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June  2019, 12(3): 513-531. doi: 10.3934/dcdss.2019034

Weak Galerkin mixed finite element methods for parabolic equations with memory

Xiaomeng Li , Qiang Xu , and  Ailing Zhu ,

School of Mathematical and Statistics, Shandong Normal University, Jinan 250014, China

* Corresponding author: Qiang Xu, Ailing Zhu.

Received  June 2017 Revised  September 2017 Published  September 2018

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

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.

Keywords: Weak Galerkin, mixed finite element, discrete weak divergence, integro-differential equations, error estimates.
Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 65R20.
Citation: Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

show all references

References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

Figure 1.  A typical uniform mesh on $(0, 1)\times(0, 1)$ with $h = 1/8$
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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
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Download as excel

$h$ $|||e_{h}|||$ $\mbox{order}\approx$$\|\varepsilon_{h}\|$ $\mbox{order}\approx$
$2^{-3}$4.8132e-002-1.7834e-003-
$2^{-4}$2.3657e-0021.02474.3564e-0042.0334
$2^{-5}$1.1823e-0021.00071.0872e-0042.0026
$2^{-6}$5.9209e-0030.99772.7312e-0051.9929
$2^{-7}$2.9583e-0031.00106.8160e-0062.0026
Table 2.  Error behaviors of FWG-MFEM for the second example with $\Delta t = 4h^2$
$h$ $|||e_{h}|||$ $\mbox{order}\approx$$\|\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
Table Options

Download as excel

$h$ $|||e_{h}|||$ $\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
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