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A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem

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Abstract

A new \(C^0\) nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space \(P_3\) and some bubble functions. Different from non-\(C^0\) nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution \(u\in H_0^2(\Omega )\cap H^4(\Omega )\), the scheme is of linear order convergence by the measurement of discrete norm \(\Vert \cdot \Vert _h\). Some numerical results are included to further illustrate the convergence analysis.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Shicang Song & Lijuan Lu

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  1. Shicang Song
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  2. Lijuan Lu
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Correspondence to Shicang Song or Lijuan Lu.

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This work is supported by the National Basic Research Program of China (Grant No. 2012CB025904).

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Song, S., Lu, L. A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem. J. Appl. Math. Comput. 66, 581–597 (2021). https://doi.org/10.1007/s12190-020-01429-1

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  • DOI: https://doi.org/10.1007/s12190-020-01429-1

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