Numerical Analysis and Computation of a Subgrid-Sparse-Grad-Div Stabilization Method for Incompressible Flow Problems

Yang, Yun-Bo; Huang, Bin-Chao

doi:https://doi.org/10.1155/2024/5580918

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2024 | Article ID 5580918 | https://doi.org/10.1155/2024/5580918

Numerical Analysis and Computation of a Subgrid-Sparse-Grad-Div Stabilization Method for Incompressible Flow Problems

Yun-Bo Yang^1,2,3and Bin-Chao Huang¹

Academic Editor: Francisco R. Villatoro

Received27 Jan 2023

Revised15 Feb 2024

Accepted21 Feb 2024

Published13 Mar 2024

Abstract

In this article, a subgrid-sparse-grad-div method for incompressible flow problem was proposed, which is a combination of the subgrid stabilization method and the recently proposed sparse-grad-div method. The method maintains the advantage of both methods: (i) It is robust for solving incompressible flow problem with dominance of the convection, especially when the viscosity is too small. (ii) It can keep mass conservation. Therefore, the method is very efficient for solving incompressible flow. Moreover, based on the Crank–Nicolson extrapolated scheme for temporal discretization, and mixed finite element in spatial discretization, we derive the unconditional stability and optimal convergence of the method. Finally, numerical experiments are proposed to validate the theoretical predictions and demonstrate the efficiency of the method on a test problem for incompressible flow.

1. Introduction

In this paper, we consider the incompressible time-dependent Navier–Stokes (NS) equations. For a bounded, regular domain ( = 2 or 3), we find and satisfyingwhere and represent the fluid velocity and zero-mean pressure, respectively, represents the kinematic viscosity, and and are the prescribed terms.

System (1) provides a mathematical model of incompressible Newtonian viscous fluid flow. They can describe many important physical phenomena such as weather prediction or climate modeling, flow around airfoils, ocean current, and blood flow in the arteries. Therefore, it is of practical interest to design efficient numerical methods for solving NS equations. Among the numerical methods, the Galerkin finite element method is a popular method, for more details, see [1–3] and references therein. However, the numerical solution solved by using finite methods often suffer from spurious oscillations and inaccurate approximation, the main reason for this is the fact that dominance of the convection, especially when the viscosity is too small, or the poor mass conservation. In order to deal with these effects, various stabilization techniques have been proposed.

Among these stabilization methods, one of the most popular stabilization methods applied on incompressible flow problems is the variational multiscale (VMS) method, which is based on the decomposition of the flow scales. In a type of VMS method, the flow is decomposed into the large scales and small scales, and the former is defined by projection into appropriate subspaces. For more details, we can see [4–8] and the references therein. The subgrid stabilization method is an improvement of the VMS method, which uses the two-grid techniques [9–11], and assumes that there exist fine scales and coarse scales of the flow. The main ideas of subgrid stabilization method are to first add an artificial viscosity term on a fine scales and then subtract it off only on the coarse scales, see, for example [12–14]. It has ample applications about subgrid stabilization method, such as steady-state natural convection problem [15], the incompressible magnetohydrodynamics (MHD) [16], and Darcy–Brinkman equations in double-diffusive convection [17], optimal control of the unsteady Navier–Stokes equations [18], and so on. The grad-div stabilization method is another popular stabilization method, which was studied in [19–21]. It can be look as the penalized stabilization of the divergence constraint. Since using the common finite element methods does not give divergence-free solutions, the divergence constraint condition is damaged severely [19]. When adding the grad-div term as a penalty term with respect to the continuity equation, it can improve the conservation of mass. The significant feature of the method is that it enhances the discrete mass conservation and reduces the effect of the pressure error on the velocity error.

