Abstract
In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i − 1 for the pressure and enhancing the polynomials of degree i − 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem.
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Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comp. Appl Math. 241, 103–115 (2013)
Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. 83, 2101–2126 (2014)
Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42, 155–174 (2016)
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on Polytopal Meshes. Int. J. Numer. Anal Mod. 12, 31–53 (2015)
Mu, L., Wang, J., Wang, Y., Ye, X.: A computational study of the weak Galerkin method for second-order elliptic equations. Numer. Algor. 63, 753–777 (2013)
Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comp. Appl. Math. 285, 45–58 (2015)
Girault, V., Raviart, P.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics. Springer, Berlin (1986)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. Reprint of the 1984 Edition. AMS Chelsea Publishing, Providence (2001)
Chen, Z.: Finite Element Methods and Their Applications. Scientific Computation. Springer, Berlin (2005)
Mu, L., Wang, J., Wei, G., Ye, X., Zhao, S.: Weak Galerkin methods for second order elliptic interface problems. J. Comp. Phys. 250, 106–125 (2013)
Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math Appl. 68, 2314–2330 (2014)
Wang, C., Wang, J.: A hybridized weak Galerkin finite element method for the Biharmonic equation. Int. J. Numer. Anal. Mod. 12, 302–317 (2015)
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element method for the Biharmonic equation on polytopal meshes. Numer. Meth. Part. D E 30, 1003–1029 (2014)
Chen, W., Wang, F., Wang, Y.: Weak Galerkin method for the coupled Darcy-Stokes flow. IMA. J. Numer. Anal. 36, 897–921 (2016)
Mu, L., Wang, J., Ye, X.: A new weak Galerkin finite element method for the Helmholtz equation. IMA J. Numer. Anal. 35, 1228–1255 (2015)
Lin, G., Liu, J., Mu, L., Ye, X.: Weak Galerkin finite element methods for Darcy flow Anisotropy and heterogeneity. J. Comp. Phys. 276, 422–437 (2014)
Mu, L., Wang, J., Ye, X.: A stable numerical algorithm for the Brinkman equations by weak Galerkin finite elemnt methods. J. Comp. Phys. 273, 327–342 (2014)
Gao, F., Mu, L.: On error estimate for weak Galerkin finite element methods for parabolic problems. J. Comput. Math. 32, 195–204 (2014)
Gao, F., Wang, X.: A modified weak Galerkin finite element method for a class of parabolic problems. J. Comp. Appl. Math. 271, 1–19 (2014)
Li, Q., Wang, J.: Weak Galerkin finite element methods for parabolic equations. Numer. Meth. Part. D E 29, 2004–2024 (2013)
Mu, L., Wang, J., Ye, X., Zhao, S.: A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys. 15, 1461–1479 (2014)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics. Springer, New York (1991)
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Communicated by: A. Zhou Research supported in part by NSF of China (No. 11371031), and the Key Projects of Baoji university of Arts and Sciences (No. ZK15040) and (No. ZK15033).
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Liu, X., Li, J. & Chen, Z. A weak Galerkin finite element method for the Oseen equations. Adv Comput Math 42, 1473–1490 (2016). https://doi.org/10.1007/s10444-016-9471-2
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DOI: https://doi.org/10.1007/s10444-016-9471-2