June  2020, 28(2): 935-949. doi: 10.3934/era.2020049

Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation

1. 

School of Science, Nantong University, Nantong 226019, China

2. 

Department of Public Courses, Nantong Vocational University, Nantong 226007, China

3. 

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

* Corresponding author: jiangshan@ntu.edu.cn

Received  February 2020 Revised  April 2020 Published  June 2020

Fund Project: The first author is supported by NSFC grant 11771224 and Jiangsu Province Qing Lan Project

A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.

Citation: Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049
References:
[1]

O. Abu Arqub, Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm, Calcolo, 55 (2018), 28pp. doi: 10.1007/s10092-018-0274-3.  Google Scholar

[2]

M. Ainsworth and T. Vejchodský, Fully computable robust a posterior error bounds for singularly perturbed reaction-diffusion problems, Numer. Math., 119 (2011), 219-243.  doi: 10.1007/s00211-011-0384-1.  Google Scholar

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I. Boglaev, A parameter uniform numerical method for a nonlinear elliptic reaction-diffusion problem, J. Comput. Appl. Math., 350 (2019), 178-194.  doi: 10.1016/j.cam.2018.10.017.  Google Scholar

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D. CaiZ. Cai and S. Zhang, Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems, Numer. Math., 144 (2020), 1-21.  doi: 10.1007/s00211-019-01075-1.  Google Scholar

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L. Chen and J. Xu, Stability and accuracy of adapted finite element methods for singularly perturbed problems, Numer. Math., 109 (2008), 167-191.  doi: 10.1007/s00211-007-0118-6.  Google Scholar

[6]

C. Clavero and J. C. Jorge, Solving efficiently one dimensional parabolic singularly perturbed reaction-diffusion systems: A splitting by components, J. Comput. Appl. Math., 344 (2018), 1-14.  doi: 10.1016/j.cam.2018.05.019.  Google Scholar

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P. ConstantinouS. FranzL. Ludwig and C. Xenophontos, Finite element approximation of reaction-diffusion problems using an exponentially graded mesh, Comput. Math. Appl., 76 (2018), 2523-2534.  doi: 10.1016/j.camwa.2018.08.051.  Google Scholar

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J. Du and E. Chung, An adaptive staggered discontinuous Galerkin method for the steady state convection-diffusion equation, J. Sci. Comput., 77 (2018), 1490-1518.  doi: 10.1007/s10915-018-0695-9.  Google Scholar

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R. G. Durán and A. L. Lombardi, Finite element approximation of convection diffusion problems using graded meshes, Appl. Numer. Math., 56 (2006), 1314-1325.  doi: 10.1016/j.apnum.2006.03.029.  Google Scholar

[10]

G. Janani Jayalakshmi and A. Tamilselvan, Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditions, J. Appl. Math. Comput., 62 (2020), 341-360.  doi: 10.1007/s12190-019-01287-6.  Google Scholar

[11]

S. JiangM. Presho and Y. Huang, An adapted Petrov-Galerkin multi-scale finite element for singularly perturbed reaction-diffusion problems, Int. J. Comput. Math., 93 (2016), 1200-1211.  doi: 10.1080/00207160.2015.1041935.  Google Scholar

[12]

S. JiangM. Sun and Y. Yang, Reduced multiscale computation on adapted grid for the convection-diffusion Robin problem, J. Appl. Anal. Comput., 7 (2017), 1488-1502.  doi: 10.11948/2017091.  Google Scholar

[13]

R. Lin and M. Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 50 (2012), 2729-2743.  doi: 10.1137/110837784.  Google Scholar

[14]

J. J. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814390743.  Google Scholar

[15]

H.-G. RoosL. Teofanov and Z. Uzelac, Graded meshes for higher order FEM, J. Comput. Math., 33 (2015), 1-16.  doi: 10.4208/jcm.1405-m4362.  Google Scholar

[16]

G. I. Shishkin, A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation, Numer. Math. Theory Methods Appl., 1 (2008), 214-234.   Google Scholar

[17]

M. Stynes and L. Tobiska, A finite difference analysis of a streamline diffusion method on a Shishkin mesh, Numer. Algorithms, 18 (1998), 337-360.  doi: 10.1023/A:1019185802623.  Google Scholar

[18]

L. Tobiska, Analysis of a new stabilized higher order finite element method for advection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 196 (2006), 538-550.  doi: 10.1016/j.cma.2006.05.009.  Google Scholar

[19]

J. Zhao and S. Chen, Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes, Adv. Comput. Math., 40 (2014), 797-818.  doi: 10.1007/s10444-013-9327-y.  Google Scholar

[20]

P. Zhu and S. Xie, Higher order uniformly convergent continuous/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type, Appl. Numer. Math., 76 (2014), 48-59.  doi: 10.1016/j.apnum.2013.10.001.  Google Scholar

show all references

References:
[1]

O. Abu Arqub, Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm, Calcolo, 55 (2018), 28pp. doi: 10.1007/s10092-018-0274-3.  Google Scholar

[2]

