (2015) H 2-Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier-Stokes Equations. J. Sci. Comput. , 62, 230–264

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Heywood, J. G. & Rannacher, R. (1982) Finite element approximation of the nonstationary Navier-S tokes problem: I. Regularity of solutions and second order error estimat es for spatial discretization. SIAM J. Numer. Anal., 19, 275–311

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Liu, C. & Si, Z. (2019) An incremental pressure correction ﬁnite element method for the time-dependent Oldroyd ﬂows. Appl. Math. Comput. , 351, 99-115

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(2020) Deterministic and stochastic equations of motion arising in Old royd ﬂuids of order one: existence, uniqueness, exponential stability and invariant measur es

Mohan, M.T. (2020) Deterministic and stochastic equations of motion arising in Old royd ﬂuids of order one: existence, uniqueness, exponential stability and invariant measur es. Stochastic Anal. Appl. , 38, 1-61

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(2020) Well posedness, large deviations and ergodicity of the stoc hastic 2D Oldroyd model of order one

Mohan, M.T. (2020) Well posedness, large deviations and ergodicity of the stoc hastic 2D Oldroyd model of order one. Stochastic Processes and their Applications , 130, 4513-4562

work page 2020
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& Thom ´ ee, V

McLean, W. & Thom ´ ee, V. (1993) Numerical solution of an evolution equation with a positive typ e memory term. J. Austral. Math. Soc. Ser. B , 35, 23–70

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Oldroyd, J. G. (1956) Non-Newtonian ﬂow of liquids and solids. Rheology: Theory a nd Applications, Vol. I (F. R. Eirich, Ed.). New York: Academic Press, pp. 653–682

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Pani, A. K. & Sinha, R. K. (1998) On the backward Euler method for time dependent parabolic integro- diﬀerential equations with nonsmooth initial data. J. Integral Equations Appl. , 10, 219–249

work page 1998
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Pani, A. K. & Sinha, R. K. (1998) Quadrature based ﬁnite element approximations to time dep endent parabolic equations with nonsmooth initial data. Calcolo, 35, 225–248

work page 1998
[22]

K., Thom ´ ee, V

Pani, A. K., Thom ´ ee, V. & W ahlbin, L. B. (1992) Numerical Methods for Hyperbolic and Parabolic Integro-Diﬀerential Equations. J. Integral Equations Appl. , 4, 533–584

work page 1992
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Pani, A. K. & Yuan, J. Y. (2005) Semidiscrete ﬁnite element Galerkin approximations to the eq uations of motion arising in the Oldroyd model. IMA J. Numer. Anal. , 25, 750–782

work page 2005
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K., Yuan, J

Pani, A. K., Yuan, J. Y. & Damazio, P. (2006) On a linearized backward Euler method for the equations of motion arising in the Oldroyd ﬂuids of order one. SIAM J. Numer. Anal. , 44, 804–825

work page 2006
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(1984) Navier-Stokes Equations, Theory and Numerical Analysis , Studies in Mathematics and its Application 2, Third Edition

Temam, R. (1984) Navier-Stokes Equations, Theory and Numerical Analysis , Studies in Mathematics and its Application 2, Third Edition. Amsterdam: North-Holland Publishing Co., pp. xii+526

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& Wirosoetisno, D

Tone, F. & Wirosoetisno, D. (2006) On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations. SIAM J. Numer. Anal. , 44, 29–40

work page 2006
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& Zhang, N

Thom´ ee, V. & Zhang, N. Y. (1989) Error estimates for semidiscrete ﬁnite element methods fo r parabolic integro-diﬀerential equations. Math. Comp. , 53, 121–139

work page 1989
[28]

& Shang, Y

W ang, K., He, Y. & Shang, Y. (2010) Fully discrete ﬁnite element method for the viscoelastic ﬂuid m otion equations. Disc. Cont. Dyn. Sys. Ser. B , 13, 665–684

work page 2010
[29]

Yang, Y., Lei, Y. & Si, Z. (2020) Unconditional stability and error estimates of the modiﬁed c haracteristics FEM for the time-dependent viscoelastic Oldroyd ﬂows. Adv. Appl. Math. Mech. , 13, 311–332. 29

work page 2020
[30]

& Qian, Y

Zhang, T. & Qian, Y. (2018) Stability analysis of several ﬁrst order schemes for the Old royd model with smooth and nonsmooth initial data. Numer. Meth. Partial Diﬀ. Eqn. , 34, 2180–2216

work page 2018
[31]

& Yuan, J

Zhang, T., Qian, Y., Jiang, T. & Yuan, J. (2018) Stability and convergence of the higher projection method for the time-dependent viscoelastic ﬂow problem. J. Comput. Appl. Math. , 318, 1–21

work page 2018
[32]