Recently, a new grad-div type stabilization technique, namely, the sparse-grad-div stabilization method, was derived in [22]. The stabilization term added in this method produces block upper triangular matrices in two dimensions case and in three dimensions, the 2, 1 and 3, 1 blocks are empty. Since the sparser structure of its matrices, this new grad-div method not only can achieve the same positive effect compare with the common grad-div method but also is more efficient. The new grad-div method was extended to solve Navier–Stokes equations with a projection algorithm [23] and to solve optimal control problem of stationary Navier–Stokes equations [24]. The other variant of the grad-div stabilization can be seen in [25–27].

The main contribution of this work is to derive a synthesized stabilization finite element method for solving incompressible flow problem. This method is a combination of the subgrid stabilization method and the recently proposed sparse-grad-div method and possesses the advantage of both methods. It is robust for solving incompressible flow problems with dominance of the convection, especially when the viscosity is too small. In addition, it can keep mass conservation. Therefore, the method is very efficient for solving incompressible flow. Moreover, based on the Crank–Nicolson extrapolated scheme for temporal discretization, and mixed finite element in spatial discretization, we derive the unconditional stability and optimal convergence of the method. Numerical experiments are proposed to validate the theoretical predictions and demonstrate the efficiency of the method on a test problem for incompressible flow.

The article is arranged as follows. Section 2 introduces some notations and preliminary results that will be used throughout this article, and presents the numerical algorithm. In Section 3, we prove the unconditional stability of the proposed method. In Section 4, we perform a rigorous error analysis of the presented algorithm. A series of numerical experiments are provided to verify the efficiency of the method in Section 5. Finally, we conclude the article.

2. Mathematical Preliminaries

We first generalize some notations, definitions, and preliminary lemmas which will be used in the analysis. Let ( = 2 or 3) be an open, bounded convex polygonal or polyhedra domain, with a Lipschitz-continuous boundary . The inner product on or , the norm in , and the norm in are denoted by , , and , respectively. Likewise, the norms and the Sobolev space norms are denoted by and , respectively. For the seminorm in , we denote it by . is the standard Hilbertian Sobolev space of order with norm . For the given function, defined on the entire time interval , we define the norm

For the mathematical setting of problem (1), the following Sobolev spaces for the velocity , the pressure are introduced, respectively, bywhere the space is equipped with the -scalar product and the norm , and the space is equipped with the usual -norm. Finally, the space , the dual space of , is equipped with the negative norm

In addition, the classical space of divergence-free functions is defined by

With above notations, we can get the weak formulation of (1) as follows: Find for a.e. ∈ (0, T] satisfyingwhere the skew symmetrized trilinear form is defined as follows:

By the divergence theorem, we know

We also note that

The trilinear form has the following bounds, see [1, 3] for more details,

In addition, if , we have (see e.g., Lemma 1 in [28]).

2.1. Finite Element Approximation

We now introduce the finite element discretization of (6). Let and are two uniformly regular triangulation of domain into triangular in or tetrahedral in . Thus, the computational domain is defined by

Here, (respectively, ) denotes the maximum diameter of the elements in (resp. ) and such that . Let , and be conforming finite element spaces, which defined as follows:where is the space of the -th order polynomial on K. The conforming velocity-pressure finite element space satisfies the discrete inf-sup condition [2, 3], i.e., there exists a constant independent of such that

As we know, this discrete inf-sup condition which are known to be valid for the classical Taylor-Hood element [29], the [30] element, and the Scott–Vogelius elements on appropriate meshes [19].

The coarse or large scales space used in this article is defined by , which is the coarse finite element spaces of the deformation tensor with

Suppose the finite element space pair satisfies the following approximation properties [2, 31]:

Let be the step size for , , , with . For notational clarity, let for the continuous variable, and for the discrete variable, here , in addition, we define the following additional norms:

The discrete Gronwall lemma plays an important role in the analysis, we recall from [28, 32] as follows:

Lemma 1 (Discrete Gronwall’s lemma). Let , and (for integers ) be nonnegative numbers such that

Then, for all ,

Furthermore, Young’s and Poincaré’s inequalities as follows will be used frequently.

