doi: 10.3934/era.2020096

Viscosity robust weak Galerkin finite element methods for Stokes problems

1. 

Craft & Hawkins Department of Petroleum Engineering, Louisiana State University, 2245 Patrick F Taylor Hall, Baton Rouge, LA, 70803, USA

2. 

Department of Mathematics, University of Georgia, Athens, GA, 30605, USA

* Corresponding author: Lin Mu

Received  April 2020 Revised  July 2020 Published  September 2020

In this paper, we develop a viscosity robust weak Galerkin finite element scheme for Stokes equations. The major idea for achieving pressure-independent energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization of the right hand side body force. The optimal convergence results for velocity and pressure have been established in this paper. Finally, numerical examples are presented for validating the theoretical conclusions.

Citation: Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, doi: 10.3934/era.2020096
References:
[1]

C. BrenneckeA. LinkeC. Merdon and J. Schöberl, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions, J. Comput. Math., 33 (2015), 191-208.  doi: 10.4208/jcm.1411-m4499.  Google Scholar

[2]

D. A. Di PietroA. ErnA. Linke and F. Schieweck, A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Engrg., 306 (2016), 175-195.  doi: 10.1016/j.cma.2016.03.033.  Google Scholar

[3]

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp., 83 (2014), 15-36.  doi: 10.1090/S0025-5718-2013-02753-6.  Google Scholar

[4]

C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307 (2016), 339-361.  doi: 10.1016/j.cma.2016.04.025.  Google Scholar

[5]

A. Linke, A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, 350 (2012), 837-840.  doi: 10.1016/j.crma.2012.10.010.  Google Scholar

[6]

A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., 268 (2014), 782-800.  doi: 10.1016/j.cma.2013.10.011.  Google Scholar

[7]

A. LinkeG. Matthies and L. Tobiska, Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors, ESAIM Math. Model. Numer. Anal., 50 (2016), 289-309.  doi: 10.1051/m2an/2015044.  Google Scholar

[8]

Y. Liu and J. Wang, Simplified weak Galerkin and new finite difference schemes for the Stokes equation, J. Comput. Appl. Math., 361 (2019), 176-206.  doi: 10.1016/j.cam.2019.04.024.  Google Scholar

[9]

L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, in Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, Springer Proc. Math. Stat., 45, Springer, New York, 2013,247-277. doi: 10.1007/978-1-4614-7172-1_13.  Google Scholar

[10]

L. MuJ. Wang and X. Ye, Effective implementation of the weak Galerkin finite element methods for the biharmonic equation, Comput. Math. Appl., 74 (2017), 1215-1222.  doi: 10.1016/j.camwa.2017.06.002.  Google Scholar

[11]

T. TianQ. Zhai and R. Zhang, A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 329 (2018), 268-279.  doi: 10.1016/j.cam.2017.01.021.  Google Scholar

[12]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[13]

J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.  Google Scholar

[14]

S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., 74 (2005), 543-554.  doi: 10.1090/S0025-5718-04-01711-9.  Google Scholar

show all references

References:
[1]

C. BrenneckeA. LinkeC. Merdon and J. Schöberl, Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions, J. Comput. Math., 33 (2015), 191-208.  doi: 10.4208/jcm.1411-m4499.  Google Scholar

[2]

D. A. Di PietroA. ErnA. Linke and F. Schieweck, A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Engrg., 306 (2016), 175-195.  doi: 10.1016/j.cma.2016.03.033.  Google Scholar

[3]

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp., 83 (2014), 15-36.  doi: 10.1090/S0025-5718-2013-02753-6.  Google Scholar

[4]

C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307 (2016), 339-361.  doi: 10.1016/j.cma.2016.04.025.  Google Scholar

[5]

A. Linke, A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, 350 (2012), 837-840.  doi: 10.1016/j.crma.2012.10.010.  Google Scholar

[6]

A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., 268 (2014), 782-800.  doi: 10.1016/j.cma.2013.10.011.  Google Scholar

[7]

