- Research
- Open Access
Stabilized lowest equal-order mixed finite element method for the Oseen viscoelastic fluid flow
- Shahid Hussain^{1},
- Md. Abdullah Al Mahbub^{1, 2},
- Nasrin Jahan Nasu^{1} and
- Haibiao Zheng^{1}Email author
https://doi.org/10.1186/s13662-018-1916-0
© The Author(s) 2018
- Received: 21 June 2018
- Accepted: 4 December 2018
- Published: 12 December 2018
Abstract
In this paper, we present a stabilized lowest equal-order mixed finite element (FE) method for the Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law. To approximate the velocity, pressure, and stress tensor, we choose lowest equal-order FE triples \(p1-p1-p1_{\mathrm{dg}}\) respectively. It is well known that these elements don’t satisfy the inf–sup (or LBB) condition. Owing to the violation of the essential stability condition, the system became unstable. To overcome this difficulty, a standard pressure stabilization term is added to the discrete variational formulation, which ensures the well-posedness of the FE scheme. The existences and uniqueness of the FE scheme are derived. The desired optimal error bound is obtained. Three numerical experiments are executed to illustrate the validity and efficiency of the numerical method. The stabilized method provides attractive computational advantages, such as simpler data structures, parameter-free, no calculations of higher-order derivatives, and fast solver in simulations.
Keywords
- Lowest equal-order FE
- Oseen viscoelastic fluid
- DG method
- Stabilized method
1 Introduction
Solving the viscoelastic fluid flow model is a great challenge due to the slow flow and the hyperbolic nature of the constitutive equation [1]. Owing to the complex structure of viscoelastic fluid, it is not solvable similarly to the Navier–Stokes equation. The difficulty arises in performing correct numerical computations due to the hyperbolic character of the constitutive equation, which does not include a dissipative (stabilizing) term for the stress. As a result, a certain technique must be used to discretize the constitutive equation for approximation. It was a great challenge for scientists and researchers to formulate a new mathematical model that can describe the large deformation of the viscoelastic fluid flow. In 1950, James G. Oldroyd was the first to develop a constitutive equation to model the large deformation of the viscoelastic fluids in [2]. By using Oldroyd’s original work many other constitutive equations have been formulated to describe different features of the viscoelastic fluids, for example, the Phan–Thien–Tanner model, the Maxwell model, the Jeffrey model, the Johnson–Segalman model, and so on [3–7].
Over the last decades, significant progress has been made in the development of numerical approximation for the stable and accurate solutions of the viscoelastic flow problems. Currently, the literature on the FE method is burgeoning to approximate the viscoelastic fluid flow model equations by using a variety of alternative stabilization formulations. The most common among them are streamline-upwind Petrov–Galerkin (SUPG) methods [8], discontinuous Galerkin (DG) methods [9], decoupled FE methods [10], multigrid methods [11], variational multiscale methods [12], and so on.
We consider the DG method for the mixed FE approximation of the viscoelastic fluid flow. To the best of our knowledge, Reed and Hill [13] were the first who studied the DG technique. To deal with the hyperbolic nature of the constitutive equation, Lesaint and Raviart [14] analyzed the DG method on the neutron transport equation. Fortin and Fortin [15] first introduced the DG method for the viscoelastic fluid flow. Baranger and Sandri [16] analyzed the stability and error estimates for the steady-state viscoelastic flow by using the DG method. Zhang et al. [17] studied and obtained unconditional error estimate for the viscoelastic fluid flow with the DG method.
The Oseen fluid flow model is the reduced linearized form of the Newtonian fluid described by the Navier–Stokes equation [18]. The nonlinear convective term of the Navier–Stokes equation can be reduced to a linear system by replacing the unknown velocity with a known velocity field. The non-Newtonian fluid flow obeying the Oldroyd-B model is the combination of conservation of momentum equation and constitutive equation. Under the assumption of the creeping flow, the nonlinearity vanishes in the momentum equation of the Oldroyd-B model. So in the viscoelastic fluid flow model, the nonlinearity occurs only in the constitutive equation [19], which can be reduced to a linear form by fixing velocity u with a known velocity field \(\mathbf{b}(x)\). The resulting system of equations can be explicitly described with the parameter space for α, λ, and \({\|\nabla\mathbf{b}\|}_{\infty}\), which guarantee the existence and uniqueness condition for the solution of the continuous problem and its numerical approximation [20–22]. To solve the Oseen viscoelastic fluid flow, many methods have been formulated and discussed; the reader can see [23–26] and the references therein.
