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A Uniformly Convergent Weak Galerkin Finite Element Method on Shishkin Mesh for 1d Convection–Diffusion Problem

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Abstract

In this paper, a weak Galerkin finite element method is proposed and analyzed for one-dimensional singularly perturbed convection–diffusion problems. This finite element scheme features piecewise polynomials of degree \(k\ge 1\) on interior of each element plus piecewise constant on the node of each element. Our WG scheme is parameter-free and has competitive number of unknowns since the interior unknowns can be eliminated efficiently from the discrete linear system. An \(\varepsilon \)-uniform error bound of \(\mathcal {O}((N^{-1}\ln N)^k)\) in the energy-like norm is established on Shishkin mesh, where N is the number of elements. Finally, the numerical experiments are carried out to confirm the theoretical results. Moreover, the numerical results show that the present method has the optimal convergence rate of \(\mathcal {O}(N^{-(k+1)})\) in the \(L^2\)-norm and the superconvergence rates of \(\mathcal {O}((N^{-1}\ln N)^{2k})\) in the discrete \(L^{\infty }\)-norm.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang, 314001, China

    Peng Zhu

  2. Nanhu College, Jiaxing University, Jiaxing, Zhejiang, 314001, China

    Shenglan Xie

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  1. Peng Zhu
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  2. Shenglan Xie
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Correspondence to Peng Zhu.

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Supported by Natural Science Foundation of Zhejiang province, China (Grant No.LY19A010008).

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Zhu, P., Xie, S. A Uniformly Convergent Weak Galerkin Finite Element Method on Shishkin Mesh for 1d Convection–Diffusion Problem. J Sci Comput 85, 34 (2020). https://doi.org/10.1007/s10915-020-01345-3

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  • DOI: https://doi.org/10.1007/s10915-020-01345-3

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