Defining to be the projection into , satisfying for

In the next lemma, we give out the important properties of the sparse-grad-div operator [22]:

Lemma 2. Let be a bounded regular domain, and ( or 3). Defining the sparse-grad-div operator by

Then, the operator has the following properties:(1)The operator can be written as(2)Similar to grad-div stabilization, satisfies in or ,.(3)If , then in or ,.

Now, we give out the subgrid-sparse-grad-div algorithm as follows:

Algorithm 3. (Subgrid-Sparse-Grad-Div). The full discrete approximation of (2) is: Let , given , find satisfying, for

Remark 4. In Algorithm 3, we first add an artificial viscosity term on a fine scales, and then subtract it off only on the coarse scales, which leads to a much better conditioning of linear system is achieved while only altering the NS equations on fine scales. This allows us to solve incompressible flow problem with dominance of the convection, especially when the viscosity is too small.

Remark 5. We assume that is known, positive, element-wise constants. When , Algorithm 3 become the Sparse-Grad-Div algorithm [22].

Remark 6. Combining the sparse-grad-div stabilization term with the pressure term, we can derive a modified pressure , which will be used in the numerical experiments.

3. Stability Analysis

In this section, we establish the unconditional stability of Algorithm 3.

Theorem 7. Assume that , , the solution of numerical scheme (25) satisfies the following energy estimatesand

Proof. Setting in the third equation of (25), then we obtainwhich yieldsChoosing in (25), making use of Cauchy–Schwarz and Young’s inequality to the right hand side, and thanks to Lemma 2, we arrive atMultiplying through by and applying (29), it givesSumming up from to , we obtainand then, we obtain the estimate (26). By (29) and (32), we get the bound of as follows:which we get the bound (27), and complete the proof.

4. Error Analysis

In this section, we devote to derive the error estimate of the presented numerical scheme (25). For convenience, we denote a generic constant whose value may depends only on and system parameters but is independent of the solutions, time step size and mesh width, and it may take on different value from place to place.

To establish the optimal asymptotic error estimate for the approximation, we assume that the true solution of problem (1) satisfies the following regularity assumptions:

Theorem 8. Let be the solution of Navier–Stokes equations (1) and satisfy the regularity assumptions (18), is given by the scheme (10), then we have the following error estimate:

Remark 9. From the right hand side of estimate (19), we can see that the sparse-grad-div operator act to reduce the effect of pressure discretization error on the velocity error and increase the effect of the velocity discretization error.

Proof. We rewrite the continuous variational formulations of equation (2) at . For , adding and subtracting some terms yieldDenoting by , after subtracting (25) from (36) and performing some simple algebraic manipulations, we obtained the error equation as followsNext, we decompose the errors into interpolation error and approximation error terms as follows:and rewrite (37) as follows:for all .
Setting in (39) and using Lemma 2, we arrived atDue to the orthogonality of projection operator , that , so . Using the property (9) of the trilinear , which implies . Next, we estimate the rest terms on the RHS of (40) one by one. First, the applications of Cauchy–Schwarz and Young’s inequality lead toandalsoFor the nonlinear term, using estimates (10) and (12), it givesandFor the next nonlinear term, we have the identityThe right hand side of (46) can be bounded as follows. First,Next,The last term in (46) can be bounded asCombining estimates (47)–(49), we can get the bound of the RHS of (46). The rest of terms of (40) can be estimated as follows:andandandFor the pressure term, we haveFinally, we estimate the coarse mesh projection term, by the third equation of (25), we know for We denote , ; here, is projection onto . Thanks to the orthogonality of projection operator , that , soChoosing in above equation, it giveswhich yieldsandAfter inserting the bound (41)–(59) into (40), multiplying by on both side, and taking the sum from to , noting that , we haveAbsorbing constants into , it givesHence, with , from the discrete Gronwall lemma yieldsBy applying the triangle inequality, we obtain the error estimate (35).