We study the mixed FE method to approximate the Oseen viscoelastic fluid flow, which is developed to approximate both scalar (pressure) variables and vector (velocity) variables simultaneously. The mixed FE method in viscoelastic fluid flow [27] introduces two spaces for the approximation of pressure and velocity. These two spaces must satisfy the inf–sup or the Ladyzhenskaya–Babuska–Breezi (LBB) condition for the stability [28, 29]. Here in our interest, during the implementation of the mixed FE method, we prefer to choose lowest equal-order FE triples \(p1-p1-p1_{\mathrm{dg}}\) to approximate the solution of linear velocity, linear pressure, and discontinues stress. We claim that our choice in the sense of FE will be more convenient to approximate the unknowns. However, these elements fail to satisfy the inf–sup condition [30]. The violation of the inf–sup condition often leads to nonphysical pressure oscillation. Therefore, a stabilization term might be introduced. Stabilization methods are often used to overcome the difficulty associated with the stability of the lowest order mixed FE method. The discussion about stabilization of mixed FE methods and the stability term is analyzed briefly in [31]. Moreover, the lowest equal-order FE methods are easily implementable in a scientific computational sense as compared to the higher-order FE. The lowest order FE and stabilization methods have been considered in the Stokes equation [32], the Navier–Stokes equation [33, 34], and the Stokes–Darcy fluid flow model [35]. So far, this method is novel to solve the Oseen viscoelastic fluids with the lowest equal-order FE.
This paper focuses on stabilization of lowest equal-order FE triples for the approximate solution of unknowns in the Oseen viscoelastic fluid flow model with the DG method. The method introduces a stabilized term with respect to the pressure space to get around the inf–sup condition. It has several important aspects; most notably, the new method is free of nonstandard data structures, approximation of higher-order derivatives, and specification of mesh-dependent parameters. Furthermore, the stabilized lowest equal-order method can be cast in the framework of the Oseen viscoelastic fluid flow with all the advantages discussed. The stability and optimal convergence order of the temporal discretized scheme are derived. To show the validation of the theoretical analysis, three numerical tests are executed, which reveal the efficiency of the Oseen viscoelastic fluid flow model.
The rest of the paper is organized as follows. In Sect. 2, we introduce the governing equations for Oseen viscoelastic fluid flow model, the notations, and preliminaries. The variational formulation, spatial discretization, and some well-known lemmas are discussed in Sect. 3. To justify the proposed lowest equal-order FE algorithm, the well-posedness and optimal convergence analysis are derived in Sect. 4. The results of the numerical simulations of three different experiments are illustrated in Sect. 5 to validate the efficiency and accuracy of the stabilization method. Finally, in Sect. 6, we summarize this work by a short conclusion.
2 Model equations
2.1 Model problem
In the analysis, we consider the Oseen system as a linearization of viscoelastic model equations. For the ease of presentation, we suppose homogeneous Dirichlet boundary conditions with given velocity \(\mathbf{b}(x)\).
Problem \((O)\)
2.2 Variational formulation
3 Discontinuous FE approximation
Lemma 3.1
Proof
Thanks to [29]. □
Lemma 3.2
Proof
Thanks to [29]. □
Now, the stabilized scheme for the approximation of the Oseen viscoelastic fluid flow problem for the lowest equal-order triples is formed in the discrete way as
Problem \((O_{Dg})\)
4 Existence and uniqueness of Problem \((O_{Dg})\) and error bound
In this section, we analyze the existence and uniqueness of the developed stabilized scheme with the lowest equal-order triples for the FE approximation of the Oseen viscoelastic problem.
Theorem 4.1
Let \(\mathbf{f}\in\mathbf{H}^{-1}(\varOmega)\). If \(1-2\lambda Md >0\). Then there exists a unique solution \((\tau ^{h},\mathbf{u}^{h},p^{h})\in(S^{h}\times X^{h} \times Q^{h})\) of (3.6).