& Yuan, J

Zhang, T. & Yuan, J. (2015) A stabilized characteristic ﬁnite element method for the visc oelastic Oldroyd ﬂuid motion problem. Int. J. Numer. Anal. Model. , 12, 617–635

work page 2015
[33]

& Qian, Y

Zhao, J., Zhang, T. & Qian, Y. (2018) Stability and convergence of second order time discrete pr ojection method for the linearized Oldroyd model. Appl. Math. Comput. , 316, 342–635. 30

work page 2018

[1] [1]

& Dehghan, M

Abbaszadeh, M. & Dehghan, M. (2020) Investigation of the Oldroyd model as a generalized incompr essible Navier-Stokes equation via the interpolating stabilized element free Galerkin technique. Appl. Numer. Math. , 150, 274-294. 27 ❍ ❍ ❍ ❍ ❍h k 0.1 0.5 1 1.3 1/10 0.04066058 0.04013758 0.03994627 0.03431507 P2-P0 1/20 0.04060207 0.04007989 0.03988995 0.0342...

work page 2020

[2] [2]

Akhmatov, M. M. & Oskolkov, A. P. (1989) On convergent diﬀerence schemes for the equations of mo tion of an Oldroyd ﬂuid. J. Math. Sci. , 47, 2926–2933

work page 1989

[3] [3]

& Fortin, M

Brezzi, F. & Fortin, M. (1991) Mixed and Hybrid ﬁnite element methods , Springer Series in Computational Mathematics 15. New York: Springer-Verlag

work page 1991

[4] [4]

& Pironneau, O

Bercovier, M. & Pironneau, O. (1979) Error estimates for ﬁnite element solution of the Stokes pr oblem in the primitive variables. Numer. Math. 33, 211–224

work page 1979

[5] [5]

& Raviart, P

Girault, V. & Raviart, P. A. (1980) Finite element approximation of the Navier-Stokes equatio ns, Lecture Notes in Mathematics, 749. Berlin-New York: Springer-Verlag

work page 1980

[6] [6]

(2011) Finite element approximation to the equations of motion arisin g in Oldroyd viscoelastic model of order one

Goswami, D. (2011) Finite element approximation to the equations of motion arisin g in Oldroyd viscoelastic model of order one. Ph.D. Thesis , Department of Mathematics, IIT Bombay

work page 2011

[7] [7]

& Pani, A

Goswami, D. & Pani, A. K. (2011) A priori error estimates for semidiscrete ﬁnite element app roximations to the equations of motion arising in Oldroyd ﬂuids of order one. Int. J. Numer. Anal. Model. , 8, 324–352

work page 2011

[8] [8]

Guo, Y. & He, Y. (2016) An eﬃcient and accurate fully discrete ﬁnite element method for unsteady incompressible Oldroyd ﬂuids with large time step. Internat. J. Numer. Methods Fluids , 80, 375–394. 28

work page 2016

[9] [9]

& He, Y., (2018) On the Euler implicit/explicit iterative scheme for the stationa ry Oldroyd ﬂuid

Guo, Y. & He, Y., (2018) On the Euler implicit/explicit iterative scheme for the stationa ry Oldroyd ﬂuid. Numer. Meth. Partial Diﬀ. Eqn. , 34, 906–937

work page 2018

[10] [10]

& Feng, X

He, Y., Huang, P. & Feng, X. (2015) H 2-Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier-Stokes Equations. J. Sci. Comput. , 62, 230–264

work page 2015

[11] [11]

He, Y. & Li, K. (1998) Convergence and stability of ﬁnite element nonlinear Galerkin method for the Navier- Stokes equations. Numer. Math. , 79, 77–106

work page 1998

[12] [12]

He, Y., Lin, Y., Shen, S. S. P., Sun, W. & Tait, R. (2003) Finite element approximation for the viscoelastic ﬂuid motion problem. J. Comput. Appl. Math. , 155, 201–222

work page 2003

[13] [13]

Heywood, J. G. & Rannacher, R. (1982) Finite element approximation of the nonstationary Navier-S tokes problem: I. Regularity of solutions and second order error estimat es for spatial discretization. SIAM J. Numer. Anal., 19, 275–311

work page 1982

[14] [14]

Heywood, J. G. & Rannacher, R. (1990) Finite element approximation of the nonstationary Navier-S tokes problem: I. Error analysis for second-order time discretization. SIAM J. Numer. Anal. , 27, 353–384

work page 1990

[15] [15]

Liu, C. & Si, Z. (2019) An incremental pressure correction ﬁnite element method for the time-dependent Oldroyd ﬂows. Appl. Math. Comput. , 351, 99-115

work page 2019

[16] [16]

(2020) Deterministic and stochastic equations of motion arising in Old royd ﬂuids of order one: existence, uniqueness, exponential stability and invariant measur es

