Simulation of multiscale finite element method on graded meshes for two-dimensional singularly perturbed twin boundary layers problems

doi:10.3785/j.issn.1008-9497.2022.05.007

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (5): 564-569 DOI: 10.3785/j.issn.1008-9497.2022.05.007

Mathematics and Computer Science

Simulation of multiscale finite element method on graded meshes for two-dimensional singularly perturbed twin boundary layers problems

Meiling SUN^1,²(

),Shan JIANG²(

)

^1.Department of Mathematics，Nantong Vocational University，Nantong 226007，Jiangsu Province，China
^2.School of Science，Nantong University，Nantong 226019，Jiangsu Province，China

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Abstract

To solve a two-dimensional singularly perturbed model, a multiscale finite element method on graded meshes built from the perturbed parameter is presented for capturing the twin boundary layers of convection-diffusion equations effectively. Based on the graded meshes, the multiscale basis functions are capable of subtly describing the microscopic information in the boundary layers. No wonder, it just costs a handful of computing resource and short time to achieve the accurate and efficient results, and the results are independent of the perturbed parameter with uniform stability.

Key words： singular perturbation adaptive meshes twin boundary layers multiscale finite element uniform stability

Received: 24 August 2021 Published: 14 September 2022

CLC:

O 241.82

Corresponding Authors: Shan JIANG E-mail: sunmeiling81@163.com;jiangshan@ntu.edu.cn

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Cite this article:

Meiling SUN,Shan JIANG. Simulation of multiscale finite element method on graded meshes for two-dimensional singularly perturbed twin boundary layers problems. Journal of Zhejiang University (Science Edition), 2022, 49(5): 564-569.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I5/564

多尺度有限元法结合分层网格模拟二维奇异摄动的两端边界层问题

通过摄动系数建立分层网格，用多尺度有限元法捕捉对流扩散方程的两端边界层，研究二维奇异摄动模型。基于分层网格并利用多尺度基函数刻画了边界层的微观信息，用有限的计算资源、较短的计算时间，得到了不依赖于摄动系数、一致稳定的模拟结果。

关键词： 奇异摄动, 自适应网格, 两端边界层, 多尺度有限元, 一致稳定

Fig.1 Sub-domains of domain

Ω

Fig.2 Exact solutions when

ε = 10 - 1

and

10 - 5

，respectively

Fig.3 The solutions of FEM（G） and MsFEM（G） when

ε = 10 - 5

h	$ε = 10 - 5$		$ε = 10 - 6$		$ε = 10 - 7$
h	$N F E M$	误差	$N F E M$	误差	$N F E M$	误差
$2 + 0$	136	4.510×10^-3	160	4.664×10^-3	192	4.748×10^-3
$2 - 1$	240	1.310×10^-3	288	1.362×10^-3	328	1.402×10^-3
$2 - 2$	448	3.870×10^-4	536	4.054×10^-4	616	4.189×10^-4
$2 - 3$	888	1.083×10^-4	1 048	1.142×10^-4	1 200	1.186×10^-4

Table 1 FEM（G） with different parameters for errors of energy norm

h	$ε = 10 - 5$		$ε = 10 - 6$		$ε = 10 - 7$
h	$N M s F E M$	误差	$N M s F E M$	误差	$N M s F E M$	误差
$2 + 0$	34	8.866×10^-2	40	9.084×10^-2	48	9.083×10^-2
$2 - 1$	60	2.270×10^-2	72	2.303×10^-2	82	2.329×10^-2
$2 - 2$	112	4.686×10^-3	134	4.938×10^-3	154	4.955×10^-3
$2 - 3$	222	9.228×10^-4	262	9.565×10^-4	300	9.627×10^-4

Table 2 MsFEM（G） with different parameters for errors of energy norm

Fig.4 Errors of FEM（G） on

N F E M = 136

and MsFEM（G） on

N M s F E M = 112

when

ε = 10 - 5

Fig.5 Errors of FEM（G） on

N F E M = 240

and MsFEM（G） on

N M s F E M = 222

when

ε = 10 - 5

$N F E M 2$	FEM（G）CPU时间/s	$N M s F E M 2$	MsFEM（G）CPU时间/s
$1362$	4.1	$342$	3.3
$2402$	49	$602$	12
$4482$	659	$1122$	84
$8882$	10 277	$2222$	1 106

Table 3 FEM（G） and MsFEM（G）'s CPU time when

ε = 10 - 5

Fig.6 Two methods' log-log on partition

N

and CPU time when

ε = 10 - 6

and

10 - 7


[1]	苏煜城，吴启光. 奇异摄动问题数值方法引论［M］. 重庆：重庆出版社， 1991. SU Y C， WU Q G. An Introduction to Numerical Methods for the Singular Perturbation Problems［M］. Chongqing： Chongqing Publishing House， 1991.

