Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T01:00:27.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

Published online by Cambridge University Press:  03 June 2015

Mei-Ling Sun  and
Shan Jiang
Mei-Ling Sun
Affiliation:
College of Mathematics Science, Yangzhou University, Yangzhou 225002, China Education and Technology Center, Nantong Vocational College, Nantong 226007, China
Shan Jiang*
Affiliation:
College of Mathematics Science, Yangzhou University, Yangzhou 225002, China Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
*
*Corresponding author. Email: [email protected]
Get access

Abstract

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L2 norm and first order convergence in the energy norm on graded meshes, which is independent of ɛ. In contrast with the conventional methods, our method is much more accurate and effective.

Keywords

35J2565N1265N30Multiscale basis functionssingular perturbationboundary layeradaptively graded meshes
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 5 , October 2014 , pp. 604 - 614
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Araya, R. and Valentin, F., A multiscale a posterior error estimate, Comput. Method. Appl. Mech. Eng., 194(18-20) (2005), pp. 2077–2094.CrossRefGoogle Scholar
[2]Chang, L. L., Gong, W. and Yan, N. N., Finite element method for a nonsmooth elliptic equa-tion, Front. Math. China, 5(2) (2010), pp. 191–209.Google Scholar
[3]Chen, L. and Xu, J. C., Stability and accuracy of adapted finite element methods for singularly perturbed problems, Numer. Math., 109(2) (2008), pp. 167–191.Google Scholar
[4]Efendiev, Y., Galvis, J. and Gildin, E., Local-global multiscale model reduction for flows in high-contrast heterogeneous media, J. Comput. Phys., 231(24) (2012), pp. 8100–8113.Google Scholar
[5]Efendiev, Y. and Hou, T. Y., Multiscale finite element methods for porous media flows and their applications, Appl. Numer. Math., 57(5-7) (2007), pp. 577–596.Google Scholar
[6]Frazier, J. D., Jimack, P. K. and Kirby, R.M., On the use of adjoint-based sensitivity estimates to control local mesh refinement, Commun. Comput. Phys., 7(3) (2010), pp. 631–638.CrossRefGoogle Scholar
[7]Hou, T. Y. and Wu, X. H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134(1) (1997), pp. 168–189.Google Scholar
[8]Hou, T. Y., Wu, X. H. and Cai, Z. Q., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput., 68(227) (1999), pp. 913943.Google Scholar
[9]Jiang, S. and Huang, Y. Q., Numerical investigation on the boundary conditions for the multiscale base functions, Commun. Comput. Phys., 5(5) (2009), pp. 928–941.Google Scholar
[10]Jiang, S. and Sun, M. L., Multiscale finite element method for the singularly perturbed reaction- diffusion problem, J. Basic Sci. Eng., 17(5) (2009), pp. 756–764.Google Scholar
[11]Lin, Q. and Lin, J., Finite Element Methods: Accuracy and Improvement, Beijing, 2006.Google Scholar
[12]Llnb, T., Layer-adapted meshes for convection-diffusion problems, Comput. Method. Appl. Mech. Eng., 192(9-10) (2003), pp. 1061–1105.Google Scholar
[13]Miller, J. J., O’Riordan, E. and Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.CrossRefGoogle Scholar
[14]Roos, H. G., Stabilized FEM for convection-diffusion problems on layer-adapted meshes, J. Comput. Math., 27(2-3) (2009), pp. 266–279.Google Scholar
[15]Shishkin, G. I., A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation, Numer. Math. Theor. Meth. Appl., 1(2) (2008), pp. 214234.Google Scholar
[16]Su, Y. C. and Q. G., Wu, The Introduction of Numerical Methods for the Singular Perturbed Problems, Chongqing Publishing House, Chongqing, 1992.Google Scholar
[17]Tang, H. Z. and Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41(2) (2003), pp. 487–515.Google Scholar
[18]Xenophontos, C. and Oberbroeckling, L., On the finite element approximation of systems of reaction-diffusion equations by p/hp methods, J. Comput. Math., 28(3) (2010), pp. 386–400.Google Scholar
[19]Xie, Z. Q., Zhang, Z. Z. and Zhang, Z. M., A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comput. Math., 27(2-3) (2009), pp. 280–298.Google Scholar
[20]Zienkiewicz, O. C., The background of error estimation and adaptivity infinite element computations, Comput. Method. Appl. Mech. Eng., 195(4-6) (2006), pp. 207–213.CrossRefGoogle Scholar