Mohan, M.T. (2020) Deterministic and stochastic equations of motion arising in Old royd ﬂuids of order one: existence, uniqueness, exponential stability and invariant measur es. Stochastic Anal. Appl. , 38, 1-61

work page 2020

[17] [17]

(2020) Well posedness, large deviations and ergodicity of the stoc hastic 2D Oldroyd model of order one

Mohan, M.T. (2020) Well posedness, large deviations and ergodicity of the stoc hastic 2D Oldroyd model of order one. Stochastic Processes and their Applications , 130, 4513-4562

work page 2020

[18] [18]

& Thom ´ ee, V

McLean, W. & Thom ´ ee, V. (1993) Numerical solution of an evolution equation with a positive typ e memory term. J. Austral. Math. Soc. Ser. B , 35, 23–70

work page 1993

[19] [19]

Oldroyd, J. G. (1956) Non-Newtonian ﬂow of liquids and solids. Rheology: Theory a nd Applications, Vol. I (F. R. Eirich, Ed.). New York: Academic Press, pp. 653–682

work page 1956

[20] [20]

Pani, A. K. & Sinha, R. K. (1998) On the backward Euler method for time dependent parabolic integro- diﬀerential equations with nonsmooth initial data. J. Integral Equations Appl. , 10, 219–249

work page 1998

[21] [21]

Pani, A. K. & Sinha, R. K. (1998) Quadrature based ﬁnite element approximations to time dep endent parabolic equations with nonsmooth initial data. Calcolo, 35, 225–248

work page 1998

[22] [22]

K., Thom ´ ee, V

Pani, A. K., Thom ´ ee, V. & W ahlbin, L. B. (1992) Numerical Methods for Hyperbolic and Parabolic Integro-Diﬀerential Equations. J. Integral Equations Appl. , 4, 533–584

work page 1992

[23] [23]

Pani, A. K. & Yuan, J. Y. (2005) Semidiscrete ﬁnite element Galerkin approximations to the eq uations of motion arising in the Oldroyd model. IMA J. Numer. Anal. , 25, 750–782

work page 2005

[24] [24]

K., Yuan, J

Pani, A. K., Yuan, J. Y. & Damazio, P. (2006) On a linearized backward Euler method for the equations of motion arising in the Oldroyd ﬂuids of order one. SIAM J. Numer. Anal. , 44, 804–825

work page 2006

[25] [25]

(1984) Navier-Stokes Equations, Theory and Numerical Analysis , Studies in Mathematics and its Application 2, Third Edition

Temam, R. (1984) Navier-Stokes Equations, Theory and Numerical Analysis , Studies in Mathematics and its Application 2, Third Edition. Amsterdam: North-Holland Publishing Co., pp. xii+526

work page 1984

[26] [26]

& Wirosoetisno, D

Tone, F. & Wirosoetisno, D. (2006) On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations. SIAM J. Numer. Anal. , 44, 29–40

work page 2006

[27] [27]

& Zhang, N

Thom´ ee, V. & Zhang, N. Y. (1989) Error estimates for semidiscrete ﬁnite element methods fo r parabolic integro-diﬀerential equations. Math. Comp. , 53, 121–139

work page 1989

[28] [28]

& Shang, Y

W ang, K., He, Y. & Shang, Y. (2010) Fully discrete ﬁnite element method for the viscoelastic ﬂuid m otion equations. Disc. Cont. Dyn. Sys. Ser. B , 13, 665–684

work page 2010

[29] [29]

Yang, Y., Lei, Y. & Si, Z. (2020) Unconditional stability and error estimates of the modiﬁed c haracteristics FEM for the time-dependent viscoelastic Oldroyd ﬂows. Adv. Appl. Math. Mech. , 13, 311–332. 29

work page 2020

[30] [30]

& Qian, Y

Zhang, T. & Qian, Y. (2018) Stability analysis of several ﬁrst order schemes for the Old royd model with smooth and nonsmooth initial data. Numer. Meth. Partial Diﬀ. Eqn. , 34, 2180–2216

work page 2018

[31] [31]

& Yuan, J

Zhang, T., Qian, Y., Jiang, T. & Yuan, J. (2018) Stability and convergence of the higher projection method for the time-dependent viscoelastic ﬂow problem. J. Comput. Appl. Math. , 318, 1–21

work page 2018

[32] [32]

& Yuan, J

Zhang, T. & Yuan, J. (2015) A stabilized characteristic ﬁnite element method for the visc oelastic Oldroyd ﬂuid motion problem. Int. J. Numer. Anal. Model. , 12, 617–635

work page 2015

[33] [33]

& Qian, Y

Zhao, J., Zhang, T. & Qian, Y. (2018) Stability and convergence of second order time discrete pr ojection method for the linearized Oldroyd model. Appl. Math. Comput. , 316, 342–635. 30

work page 2018