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

Shan Jiang; Li Liang; Meiling Sun; Fang Su; Shan Jiang; Li Liang; Meiling Sun; Fang Su

doi:10.3934/era.2020049

Electronic Research Archive

2020, Volume 28, Issue 2: 935-949. doi: 10.3934/era.2020049

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

Shan Jiang ^{1
,
,},
Li Liang ^{1
,},
Meiling Sun ^{2
,},
Fang Su ^{3
,}

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

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.
- Convection-diffusion,
- small parameter,
- multiscale finite element,
- graded mesh,
- higher-order convergence
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

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Abstract

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.

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
[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.

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