A. LinkeG. Matthies and L. Tobiska, Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors, ESAIM Math. Model. Numer. Anal., 50 (2016), 289-309.  doi: 10.1051/m2an/2015044.  Google Scholar

[8]

Y. Liu and J. Wang, Simplified weak Galerkin and new finite difference schemes for the Stokes equation, J. Comput. Appl. Math., 361 (2019), 176-206.  doi: 10.1016/j.cam.2019.04.024.  Google Scholar

[9]

L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, in Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, Springer Proc. Math. Stat., 45, Springer, New York, 2013,247-277. doi: 10.1007/978-1-4614-7172-1_13.  Google Scholar

[10]

L. MuJ. Wang and X. Ye, Effective implementation of the weak Galerkin finite element methods for the biharmonic equation, Comput. Math. Appl., 74 (2017), 1215-1222.  doi: 10.1016/j.camwa.2017.06.002.  Google Scholar

[11]

T. TianQ. Zhai and R. Zhang, A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 329 (2018), 268-279.  doi: 10.1016/j.cam.2017.01.021.  Google Scholar

[12]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[13]

J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.  Google Scholar

[14]

S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., 74 (2005), 543-554.  doi: 10.1090/S0025-5718-04-01711-9.  Google Scholar

Figure 1.  Example 4.1: solution from Algorithm 2.2 (top); Algorithm 2.1 (bottom)
Figure 2.  Example 4.2: Error profiles and convergence results for $ {{|||}}\cdot{{|||}} $-norm on triangular mesh: standard scheme Algorithm 2.2 (left); new scheme Algorithm 2.1 (right)
Figure 3.  Example 4.2: Error profiles and convergence results for $ L^2 $-norm on triangular mesh: standard scheme Algorithm 2.2 (left); new scheme Algorithm 2.1 (right)
Figure 4.  Example 4.2: Error profiles and convergence results for $ \|p-p_h\| $ on triangular mesh: standard scheme Algorithm 2.2 (left); new scheme Algorithm 2.1 (right)
Figure 5.  Example 4.3: Profile for viscosity (left); magnitude plot for velocity by Algorithm 2.1 (middle); magnitude plot for Algorithm 2.2 (right). Top row illustrates the results for Test Case 1; Bottom row illustrates the results for Test Case 2
Table 1.  Example 4.4: Numerical results and convergence test for $ k = 0 $
Algorithm 2.1 Algorithm 2.2
$ 1/h $ $ {{|||}} {\bf e}_h{{|||}} $ Rate $ \| {\bf e}_h\| $ Rate $ \|\epsilon_h\| $ Rate $ {{|||}} {\bf e}_h{{|||}} $ Rate $ \| {\bf e}_h\| $ Rate $ \|\epsilon_h\| $ Rate
$ \nu=1 $
2 2.69E+2 7.22 1.11E+2 2.69E+2 7.22E 1.11E+2
4 2.35E+2 0.2 3.46 1.1 5.70E+1 1.0 2.35E+2 0.2 3.46 1.1 5.70E+1 1.0
8 1.54E+2 0.6 1.14 1.6 2.81E+1 1.0 1.54E+2 0.6 1.14 1.60 2.81E+1 1.0
16 7.65E+1 1.0 2.95E-1 2.0 1.31E+1 1.1 7.65E+1 1.0 2.95E-1 2.0 1.31E+1 1.1
32 3.83E+1 1.0 7.39E-2 2.0 6.55 1.0 3.83E+1 1.0 7.38E-2 2.0 6.55 1.0
$ \nu=1e-2 $
2 2.69E+2 7.22 1.19 2.75E+2 7.74 1.12
4 2.35E+2 0.2 3.46 1.1 5.77E-1 1.1 2.35E+2 0.2 3.46 1.2 5.70E-1 1.0