M. Ainsworth and T. Vejchodský, Fully computable robust a posterior error bounds for singularly perturbed reaction-diffusion problems, Numer. Math., 119 (2011), 219-243.  doi: 10.1007/s00211-011-0384-1.  Google Scholar

[3]

I. Boglaev, A parameter uniform numerical method for a nonlinear elliptic reaction-diffusion problem, J. Comput. Appl. Math., 350 (2019), 178-194.  doi: 10.1016/j.cam.2018.10.017.  Google Scholar

[4]

D. CaiZ. Cai and S. Zhang, Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems, Numer. Math., 144 (2020), 1-21.  doi: 10.1007/s00211-019-01075-1.  Google Scholar

[5]

L. Chen and J. Xu, Stability and accuracy of adapted finite element methods for singularly perturbed problems, Numer. Math., 109 (2008), 167-191.  doi: 10.1007/s00211-007-0118-6.  Google Scholar

[6]

C. Clavero and J. C. Jorge, Solving efficiently one dimensional parabolic singularly perturbed reaction-diffusion systems: A splitting by components, J. Comput. Appl. Math., 344 (2018), 1-14.  doi: 10.1016/j.cam.2018.05.019.  Google Scholar

[7]

P. ConstantinouS. FranzL. Ludwig and C. Xenophontos, Finite element approximation of reaction-diffusion problems using an exponentially graded mesh, Comput. Math. Appl., 76 (2018), 2523-2534.  doi: 10.1016/j.camwa.2018.08.051.  Google Scholar

[8]

J. Du and E. Chung, An adaptive staggered discontinuous Galerkin method for the steady state convection-diffusion equation, J. Sci. Comput., 77 (2018), 1490-1518.  doi: 10.1007/s10915-018-0695-9.  Google Scholar

[9]

R. G. Durán and A. L. Lombardi, Finite element approximation of convection diffusion problems using graded meshes, Appl. Numer. Math., 56 (2006), 1314-1325.  doi: 10.1016/j.apnum.2006.03.029.  Google Scholar

[10]

G. Janani Jayalakshmi and A. Tamilselvan, Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditions, J. Appl. Math. Comput., 62 (2020), 341-360.  doi: 10.1007/s12190-019-01287-6.  Google Scholar

[11]

S. JiangM. Presho and Y. Huang, An adapted Petrov-Galerkin multi-scale finite element for singularly perturbed reaction-diffusion problems, Int. J. Comput. Math., 93 (2016), 1200-1211.  doi: 10.1080/00207160.2015.1041935.  Google Scholar

[12]

S. JiangM. Sun and Y. Yang, Reduced multiscale computation on adapted grid for the convection-diffusion Robin problem, J. Appl. Anal. Comput., 7 (2017), 1488-1502.  doi: 10.11948/2017091.  Google Scholar

[13]

R. Lin and M. Stynes, A balanced finite element method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 50 (2012), 2729-2743.  doi: 10.1137/110837784.  Google Scholar

[14]

J. J. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814390743.  Google Scholar

[15]

H.-G. RoosL. Teofanov and Z. Uzelac, Graded meshes for higher order FEM, J. Comput. Math., 33 (2015), 1-16.  doi: 10.4208/jcm.1405-m4362.  Google Scholar

[16]

G. I. Shishkin, A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation, Numer. Math. Theory Methods Appl., 1 (2008), 214-234.   Google Scholar

[17]

M. Stynes and L. Tobiska, A finite difference analysis of a streamline diffusion method on a Shishkin mesh, Numer. Algorithms, 18 (1998), 337-360.  doi: 10.1023/A:1019185802623.  Google Scholar

[18]

L. Tobiska, Analysis of a new stabilized higher order finite element method for advection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 196 (2006), 538-550.  doi: 10.1016/j.cma.2006.05.009.  Google Scholar

[19]

J. Zhao and S. Chen, Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes, Adv. Comput. Math., 40 (2014), 797-818.  doi: 10.1007/s10444-013-9327-y.  Google Scholar

[20]

P. Zhu and S. Xie, Higher order uniformly convergent continuous/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type, Appl. Numer. Math., 76 (2014), 48-59.  doi: 10.1016/j.apnum.2013.10.001.  Google Scholar

