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Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions

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Abstract

We consider the singularly perturbed fourth-order boundary value problem ε2Δ2u −Δu = f on the unit square \({\Omega }\subset \mathbb {R}^{2}\), with boundary conditions u = ∂u/∂n = 0 on ∂Ω. Here, ε ∈ (0,1) is a small parameter. The problem is solved numerically by means of Adini finite elements—a simple nonconforming finite element method for this problem. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with N2 elements is constructed and convergence of the method is proved in a ‘broken’ version of the Sobolev norm \(v\mapsto \left (\varepsilon ^{2}|v|_{2}^{2} + |v|_{1}^{2} \right )^{1/2}\). For a particular choice of the mesh, the error in the computed solution is at most C [ε1/2(N− 1 lnN)2 + min {ε1/2,ε− 3/2N− 2} + N− 3], where the constant C is independent of ε and N. Numerical results support our theoretical convergence rates, even for an example where not all the hypotheses of our theory are satisfied.

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Acknowledgements

We thank two unknown reviewers for their extremely careful reading of our manuscript and for giving us many helpful suggestions.

Funding

The research of the second author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF U1530401.

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Authors and Affiliations

  1. Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing, China

    Xiangyun Meng & Martin Stynes

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  1. Xiangyun Meng
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  2. Martin Stynes
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Correspondence to Martin Stynes.

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Communicated by: Long Chen

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Meng, X., Stynes, M. Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions. Adv Comput Math 45, 1105–1128 (2019). https://doi.org/10.1007/s10444-018-9646-0

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  • DOI: https://doi.org/10.1007/s10444-018-9646-0

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