Corollary 10. Consider the numerical method (10), under the assumptions of Theorem 8, with given by the Taylor-Hood approximation element , and choose , then there exist a positive constant such that

5. Numerical Experiments

Herein, we perform some numerical experiments to test the effectiveness of the presented method. We choose finite element pair to approximate the velocity and pressure. The domain is subdivided into triangles. All the numerics were implemented by using the public domain finite element software package Freefem++ [33].

There are two test problems are performed. First, we perform a problem with known analytical solution to validate the convergence rates of Algorithm 3. The next experiment is for simulating the benchmark problem of flow around a cylinder [34, 35]. In all cases, the proposed methods perform very effectively.

5.1. Convergence Rate Verification

Our first example is designed to verify the predicted convergence rates of Algorithm 3. The problem is an interesting test problem for simulating decay of Green–Taylor vortex [36–38], which has following the analytic solution.

This is a solution of the NSE with , consisting of an array of oppositely signed vortices that decay in time. The initial condition at is above true solution. We consider the convergence rates of Algorithm 3 to this solution on the domain and time interval . Taking , , , final time , and setting the values of the time step size and the mesh width so that for a refinement, each of and gets cut in a half. We choose and 10000. The convergence rates are calculated from the errors at two successive values of in the usual manner by postulating and solving the formula.where and are the errors corresponding to the mesh size and , respectively. The results are shown in Tables 1–3. We can see from that, the proposed Subgrid-Sparse-Div-Grad stabilization method can keep the rates of convergence excellently for different values, which are in good agreement with the theoretical convergence rates predictions.

5.2. Flow around a Cylinder

The second example is the two-dimensional flow around a cylinder, which is a popular benchmark problem for testing numerical schemes of time-dependent Navier–Stokes equations. The domain contains a cylinder centered at (0.2, 0.2) with radius 0.05, we can see Figure 1 for more details.

No slip boundary conditions are endowed on the top and bottom walls, and the inflow and outflow profiles are set as follows:

Taking the initial condition , the viscosity , and the external force . Computations are implemented with final time . A fine mesh with 14508 triangulations and a coarse mesh with 3609 triangulations are given in Figure 2. And these meshes are used in this test.

(a)

(b)

The difference of pressure is defined by and the drag and lift coefficients are defined as follows [34]:where on the nodes around the cylinder and vanishes everywhere else and on the nodes around the cylinder and vanishes everywhere else.

The following reference intervals for the benchmark drag and lift coefficients are given in [34, 35] as:

Table 4 shows and are exactly in the interval of reference values, is exactly in the benchmark interval when the time step size not too large. The evolutions of and with , , by the Subgrid-Sparse-Div-Grad stabilization method are shown in Figure 3, which is in perfect agreement with with the results provided in [34]. The velocity field at times with , are shown in Figure 4, we can see the vortex street forms successfully. Finally, a plot of vs. time for different methods is given in Figure 5, we can see that our method can keep mass conservation with time evolve.

6. Conclusions

In this article, we considered the synthesis of subgrid stabilization method with sparse div-grad stabilization method, derived the subgrid-sparse-grad-div stabilization method, which is particularly efficient and maintains the best algorithmic features of each. The stability and convergence of this method were given. Numerical test verified the theoretical preconditions and demonstrated the effectiveness of the method. The application of the method to other coupled equations [39, 40] will be consider in the future.

Data Availability

All data that support the findings of this study can be obtained from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (NSFC) under grant no. 12161095, the Project of Yunnan Basic Research Program (Grant nos. 202001AU070068, 202201AT070032, and 202001AU070066), and the Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (no. 202302AN360007).

