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Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

  • Mei-Ling Sun (a1) (a2) and Shan Jiang (a1) (a3)
Abstract
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.

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Corresponding author
*Corresponding author. Email: jiangshan@yzu.edu.cn
References
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[1]Araya R. and Valentin F., A multiscale a posterior error estimate, Comput. Method. Appl. Mech. Eng., 194(18-20) (2005), pp. 20772094.
[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. 191209.
[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. 167191.
[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. 81008113.
[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. 577596.
[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. 631638.
[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. 168189.
[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.
[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. 928941.
[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. 756764.
[11]Lin Q. and Lin J., Finite Element Methods: Accuracy and Improvement, Beijing, 2006.
[12]Llnb T., Layer-adapted meshes for convection-diffusion problems, Comput. Method. Appl. Mech. Eng., 192(9-10) (2003), pp. 10611105.
[13]Miller J. J., O’Riordan E. and Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
[14]Roos H. G., Stabilized FEMfor convection-diffusion problems on layer-adapted meshes, J. Comput. Math., 27(2-3) (2009), pp. 266279.
[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.
[16]Su Y. C. and Q. G. Wu, The Introduction of Numerical Methods for the Singular Perturbed Problems, Chongqing Publishing House, Chongqing, 1992.
[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. 487515.
[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. 386400.
[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. 280298.
[20]Zienkiewicz O. C., The background of error estimation and adaptivity infinite element computations, Comput. Method. Appl. Mech. Eng., 195(4-6) (2006), pp. 207213.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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