[2]	MILLER J J H， O’RIORDAN E， SHISHKIN G I. Fitted Numerical Methods for Singular Perturbation Problems （Revised Edition）［M］. Singapore： World Scientific， 2012. doi:10.1142/8410 doi: 10.1142/8410

[3]	KADALBAJOO M K， GUPTA V. A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers［J］. International Journal of Computer Mathematics， 2010， 87（14）： 3218-3235. doi:10.1080/00207160902980492 doi: 10.1080/00207160902980492

[4]	GENG F Z， QIAN S P. Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers［J］. Applied Mathematics Letters， 2013， 26（10）： 998-1004. DOI：10.1016/j.aml.2013.05.006 doi: 10.1016/j.aml.2013.05.006

[5]	杨宇博，祝鹏，尹云辉. 分层网格上奇异摄动问题的一致NIPG分析［J］. 计算数学， 2014， 36（4）： 437-448. doi:10.12286/jssx.2014.4.437 YANG Y B， ZHU P， YIN Y H. Uniform analysis of the NIPG method on graded meshes for singularly perturbed convection-diffusion problems［J］. Mathematica Numerica Sinica， 2014， 36（4）： 437-448. doi:10.12286/jssx.2014.4.437 doi: 10.12286/jssx.2014.4.437

[6]	ZHENG Q， LI X Z， GAO Y. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs［J］. Applied Numerical Mathematics， 2015， 91： 46-59. DOI：10. 1016/j.apnum.2014.12.010 doi: 10. 1016/j.apnum.2014.12.010

[7]	江山，孙美玲. 多尺度有限元结合Bakhvalov-Shishkin网格法高效处理边界层问题［J］. 浙江大学学报（理学版）， 2015， 42（2）： 142-146. DOI：10. 3785/j.issn.1008-9497.2015.02.004 JIANG S， SUN M L. Combining the multiscale finite element and Bakhvalov-Shishkin grid to solve the boundary layer problems［J］. Journal of Zhejiang University （Science Edition）， 2015， 42（2）： 142-146. DOI：10.3785/j.issn.1008-9497.2015.02.004 doi: 10.3785/j.issn.1008-9497.2015.02.004

[8]	郑权，刘颖，刘忠礼. 奇异摄动问题在修正的Bakhvalov-Shishkin网格上的混合差分格式［J］. 浙江大学学报（理学版）， 2020， 47（4）： 460-468. DOI：10. 3785/j.issn.1008-9497.2020.04.009 ZHENG Q， LIU Y， LIU Z L. The hybrid finite difference schemes on the modified Bakhvalov-Shishkin mesh for the singularly perturbed problem［J］. Journal of Zhejiang University （Science Edition）， 2020， 47（4）： 460-468. DOI：10.3785/j.issn.1008-9497.2020. 04.009 doi: 10.3785/j.issn.1008-9497.2020. 04.009

[9]	CHENG Y. On the local discontinuous Galerkin method for singularly perturbed problem with two parameters［J］. Journal of Computational and Applied Mathematics， 2021， 392： 113485. DOI：10. 1016/j.cam.2021.113485 doi: 10. 1016/j.cam.2021.113485

[10]	FRANZ S， LINß T， ROOS H G， et al. Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems［J］. Journal of Computational Mathematics， 2010， 28（1）： 32-44. DOI：10.4208/jcm.2009.09-m1005 doi: 10.4208/jcm.2009.09-m1005

[11]	BRDAR M， ZARIN H， TEOFANOV L. A singularly perturbed problem with two parameters in two dimensions on graded meshes［J］. Computers and Mathematics with Applications， 2016， 72（10）： 2582-2603. DOI：10.1016/j.camwa.2016.09.021 doi: 10.1016/j.camwa.2016.09.021

[12]	JIANG S， PRESHO M， HUANG Y Q. An adapted Petrov-Galerkin multi-scale finite element method for singularly perturbed reaction-diffusion problems［J］. International Journal of Computer Mathematics， 2016， 93（7）： 1200-1211. DOI：10.1080/00207160. 2015.1041935 doi: 10.1080/00207160. 2015.1041935

[13]	LI Z W， WU B， XU Y S. High order Galerkin methods with graded meshes for two-dimensional reaction-diffusion problems［J］. International Journal of Numerical Analysis and Modeling， 2016， 13（3）： 319-343.

[14]	XU S P， DENG W B， WU H J. A combined finite element method for elliptic problems posted in domains with rough boundaries［J］. Journal of Computational and Applied Mathematics， 2018， 336： 235-248. DOI：10.1016/j.cam.2017.12.049 doi: 10.1016/j.cam.2017.12.049

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