Figure 1.  For Example 1, partition number $ N = 160 $ for Shishkin zoom, Bakhvalov zoom, and $ N = 37 $ for Graded zoom (from left to right)
Figure 2.  For Example 1 with $ \varepsilon = 2^{-20} $, exact solution (blue line), FEM solution (green star) and MsFEM solution (red star) on different meshes (from left upper to right lower)
Figure 3.  For Example 1 with $ \varepsilon = 2^{-20} $, corresponding error of FEM and MsFEM on different meshes (from left upper to right lower)
Figure 4.  For Example 1 with different $ \varepsilon $, MsFEM(G) error to partition number $ N $ on semilog scale
Figure 5.  For Example 2 with $ \varepsilon = 2^{-30} $, exact solution (blue line), FEM solution (green star) and MsFEM solution (red star) on different meshes (from left upper to right lower)
Figure 6.  For Example 2 with $ \varepsilon = 2^{-30} $, corresponding error of FEM and MsFEM on different meshes (from left upper to right lower)
Table 1.  For Example 1 with $ \varepsilon = 2^{-20} $, corresponding energy norm error and convergence order of FEM and MsFEM on different meshes
$ N $ FEM(U) order FEM(S) order FEM(B) order $ N $ MsFEM(G) order
160 1.568e+0 - 1.080e-4 - 7.352e-5 - 37 3.235e-4 -
320 1.243e+0 0.34 3.255e-5 1.73 1.782e-5 2.04 70 8.357e-5 1.95
640 6.232e-1 1.00 9.591e-6 1.76 3.939e-6 2.18 137 1.686e-5 2.31
1280 2.158e-1 1.53 2.771e-6 1.79 6.235e-7 2.66 276 2.908e-6 2.54
2560 1.032e-1 1.06 7.910e-7 1.81 5.356e-8 3.54 565 4.650e-7 2.64
5120 4.957e-2 1.06 2.244e-7 1.82 1.069e-8 2.32 1164 7.249e-8 2.68
10240 2.315e-2 1.10 6.307e-8 1.83 2.678e-9 2.00 2406 1.139e-8 2.67
$ N $ FEM(U) order FEM(S) order FEM(B) order $ N $ MsFEM(G) order
160 1.568e+0 - 1.080e-4 - 7.352e-5 - 37 3.235e-4 -
320 1.243e+0 0.34 3.255e-5 1.73 1.782e-5 2.04 70 8.357e-5 1.95
640 6.232e-1 1.00 9.591e-6 1.76 3.939e-6 2.18 137 1.686e-5 2.31
1280 2.158e-1 1.53 2.771e-6 1.79 6.235e-7 2.66 276 2.908e-6 2.54
2560 1.032e-1 1.06 7.910e-7 1.81 5.356e-8 3.54 565 4.650e-7 2.64
5120 4.957e-2 1.06 2.244e-7 1.82 1.069e-8 2.32 1164 7.249e-8 2.68
10240 2.315e-2 1.10 6.307e-8 1.83 2.678e-9 2.00 2406 1.139e-8 2.67
Table 2.  For Example 1 with different $ \varepsilon $, refinement history of $ N $ to the same magnitude of error on Graded mesh
MsFEM(G) order $ N $($ \varepsilon= $ 1e-7) $ N $($ \varepsilon= $ 1e-8) $ N $($ \varepsilon= $ 1e-9)
2.811e-4 - 43 49 54
7.371e-5 1.93 80 90 101
1.488e-5 2.31 156 176 195
2.578e-6 2.53 313 351 389
4.139e-7 2.64 638 713 788
6.477e-8 2.68 1309 1458 1606
1.021e-8 2.67 2696 2992 3288
MsFEM(G) order $ N $($ \varepsilon= $ 1e-7) $ N $($ \varepsilon= $ 1e-8) $ N $($ \varepsilon= $ 1e-9)
2.811e-4 - 43 49 54
7.371e-5 1.93 80 90 101
1.488e-5 2.31 156 176 195
2.578e-6 2.53 313 351 389
4.139e-7 2.64 638 713 788
6.477e-8 2.68 1309 1458 1606
1.021e-8 2.67 2696 2992 3288
Table 3.  For Example 2 with $ \varepsilon = 2^{-30} $, corresponding energy norm error and convergence order of FEM and MsFEM on different meshes
$ N $ FEM(S) order FEM(B) order $ N $ MsFEM(G) order
320 1.069e-4 - 2.086e-5 - 106 2.200e-5 -
640 5.215e-5 1.04 5.567e-6 1.91 196 5.781e-6 1.93
1280 2.479e-5 1.07 1.486e-6 1.91 380 1.376e-6 2.07
2560 1.161e-5 1.09 3.962e-7 1.91 758 3.111e-7 2.15
5120 5.337e-6 1.12 1.052e-7 1.91 1534 6.949e-8 2.16
10240 2.320e-6 1.20 2.786e-8 1.92 3132 1.560e-8 2.16
20480 8.097e-7 1.52 7.173e-9 1.96 6416 3.543e-9 2.14
$ N $ FEM(S) order FEM(B) order $ N $ MsFEM(G) order
320 1.069e-4 - 2.086e-5 - 106 2.200e-5 -
640 5.215e-5 1.04 5.567e-6 1.91 196 5.781e-6 1.93
1280 2.479e-5 1.07 1.486e-6 1.91 380 1.376e-6 2.07
2560 1.161e-5 1.09 3.962e-7 1.91 758 3.111e-7 2.15
5120 5.337e-6 1.12 1.052e-7 1.91 1534 6.949e-8 2.16
10240 2.320e-6 1.20 2.786e-8 1.92 3132 1.560e-8 2.16
20480 8.097e-7 1.52 7.173e-9 1.96 6416 3.543e-9 2.14
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