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On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation
Weak Galerkin mixed finite element methods for parabolic equations with memory
School of Mathematical and Statistics, Shandong Normal University, Jinan 250014, China |
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.
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. |
| | | | |
| 4.8132e-002 | - | 1.7834e-003 | - |
| 2.3657e-002 | 1.0247 | 4.3564e-004 | 2.0334 |
| 1.1823e-002 | 1.0007 | 1.0872e-004 | 2.0026 |
| 5.9209e-003 | 0.9977 | 2.7312e-005 | 1.9929 |
| 2.9583e-003 | 1.0010 | 6.8160e-006 | 2.0026 |
| | | | |
| 4.8132e-002 | - | 1.7834e-003 | - |
| 2.3657e-002 | 1.0247 | 4.3564e-004 | 2.0334 |
| 1.1823e-002 | 1.0007 | 1.0872e-004 | 2.0026 |
| 5.9209e-003 | 0.9977 | 2.7312e-005 | 1.9929 |
| 2.9583e-003 | 1.0010 | 6.8160e-006 | 2.0026 |
| | | ||
| 1.0576e-000 | - | 2.1024e-002 | - |
| 5.1613e-001 | 1.0350 | 4.8443e-003 | 2.1177 |
| 2.5868e-001 | 0.9966 | 1.2043e-003 | 2.0081 |
| 1.2929e-001 | 1.0006 | 3.0093e-004 | 2.0007 |
| 6.4627e-002 | 1.0004 | 7.5140e-005 | 2.0018 |
| | | ||
| 1.0576e-000 | - | 2.1024e-002 | - |
| 5.1613e-001 | 1.0350 | 4.8443e-003 | 2.1177 |
| 2.5868e-001 | 0.9966 | 1.2043e-003 | 2.0081 |
| 1.2929e-001 | 1.0006 | 3.0093e-004 | 2.0007 |
| 6.4627e-002 | 1.0004 | 7.5140e-005 | 2.0018 |
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