Special Issues

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

  • Received: 01 February 2020 Revised: 01 April 2020
  • Primary: 65L11; Secondary: 65L50, 65L60

  • 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[J]. Electronic Research Archive, 2020, 28(2): 935-949. doi: 10.3934/era.2020049

    Related Papers:

  • 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.



    加载中


    [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
    [2] Fully computable robust a posterior error bounds for singularly perturbed reaction-diffusion problems. Numer. Math. (2011) 119: 219-243.
    [3] A parameter uniform numerical method for a nonlinear elliptic reaction-diffusion problem. J. Comput. Appl. Math. (2019) 350: 178-194.
    [4] Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems. Numer. Math. (2020) 144: 1-21.
    [5] Stability and accuracy of adapted finite element methods for singularly perturbed problems. Numer. Math. (2008) 109: 167-191.
    [6] Solving efficiently one dimensional parabolic singularly perturbed reaction-diffusion systems: A splitting by components. J. Comput. Appl. Math. (2018) 344: 1-14.
    [7] Finite element approximation of reaction-diffusion problems using an exponentially graded mesh. Comput. Math. Appl. (2018) 76: 2523-2534.
    [8] An adaptive staggered discontinuous Galerkin method for the steady state convection-diffusion equation. J. Sci. Comput. (2018) 77: 1490-1518.
    [9] Finite element approximation of convection diffusion problems using graded meshes. Appl. Numer. Math. (2006) 56: 1314-1325.
    [10] Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditions. J. Appl. Math. Comput. (2020) 62: 341-360.
    [11] An adapted Petrov-Galerkin multi-scale finite element for singularly perturbed reaction-diffusion problems. Int. J. Comput. Math. (2016) 93: 1200-1211.
    [12] Reduced multiscale computation on adapted grid for the convection-diffusion Robin problem. J. Appl. Anal. Comput. (2017) 7: 1488-1502.
    [13] A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. (2012) 50: 2729-2743.
    [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
    [15] Graded meshes for higher order FEM. J. Comput. Math. (2015) 33: 1-16.
    [16] A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation. Numer. Math. Theory Methods Appl. (2008) 1: 214-234.
    [17] A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms (1998) 18: 337-360.
    [18] Analysis of a new stabilized higher order finite element method for advection-diffusion equations. Comput. Methods Appl. Mech. Engrg. (2006) 196: 538-550.
    [19] Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes. Adv. Comput. Math. (2014) 40: 797-818.
    [20] Higher order uniformly convergent continuous/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type. Appl. Numer. Math. (2014) 76: 48-59.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1257) PDF downloads(168) Cited by(1)

Article outline

Figures and Tables

Figures(6)  /  Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog