-
Previous Article
A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes
- ERA Home
- This Issue
-
Next Article
Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data
Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations
School of Mathematics and Statistics, Beihua University, Jilin 132013, Jilin, China |
In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the $ H^1 $-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.
References:
[1] |
J. R. Cannon and Y. P. Lin,
A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.
doi: 10.1137/0727036. |
[2] |
L. P. Chen and Y. P. Chen,
Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.
doi: 10.1007/s10915-011-9469-3. |
[3] |
K. Eriksson and C. Johnson,
Adaptive finite element methods for parabolic problems Ⅳ: Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749.
doi: 10.1137/0732078. |
[4] |
S. M. F. Garcia,
Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: The discrete-time case, Numer. Methods Partial Differ. Equ., 10 (1994), 149-169.
doi: 10.1002/num.1690100203. |
[5] |
T. L. Hou, W. Z. Jiang, Y. T. Yang and H. T. Leng,
Two-grid $P^2_0$-$P_1$ mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations, Appl. Numer. Math., 137 (2019), 136-150.
doi: 10.1016/j.apnum.2018.11.009. |
[6] |
M.-N. Le Roux and V. Thomée,
Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal., 26 (1989), 1291-1309.
doi: 10.1137/0726075. |
[7] | Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006. Google Scholar |
[8] |
W. Liu, H. Rui and Y. P. Bao,
Two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations, J. Sys. Sci. Math. Sci., 30 (2010), 181-190.
|
[9] |
P. K. Moore,
A posterior error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. Numer. Anal., 31 (1994), 149-169.
doi: 10.1137/0731008. |
[10] |
D. Y. Shi and P. C. Mu,
Superconvergence analysis of a two-grid method for semilinear parabolic equations, Appl. Math. Lett., 84 (2018), 34-41.
doi: 10.1016/j.aml.2018.04.012. |
[11] |
J. C. Xu,
A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.
doi: 10.1137/0915016. |
[12] |
J. C. Xu,
Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777.
doi: 10.1137/S0036142992232949. |
[13] |
J. M. Yang and X. Q. Xing,
A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems, Appl. Math. Comput., 346 (2019), 96-108.
doi: 10.1016/j.amc.2018.09.067. |
show all references
References:
[1] |
J. R. Cannon and Y. P. Lin,
A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory, SIAM J. Numer. Anal., 27 (1990), 595-607.
doi: 10.1137/0727036. |
[2] |
L. P. Chen and Y. P. Chen,
Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49 (2011), 383-401.
doi: 10.1007/s10915-011-9469-3. |
[3] |
K. Eriksson and C. Johnson,
Adaptive finite element methods for parabolic problems Ⅳ: Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), 1729-1749.
doi: 10.1137/0732078. |
[4] |
S. M. F. Garcia,
Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: The discrete-time case, Numer. Methods Partial Differ. Equ., 10 (1994), 149-169.
doi: 10.1002/num.1690100203. |
[5] |
T. L. Hou, W. Z. Jiang, Y. T. Yang and H. T. Leng,
Two-grid $P^2_0$-$P_1$ mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations, Appl. Numer. Math., 137 (2019), 136-150.
doi: 10.1016/j.apnum.2018.11.009. |
[6] |
M.-N. Le Roux and V. Thomée,
Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal., 26 (1989), 1291-1309.
doi: 10.1137/0726075. |
[7] | Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006. Google Scholar |
[8] |
W. Liu, H. Rui and Y. P. Bao,
Two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations, J. Sys. Sci. Math. Sci., 30 (2010), 181-190.
|
[9] |
P. K. Moore,
A posterior error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. Numer. Anal., 31 (1994), 149-169.
doi: 10.1137/0731008. |
[10] |
D. Y. Shi and P. C. Mu,
Superconvergence analysis of a two-grid method for semilinear parabolic equations, Appl. Math. Lett., 84 (2018), 34-41.
doi: 10.1016/j.aml.2018.04.012. |
[11] |
J. C. Xu,
A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15 (1994), 231-237.
doi: 10.1137/0915016. |
[12] |
J. C. Xu,
Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759-1777.
doi: 10.1137/S0036142992232949. |
[13] |
J. M. Yang and X. Q. Xing,
A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems, Appl. Math. Comput., 346 (2019), 96-108.
doi: 10.1016/j.amc.2018.09.067. |
order | ||
9.6461e-04 | - | |
2.4062e-04 | 2.00 | |
6.0130e-05 | 2.00 | |
1.5037e-05 | 2.00 |
order | ||
9.6461e-04 | - | |
2.4062e-04 | 2.00 | |
6.0130e-05 | 2.00 | |
1.5037e-05 | 2.00 |
order | ||
1.9302e-02 | - | |
4.8349e-03 | 2.00 | |
1.2096e-03 | 2.00 | |
3.0244e-04 | 2.00 |
order | ||
1.9302e-02 | - | |
4.8349e-03 | 2.00 | |
1.2096e-03 | 2.00 | |
3.0244e-04 | 2.00 |
|
order | |
8.7973e-04 | - | |
7.1420e-05 | 3.51 | |
4.6798e-06 | 3.91 | |
2.9328e-07 | 3.99 |
|
order | |
8.7973e-04 | - | |
7.1420e-05 | 3.51 | |
4.6798e-06 | 3.91 | |
2.9328e-07 | 3.99 |
two-grid time (s) | Crank-Nicolson time (s) | |
0.0998 | 0.1164 | |
0.9118 | 1.2019 | |
13.6126 | 17.9624 |
two-grid time (s) | Crank-Nicolson time (s) | |
0.0998 | 0.1164 | |
0.9118 | 1.2019 | |
13.6126 | 17.9624 |
[1] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[2] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440 |
[3] |
Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 |
[4] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[5] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[6] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[7] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[8] |
Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 |
[9] |
Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 |
[10] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282 |
[11] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[12] |
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 |
[13] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
[14] |
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 |
[15] |
Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093 |
[16] |
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 |
[17] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[18] |
Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 |
[19] |
Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 |
[20] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
Impact Factor: 0.263
Tools
Metrics
Other articles
by authors
[Back to Top]