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

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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

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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
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