References

R. Temam, Navie-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, Netherlands, 1984.
V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Germany, 1986.
W. Layton, Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Thailand, Asia, 2008.
T. J. R. Hughes, L. Mazzei, and K. E. Jansen, “Large eddy simulation and the variational multiscale method,” Computer Vision and Image Understanding, vol. 3, no. 1-2, pp. 47–59, 2000.
View at: Google Scholar
V. John and S. Kaya, “Finite element error analysis of a variational multiscale method for the Navier-Stokes equations,” Advances in Computational Mathematics, vol. 28, pp. 43–61, 2008.
View at: Google Scholar
V. John and S. Kaya, “A finite element variational multiscale method for the Navier-Stokes equations,” SIAM Journal on Scientific Computing, vol. 26, pp. 1485–1503, 2005.
View at: Google Scholar
H. Zheng, Y. Hou, F. Shi, and L. Song, “A finite element variational multiscale method for incompressible flows based on two local Gauss integrations,” Journal of Computational Physics, vol. 228, pp. 5961–5977, 2009.
View at: Google Scholar
H. Zheng, Y. Hou, and F. Shi, “Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations,” Journal of Computational Physics, vol. 229, pp. 7030–7041, 2010.
View at: Google Scholar
J. Xu, “A novel two-grid method for semilinear elliptic equations,” SIAM Journal on Scientific Computing, vol. 15, pp. 231–237, 1994.
View at: Google Scholar
J. Xu, “Two-grid discretization techniques for linear and nonlinear PDEs,” SIAM Journal on Numerical Analysis, vol. 33, pp. 1759–1777, 1996.
View at: Google Scholar
S. Kaya and B. Riviere, “A two-grid stabilization method for solving the steady-state Navier-Stokes equations,” Numerical Methods for Partial Differential Equations, vol. 22, pp. 728–743, 2006.
View at: Google Scholar
J. L. Guermond, A. Marra, and L. Quartapelle, “Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers,” Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 5857–5876, 2006.
View at: Google Scholar
W. Layton, “A connection between subgrid scale eddy viscosity and mixed methods,” Applied Mathematics and Computation, vol. 133, pp. 147–157, 2002.
View at: Google Scholar
S. Kaya, W. Layton, and B. Riviére, “Subgrid stabilized defect correction methods for the Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, pp. 1639–1654, 2006.
View at: Google Scholar
A. Çibik and S. Kaya, “A projection-based stabilized finite element method for steady-state natural convection problem,” Journal of Mathematical Analysis and Applications, vol. 381, pp. 469–484, 2011.
View at: Google Scholar
M. A. Belenli, S. Kaya, L. Rebholz, and N. E. Wilson, “A subgrid stabilization finite element method for incompressible magnetohydrodynamics,” International Journal of Computer Mathematics, vol. 90, pp. 1506–1523, 2013.
View at: Google Scholar
A. Çibik and S. Kaya, “Finite element analysis of a projection-based stabilization method for the darcy-brinkman equations in double-diffusive convection,” Applied Numerical Mathematics, vol. 64, pp. 35–49, 2013.
View at: Google Scholar
F. Yılmaz, “Semi-discrete a priori error analysis for the optimal control of the unsteady navier-stokes equations with variational multiscale stabilization,” Applied Mathematics and Computation, vol. 276, pp. 127–142, 2016.
View at: Google Scholar
M. Case, V. Ervin, A. Linke, and L. Rebholz, “A connection between scott-vogelius elements and grad-div stabilization taylor-hood FE approximations of the navier-stokes equations,” SIAM Journal on Numerical Analysis, vol. 49, pp. 1461–1481, 2011.
View at: Google Scholar
M. A. Olshanskii and A. Reusken, “Grad-Div stabilization for the stokes equations,” Mathematics of Computation, vol. 73, pp. 1699–1718, 2004.
View at: Google Scholar
E. W. Jenkins, V. John, A. Linke, and L. Rebholz, “On the parameter choice in grad-div stabilization for the Stokes equations,” Advances in Computational Mathematics, vol. 40, pp. 491–516, 2014.
View at: Google Scholar