8 1.54E+2 0.6 1.14 1.6 2.81E-1 1.0 1.68E+2 0.5 1.27 1.5 2.88E-1 1.0
16 7.65E+1 1.0 2.95E-1 2.0 1.30E-1 1.1 7.86E+1 1.1 3.06E-1 2.1 1.33E-1 1.1
32 3.83E+1 1.0 7.38E-2 2.0 6.50E-2 1.0 3.93E+1 1.0 7.65E-2 2.0 6.65E-2 1.0
$ \nu=1e-4 $
2 2.69E+2 7.22 1.19E-2 8.92E+3 3.16E+2 1.25E-1
4 2.35E+2 0.2 3.46 1.1 5.77E-3 1.1 5.34E+3 0.7 9.40E+1 1.8 7.15E-2 0.8
8 1.54E+2 0.6 1.14 1.6 2.81E-3 1.0 4.29E+3 0.3 3.53E+1 1.4 4.53E-2 0.7
16 7.65E+1 1.0 2.95E-1 2.0 1.30E-3 1.1 2.11E+3 1.0 9.11 2.0 1.62E-2 1.5
32 3.83E+1 1.0 7.38E-2 2.0 6.50E-4 1.0 1.06E+3 1.0 2.28 2.0 8.10E-3 1.0
Algorithm 2.1 Algorithm 2.2
$ 1/h $ $ {{|||}} {\bf e}_h{{|||}} $ Rate $ \| {\bf e}_h\| $ Rate $ \|\epsilon_h\| $ Rate $ {{|||}} {\bf e}_h{{|||}} $ Rate $ \| {\bf e}_h\| $ Rate $ \|\epsilon_h\| $ Rate
$ \nu=1 $
2 2.69E+2 7.22 1.11E+2 2.69E+2 7.22E 1.11E+2
4 2.35E+2 0.2 3.46 1.1 5.70E+1 1.0 2.35E+2 0.2 3.46 1.1 5.70E+1 1.0
8 1.54E+2 0.6 1.14 1.6 2.81E+1 1.0 1.54E+2 0.6 1.14 1.60 2.81E+1 1.0
16 7.65E+1 1.0 2.95E-1 2.0 1.31E+1 1.1 7.65E+1 1.0 2.95E-1 2.0 1.31E+1 1.1
32 3.83E+1 1.0 7.39E-2 2.0 6.55 1.0 3.83E+1 1.0 7.38E-2 2.0 6.55 1.0
$ \nu=1e-2 $
2 2.69E+2 7.22 1.19 2.75E+2 7.74 1.12
4 2.35E+2 0.2 3.46 1.1 5.77E-1 1.1 2.35E+2 0.2 3.46 1.2 5.70E-1 1.0
8 1.54E+2 0.6 1.14 1.6 2.81E-1 1.0 1.68E+2 0.5 1.27 1.5 2.88E-1 1.0
16 7.65E+1 1.0 2.95E-1 2.0 1.30E-1 1.1 7.86E+1 1.1 3.06E-1 2.1 1.33E-1 1.1
32 3.83E+1 1.0 7.38E-2 2.0 6.50E-2 1.0 3.93E+1 1.0 7.65E-2 2.0 6.65E-2 1.0
$ \nu=1e-4 $
2 2.69E+2 7.22 1.19E-2 8.92E+3 3.16E+2 1.25E-1
4 2.35E+2 0.2 3.46 1.1 5.77E-3 1.1 5.34E+3 0.7 9.40E+1 1.8 7.15E-2 0.8
8 1.54E+2 0.6 1.14 1.6 2.81E-3 1.0 4.29E+3 0.3 3.53E+1 1.4 4.53E-2 0.7
16 7.65E+1 1.0 2.95E-1 2.0 1.30E-3 1.1 2.11E+3 1.0 9.11 2.0 1.62E-2 1.5
32 3.83E+1 1.0 7.38E-2 2.0 6.50E-4 1.0 1.06E+3 1.0 2.28 2.0 8.10E-3 1.0
[1]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[2]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[3]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[4]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[5]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[6]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[7]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[8]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[10]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[11]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[12]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[14]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[15]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[16]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[17]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[18]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[19]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[20]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

 Impact Factor: 0.263

Metrics

  • PDF downloads (29)
  • HTML views (108)
  • Cited by (0)

Other articles
by authors

[Back to Top]