\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Weak Galerkin mixed finite element methods for parabolic equations with memory

  • * Corresponding author: Qiang Xu, Ailing Zhu.

    * Corresponding author: Qiang Xu, Ailing Zhu. 

Project supported by the Natural Science Foundation of Shandong Province (No. ZR2014AM033)

Abstract Full Text(HTML) Figure(1) / Table(2) Related Papers Cited by
  • We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both $ |\|·|\| $ and $ L^2 $ norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces $ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $ for $ k = 0 $ and the numerical convergence rates confirm the theoretical results.

    Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 65R20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A typical uniform mesh on $(0, 1)\times(0, 1)$ with $h = 1/8$

    Table 1.  Error behaviors of FWG-MFEM for the first example with $\Delta t = 4h^2$

    $h$ $|||e_{h}|||$ $\mbox{order}\approx$$\|\varepsilon_{h}\|$ $\mbox{order}\approx$
    $2^{-3}$4.8132e-002-1.7834e-003-
    $2^{-4}$2.3657e-0021.02474.3564e-0042.0334
    $2^{-5}$1.1823e-0021.00071.0872e-0042.0026
    $2^{-6}$5.9209e-0030.99772.7312e-0051.9929
    $2^{-7}$2.9583e-0031.00106.8160e-0062.0026
     | Show Table
    DownLoad: CSV

    Table 2.  Error behaviors of FWG-MFEM for the second example with $\Delta t = 4h^2$

    $h$ $|||e_{h}|||$ $\mbox{order}\approx$$\|\varepsilon_{h}\|$ $\mbox{order}\approx$
    $2^{-3}$1.0576e-000-2.1024e-002-
    $2^{-4}$5.1613e-0011.03504.8443e-0032.1177
    $2^{-5}$2.5868e-0010.99661.2043e-0032.0081
    $2^{-6}$1.2929e-0011.00063.0093e-0042.0007
    $2^{-7}$6.4627e-0021.00047.5140e-0052.0018
     | Show Table
    DownLoad: CSV
  •   H. Che , Z. Zhou , Z. Jiang  and  Y. Wang , $ H^1 $-Galerkin expanded mixed finite element methods for nonlinear pseudo-parabolic integro-differential equations, Numer. Methods Partial Differential Equations, 29 (2013) , 799-817.  doi: 10.1002/num.21731.
      Z. Jiang , $ L^∞(L^2) $ and $ L^∞(L^∞) $ error estimates for mixed methods for integro-differential equations of parabolic type, ESAIM Math. Model. Numer. Anal., 33 (1999) , 531-546.  doi: 10.1051/m2an:1999151.
      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 Press, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13.
      L. Mu , J. Wang  and  X. Ye , A hybridized formulation for the weak Galerkin mixed finite element method, J. Comput. Appl. Math., 307 (2016) , 335-345.  doi: 10.1016/j.cam.2016.01.004.
      A. K. Pani  and  G. Fairweather , $ H^1 $-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 22 (2002) , 231-252.  doi: 10.1093/imanum/22.2.231.
      R. K. Sinha , R. E. Ewing  and  R. D. Lazarov , Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data, SIAM J. Numer. Anal., 47 (2009) , 3269-3292.  doi: 10.1137/080740490.
      J. Wang  and  X. Ye , A weak Galerkin finite element methods for elliptic problems, J. Comput. Appl. Math., 241 (2013) , 103-115.  doi: 10.1016/j.cam.2012.10.003.
      J. Wang  and  X. Ye , A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83 (2014) , 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.
      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.
      E. G. Yanik  and  G. Fairweather , Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal., 12 (1988) , 785-809.  doi: 10.1016/0362-546X(88)90039-9.
      Q. Zhang  and  R. Zhang , A weak Galerkin mixed finite element method for second-order elliptic equations with Robin boundary conditions, J. Comput. Math., 34 (2016) , 532-548.  doi: 10.4208/jcm.1604-m2015-0413.
      C. Zhou , Y. Zou , S. Chai , Q. Zhang  and  H. Zhu , Weak Galerkin mixed finite element method for heat equation, Appl. Numer. Math., 123 (2018) , 180-199.  doi: 10.1016/j.apnum.2017.08.009.
      Z. Zhou , An H1-Galerkin mixed finite element method for a class of heat transport equations, Appl. Math. Model., 34 (2010) , 2414-2425.  doi: 10.1016/j.apm.2009.11.007.
      A. Zhu, Discontinuous mixed covolume methods for linear parabolic integrodifferential problems, J. Appl. Math. , 2014 (2014), Art. ID 649468, 8 pp. doi: 10.1155/2014/649468.
      A. Zhu , T. Xu  and  Q. Xu , Weak Galerkin finite element methods for linear parabolic integro-differential equations, Numer. Methods Partial Differential Equations, 32 (2016) , 1357-1377.  doi: 10.1002/num.22053.
  • 加载中

Figures(1)

Tables(2)

SHARE

Article Metrics

HTML views(585) PDF downloads(370) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return