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A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension

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Abstract

We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials \(\{1, x_{i}, x_{i}^{2}\}\), the shear stress by bilinear polynomials \(\{1, x_{i}, x_{j}, x_{i}x_{j}\}\), and the displacement by linear polynomials \(\{1, x_{i} \}\). The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension \(n\). As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Jun Hu

  2. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Hongying Man

  3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

Authors
  1. Jun Hu
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  2. Hongying Man
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  3. Shangyou Zhang
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Corresponding author

Correspondence to Hongying Man.

Additional information

The first author was supported by the NSFC Project 11271035, and in part by the NSFC Key Project 11031006. The second author was supported by the basic research foundation of Beijing Institute of Technology, under Grant 20121742004.

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Hu, J., Man, H. & Zhang, S. A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension. J Sci Comput 58, 367–379 (2014). https://doi.org/10.1007/s10915-013-9736-6

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  • DOI: https://doi.org/10.1007/s10915-013-9736-6

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