Proof
Theorem 4.2
Proof
5 Numerical tests
This section illustrates the numerical simulation results in support of the proposed stabilized lowest equal-order FE scheme and its theoretical analysis performed in Theorem 4.2 for the Oseen viscoelastic fluid flow model. For numerical evaluation, we design and examine three different experiments, that is, a nonphysical example with an exact solution, a viscoelastic cavity flow problem, and a benchmark 4-to-1 contraction channel flow [27]. In the analytical solution test, we demonstrate the optimal convergence order by assuming an exact solution. The second experiment elucidates the viscoelastic cavity flow to show the characteristics of the pressure contour and its behavior. The flow speed, behavior of the contours, streamlines patterns, and the pressure oscillation are examined by the 4-to-1 contraction channel flow. To show the distinguishing features of the new stabilized model, we compare newly formulated method for the lowest equal-order FE triples \(p1-p1-p1_{\mathrm{dg}}\) with the standard MINI element triples \(p1b-p1-p1_{\mathrm{dg}}\). All numerical tests are performed by using public domain software Freefem++ [50]. The figures and graphs are drawn by using Tecplot and MATLAB, respectively.
5.1 Analytical solution test
The theoretical convergence rates are verified by considering fluid flow across a unit square with known solution. To test the numerical stability of the new stabilized method, we considered the lowest equal-order FE triples \(p1-p1-p1_{\mathrm{dg}}\) for velocity, pressure, and stress. Different authors used this experimental pattern for the Stokes and Navier–Stokes equations [11, 44, 51].
The right-hand sides and initial and boundary conditions are derived by model Problem \((O)\) with parameters \(a=0\), \(\lambda=5.0\), and \(\alpha=0.5\).
The illustration of the error for the Oseen viscoelastic fluid flow with standard FE \(p1b-p1-p1_{\mathrm{dg}}\) triples
h | \(\|u-u^{h}\|_{1}\) | \(\|p-p^{h}\|_{0}\) | \(\|\tau-\tau^{h}\|_{0}\) |
---|---|---|---|
1/4 | 0.22122 | 0.20562 | 0.12221 |
1/8 | 0.10267 | 0.06430 | 0.04332 |
1/16 | 0.04883 | 0.02187 | 0.01579 |
1/32 | 0.02390 | 0.00724 | 0.00621 |
1/64 | 0.01185 | 0.00247 | 0.00253 |
The illustration of the error for the Oseen viscoelastic fluid flow without stabilization term for the FE \(p1-p1-p1_{\mathrm{dg}}\) triples
h | \(\|u-u^{h}\|_{1}\) | \(\|p-p^{h}\|_{0}\) | \(\|\tau-\tau^{h}\|_{0}\) |
---|---|---|---|
1/4 | 0.23599 | 1.78225 | 0.18981 |
1/8 | 0.12375 | 0.96397 | 0.06659 |
1/16 | 0.06110 | 0.54778 | 0.02502 |
1/32 | 0.03027 | 0.34309 | 0.01015 |
1/64 | 0.01507 | 0.25734 | 0.00429 |
The illustration of the error for the Oseen viscoelastic fluid flow with stabilization term for the FE \(p1-p1-p1_{\mathrm{dg}}\) triples
h | \(\|u-u^{h}\|_{1} \) | \(\|p-p^{h}\|_{0}\) | \(\|\tau-\tau^{h}\|_{0}\) |
---|---|---|---|
1/4 | 0.44815 | 1.23944 | 0.28020 |
1/8 | 0.18149 | 0.40238 | 0.10445 |
1/16 | 0.07026 | 0.12950 | 0.03951 |
1/32 | 0.02955 | 0.04116 | 0.01484 |
1/64 | 0.01348 | 0.01318 | 0.00565 |
5.2 The viscoelastic cavity flow
In this test, we apply the stabilization method in the famous problem for testing numerical validity is known as the “lid-driven cavity.” The aforementioned problem is chosen because some standard data are available for comparison. Cavity flows has been used in many test cases for testing the incompressible fluid dynamics algorithm for the Stokes flow [52]. To investigate the pressure oscillation, we perform the viscoelastic cavity flow experiment for standard FE \(p1b-p1-p1_{\mathrm{dg}}\) triples, FE \(p1-p1-p1_{\mathrm{dg}}\) triples without stabilization, and FE \(p1-p1-p1_{\mathrm{dg}}\) triples with stabilization. Moreover, we compare the pressure lines of lowest equal-order triples with standard FE methods.
