2022 Volume 12 Issue 6
Article Contents
Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(6): 2578-2592
Next Article Previous Article

Xiu Ye, Shangyou Zhang. ORDER TWO SUPERCONVERGENCE OF THE CDG METHOD FOR THE STOKES EQUATIONS ON TRIANGLE/TETRAHEDRON[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2578-2592. doi: 10.11948/20220112
Citation: Xiu Ye, Shangyou Zhang. ORDER TWO SUPERCONVERGENCE OF THE CDG METHOD FOR THE STOKES EQUATIONS ON TRIANGLE/TETRAHEDRON[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2578-2592. doi: 10.11948/20220112
shu

ORDER TWO SUPERCONVERGENCE OF THE CDG METHOD FOR THE STOKES EQUATIONS ON TRIANGLE/TETRAHEDRON

  • 1.

    Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

  • 2.

    Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

    Abstract

  • A new conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the Stokes equations. The CDG method gets its name by combining good features of both conforming finite element method and discontinuous finite element method. It has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. This new CDG method is not only stabilizer free but also has two order higher convergence rate than the optimal order. This CDG method uses discontinuous $ P_k $ element for velocity and continuous $ P_{k+1} $ element for pressure. Order two superconvergence is derived for velocity in an energy norm and the $ L^2 $ norm. The superconvergent $ P_k $ solution is lifted elementwise to a $ P_{k+2} $ velocity which converges at the optimal order. The numerical experiments confirm the theories.
    MSC: 65N15, 65N30
    • FullText(HTML)
  • 加载中
    • References
  • [1] A. Al-Twaeel, S. Hussian and X. Wang, A stabilizer free weak Galerkin finite element method for parabolic equation, J. Comput. Appl. Math., 2021, 392, 113373. doi: 10.1016/j.cam.2020.113373

    CrossRef Google Scholar

    [2] A. AL-Taweel, X. Wang, X. Ye and S. Zhang, A stabilizer free weak Galerkin method with supercloseness of order two, Numer. Meth. PDE, 2021, 37, 1012-1029. doi: 10.1002/num.22564

    CrossRef Google Scholar

    [3] I. Babuška, The finite element method with Lagrangian multiplier, Numer. Math., 1973, 20, 179-192. doi: 10.1007/BF01436561

    CrossRef Google Scholar

    [4] F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, Anal. Numér., 1974, 2, 129-151.

    Google Scholar

    [5] G. Chen, M. Feng and X. Xie, A robust WG finite element method for convection–diffusion–reaction equations, J. Comput. Appl. Math., 2017, 315, 107-125. doi: 10.1016/j.cam.2016.10.029

    CrossRef Google Scholar

    [6] W. Chen, F. Wang and Y. Wang, Weak Galerkin method for the coupled Darcy–Stokes flow, IMA J. Numer. Anal., 2016, 36, 897-921.

    Google Scholar

    [7] B. Deka and P. Roy, Weak galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions, Numer. Funct. Anal. Optim., 2019, 40, 250-279.

    Google Scholar

    [8] V. Girault and P. Raviart, Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, 1986.

    Google Scholar

    [9] Q. Guan, M. Gunzburger and W. Zhao, Weak-Galerkin finite element methods for a second-order elliptic variational inequality, Comput. Methods Appl. Mech. and Engrg., 2018, 337, 677-688 doi: 10.1016/j.cma.2018.04.006

    CrossRef Google Scholar

    [10] M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows, A Guide to Theory, Practice and Algorithms, Academic, San Diego, 1989.

    Google Scholar

    [11] Q. Hu, Y. He and K. Wang, Weak galerkin method for the helmholtz equation with DTN boundary condition, Int. J of Numer. Anal. Model., 2020, 17, 643-661.

    Google Scholar

    [12] G. Li, Y. Chen and Y. Huang, A new weak Galerkin finite element scheme for general second-order elliptic problems, J. Comput. Appl. Math., 2018, 344, 701-715.

    Google Scholar

    [13] R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Num. Anal., 2018, 56, 1482-1497. doi: 10.1137/17M1152528

    CrossRef Google Scholar

    [14] J. Liu, S. Tavener, Z. Wang, Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes, SIAM J. Sci. Comput., 2018, 40, 1229-1252. doi: 10.1137/17M1145677

    CrossRef Google Scholar

    [15] X. Liu, J. Li and Z. Chen, A weak Galerkin finite element method for the Oseen equations, Adv. Comput. Math., 2016, 42, 1473-1490.

    Google Scholar

    [16] L. Mu, X. Ye and S. Zhang, A stabilizer free, pressure robust and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh, SIAM J. Sci. Comput., 2021, 43, A2614-A2637.

    Google Scholar

    [17] W. Qi and L. Song, Weak Galerkin method with implicit $\theta$-schemes for second-order parabolic problems, Appl. Math. and Comp., 2020, 336, 124731.

    Google Scholar

    [18] S. Shields, J. Li and E.A. Machorro, Weak Galerkin methods for time-dependent Maxwell's equations, Comput. Math. Appl., 2017, 74, 2106-2124.

    Google Scholar

    [19] M. Sun and H. Rui, A coupling of weak Galerkin and mixed finite element methods for poroelasticity, Comput. & Math. with Appl., 2017, 73, 804-823.

    Google Scholar

    [20] S. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients, Appl. Numer. Math., 2021, 168, 1-12.

    Google Scholar

    [21] S. Xie and P. Zhu, Superconvergence of a WG method for the Stokes equations with continuous pressure, preprint.

    Google Scholar

    [22] J. Wang, Q. Zhai, R. Zhang and S. Zhang, A weak Galerkin finite element scheme for the Cahn-Hilliard equation, Math. Comp., 2019, 88, 211-235.

    Google Scholar

    [23] C. Wang and J. Wang, Discretization of div–curl systems by weak Galerkin finite element methods on polyhedral partitions, J. Sci. Comput., 2016, 68, 1144-1171.

    Google Scholar

    [24] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 2013, 241, 103-115.

    Google Scholar

    [25] X. Ye and S. Zhang, A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes, SIAM J. Numerical Analysis, 2020, 58, 2572-2588,

    Google Scholar

    [26] X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. Model., 2020, 17(1), 110-117.

    Google Scholar

    [27] X. Ye and S. Zhang, Constructing a CDG element with order two superconvergence for second order problem on rectangular mesh, preprint.

    Google Scholar

    [28] H. Zhang, Y. Zou, S. Chai and H. Yue, Weak Galerkin method with $(r, r-1, r-1)$-order finite elements for second order parabolic equations, Appl. Math. and Comp., 2016, 275, 24-40.

    Google Scholar

    [29] T. Zhang and T. Lin, A posteriori error estimate for a modified weak Galerkin method solving elliptic problems, Numer. Meth. PDE, 2017, 33, 381-398.

    Google Scholar

    [30] A. Zhu, T. Xu and Q. Xu, Weak G alerkin finite element methods for linear parabolic integro-differential equations, Numer. Meth. of PDE, 2016, 32, 1357-1377.

    Google Scholar

    [31] S. Zhou, F. Gao, B. Li, and Z. Sun, Weak galerkin finite element method with second-order accuracy in time for parabolic problems, Appl. Math. Lett., 2019, 90, 118-123.

    Google Scholar

    • Supplements
    • Cited by

Figures(3)  /  Tables(8)

Export Citation
PDF
XML

Article Metrics

Article views(2196) PDF downloads(350) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Published by Wilmington Scientific Publisher