### Electronic Research Archive

2021, Issue 1: 1881-1895. doi: 10.3934/era.2020096
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# Viscosity robust weak Galerkin finite element methods for Stokes problems

• Received: 01 April 2020 Revised: 01 July 2020 Published: 16 September 2020
• Primary: 65N15, 65N30; Secondary: 35B45, 35J50, 35J35

• 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[J]. Electronic Research Archive, 2021, 29(1): 1881-1895. doi: 10.3934/era.2020096

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• 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.

 [1] Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions. J. Comput. Math. (2015) 33: 191-208. [2] A discontinuous skeletal method for the viscosity-dependent Stokes problem. Comput. Methods Appl. Mech. Engrg. (2016) 306: 175-195. [3] Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comp. (2014) 83: 15-36. [4] High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Engrg. (2016) 307: 339-361. [5] A divergence-free velocity reconstruction for incompressible flows. C. R. Math. Acad. Sci. Paris (2012) 350: 837-840. [6] On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. (2014) 268: 782-800. [7] Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math. Model. Numer. Anal. (2016) 50: 289-309. [8] Simplified weak Galerkin and new finite difference schemes for the Stokes equation. J. Comput. Appl. Math. (2019) 361: 176-206. [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 [10] Effective implementation of the weak Galerkin finite element methods for the biharmonic equation. Comput. Math. Appl. (2017) 74: 1215-1222. [11] A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. (2018) 329: 268-279. [12] A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. (2013) 241: 103-115. [13] A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. (2016) 42: 155-174. [14] A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. (2005) 74: 543-554.
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