A. Linke and L. Rebholz, “On a reduced sparsity stabilization of grad-div type for incompressible flow problems,” Computer Methods in Applied Mechanics and Engineering, vol. 261, pp. 142–153, 2013.
View at: Google Scholar
A. Bowers, S. Le Borne, and L. G. Rebholz, “Error analysis and iterative solvers for navier-stokes projection methods with standard and sparse grad-div stabilization,” Computer Methods in Applied Mechanics and Engineering, vol. 275, pp. 1–19, 2014.
View at: Google Scholar
A. Çıbık, “The effect of a sparse grad-div stabilization on control of stationary navier-sokes equations,” Journal of Mathematical Analysis and Applications, vol. 437, pp. 613–628, 2016.
View at: Google Scholar
J. A. Fiordilino, W. Layton, and Y. Rong, “An efficient and modular grad-div stabilization,” Computer Methods in Applied Mechanics and Engineering, vol. 335, pp. 327–346, 2018.
View at: Google Scholar
Y. Rong and J. A. Fiordilino, “Numerical analysis of a BDF2 modular grad-div stabilization method for the navier-stokes equations,” Journal of Scientific Computing, vol. 82, p. 66, 2020.
View at: Google Scholar
X. L. Lu and P. Z. Huang, “A modular grad-div stabilization for the 2D/3D nonstationary incompressible magnetohydrodynamic equations,” Journal of Scientific Computing, vol. 82, p. 3, 2020.
View at: Google Scholar
A. Labovschii, W. Layton, M. Manica, M. Neda, and L. Rebholz, “The stabilized extrapolated trapezoidal finite element method for the navier-stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 198, pp. 958–974, 2009.
View at: Google Scholar
P. Hood and C. Taylor, “A numerical solution of the Navier-Stokes equations using the finite element technique,” Computers & Fluids, vol. 1, pp. 73–100, 1973.
View at: Google Scholar
M. Fortin, Calcul numérique des ecoulements fluides de Bingham et des fluides Newtoniens incompressible par des méthodes d’eléments finis, Springer, Berlin, Germany, 1972.
S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, Germany, 1994.
J. G. Heywood and R. Rannacher, “Finite element approximation of the nonstationary navier-stokes problem. Part IV: error analysis for the second order time discretization,” SIAM Journal on Numerical Analysis, vol. 27, pp. 353–384, 1990.
View at: Google Scholar
F. Hecht, “New development in freefem++,” Journal of Numerical Mathematics, vol. 20, no. 3-4, pp. 251–266, 2012.
View at: Google Scholar
V. John, “Reference values for drag and lift of a two dimensional timedependent flow around a cylinder,” International Journal for Numerical Methods in Fluids, vol. 44, pp. 777–788, 2004.
View at: Google Scholar
M. Schäfer and S. Turek, “Benchmark computations of laminar flowaround a cylinder,” in Flow Simulations with High-Performance Computers II, E. H. Hirschel, Ed., Springer, Berlin, Germany, 1996.
View at: Google Scholar
G. Taylor, “On decay of vortices in a viscous fluid,” Philosophical Magazine, vol. 46, pp. 671–674, 1923.
View at: Google Scholar
G. Taylor and A. E. Green, “Mechanism of the production of small eddies from large ones,” Proceedings of the Royal Society of London Series A, vol. 158, pp. 499–521, 1937.
View at: Google Scholar
L. C. Berselli, “On the large eddy simulation of the taylor-green vortex,” Journal of Mathematical Fluid Mechanics, vol. 7, pp. S164–S191, 2005.
View at: Google Scholar
Y. B. Yang and Y. L. Jiang, “Numerical analysis and computation of a type of IMEX method for the time-dependent natural convection problem,” Computational Methods in Applied Mathematics, vol. 16, pp. 321–344, 2016.
View at: Google Scholar
Y. B. Yang, B. C. Huang, and Y. L. Jiang, “Error estimates of an operator-splitting finite element method for the time-dependent natural convection problem,” Numerical Methods for Partial Differential Equations, vol. 39, pp. 2202–2226, 2023.
View at: Publisher Site | Google Scholar

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Copyright © 2024 Yun-Bo Yang and Bin-Chao Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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