The fluid is enclosed in a unit square domain with the boundary condition for velocity \(\mathbf{u}=(1,0)\) on the upper boundary and the homogeneous Dirichlet condition on the remaining boundaries. The parameters value are chosen as follows: \(a=0\), \(\lambda=5.0\), \(\mathbf{f}=0\), and \(\alpha=0.5\). In this numerical formulation, \(\mathbf{b}(x)\) denotes a known vector function used to linearize the nonlinear viscoelastic fluid flow model into reduced Oseen model equations. We output the data by computing the velocity vector field \(\mathbf {u}=(u_{1},u_{2}) \) for there different FE triples known as \(p1b-p1-p1_{\mathrm{dg}}\) standard or MINI elements, \(p1-p1-p1_{\mathrm{dg}}\) without stabilization, and \(p1-p1-p1_{\mathrm{dg}}\) with stabilization. Furthermore, the computed solution \(\mathbf{u}=(u_{1},u_{2}) \) is used as the known solution \(\mathbf{b}(x)=(b_{1},b_{2})\) to reduce the nonlinear term as linear. The resulting effect makes the viscoelastic fluid flow model as the Oseen fluid flow model.
5.3 The 4-to-1 contraction channel flow
Figure 4(a) illustrates the streamlines and flow speed of the standard FE triples \(p1b-p1-p1_{\mathrm{dg}}\). Figure 4(b) shows the streamlines without stabilization term for the FE triples \(p1-p1-p1_{\mathrm{dg}}\), and Fig. 4(c) presents the streamlines with stabilization term for the FE \(p1-p1-p1_{\mathrm{dg}}\) triples. We can observe that without the addition of the stabilization term as shown in Fig. 4(b), the approximate value of the streamlines and vortex for the FE \(p1-p1-p1_{\mathrm{dg}}\) triples is different and shows some irregularities. However, Fig. 4(a) and Fig. 4(c) appear similar in manner. The behavior of the streamlines and vortex confirm the theoretical results for the approximation of the Oseen viscoelastic fluid flow with the lowest equal-order FE triples.
6 Conclusion and future work
In this contribution, a stabilized method for lowest equal-order finite elements (FE) triples \(p1-p1-p1_{\mathrm{dg}}\) for the Oseen viscoelastic fluid flow is presented. In the standard Galerkin finite element method, the inf–sup (or LBB) condition is substantial while we circumvent the difficulty for the FE triples \(p1-p1-p1_{\mathrm{dg}}\) by adding a stabilization term essentially in the Oseen viscoelastic fluid flow model. This technique is new for the lowest equal-order FE triples for the reduced linearized viscoelastic fluid flow model. We proposed a stabilized FE algorithm and derived the well-posedness of the scheme. The desired error estimate is proved, and the optimal convergence order is obtained. In support of the given method, three numerical tests have been successfully implemented. In the analytical solution test, we demonstrate the optimal convergence order for lowest equal order. The second experiment elucidates the viscoelastic cavity flow to show the characteristics of the pressure contour and its behavior. The flow speed, behavior of the contours, streamlines patterns, and the pressure oscillation are examined by the “4-to-1 contraction channel flow.” This method can be extended to the streamline-upwind Petrov–Galerkin (SUPG) method for approximation of the Oseen viscoelastic fluid flow in the future studies.
Declarations
Acknowledgements
The authors would like to thank the Editor and reviewers for constructive comments, which have helped to enrich the content and improve the presentation of the results in this manuscript.
Funding
This work was supported by the NSF of China with Grant Nos. 11571115, 11171269 and 11201369, Science and Technology Commission of Shanghai Municipality Grant No. 18dz2271000.
Authors’ contributions
All authors contributed equally to this work. The manuscript is approved by all authors for publication.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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