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Lower order mixed elements for the linear elasticity problem in 2D and 3D

Jun Hu, Rui Ma, Yuanxun Sun

Abstract

In this paper, we construct two lower order mixed elements for the linear elasticity problem in the Hellinger-Reissner formulation, one for the 2D problem and one for the 3D problem, both on macro-element meshes. The discrete stress spaces enrich the analogous $P_k$ stress spaces in [J. Hu and S. Zhang, arxiv, 2014, J. Hu and S. Zhang, Sci. China Math., 2015] with simple macro-element bubble functions, and the discrete displacement spaces are discontinuous piecewise $P_{k-1}$ polynomial spaces, with $k=2,3$, respectively. Discrete stability and optimal convergence is proved by using the macro-element technique. As a byproduct, the discrete stability and optimal convergence of the $P_2-P_1$ mixed element in [L. Chen and X. Huang, SIAM J. Numer. Anal., 2022] in 3D is proved on another macro-element mesh. For the mixed element in 2D, an $H^2$-conforming composite element is constructed and an exact discrete elasticity sequence is presented. Numerical experiments confirm the theoretical results.

Lower order mixed elements for the linear elasticity problem in 2D and 3D

Abstract

In this paper, we construct two lower order mixed elements for the linear elasticity problem in the Hellinger-Reissner formulation, one for the 2D problem and one for the 3D problem, both on macro-element meshes. The discrete stress spaces enrich the analogous stress spaces in [J. Hu and S. Zhang, arxiv, 2014, J. Hu and S. Zhang, Sci. China Math., 2015] with simple macro-element bubble functions, and the discrete displacement spaces are discontinuous piecewise polynomial spaces, with , respectively. Discrete stability and optimal convergence is proved by using the macro-element technique. As a byproduct, the discrete stability and optimal convergence of the mixed element in [L. Chen and X. Huang, SIAM J. Numer. Anal., 2022] in 3D is proved on another macro-element mesh. For the mixed element in 2D, an -conforming composite element is constructed and an exact discrete elasticity sequence is presented. Numerical experiments confirm the theoretical results.

Paper Structure

This paper contains 28 sections, 21 theorems, 99 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Some notation
  4. Piola transform
  5. The mixed finite element spaces
  6. Discrete stability
  7. Lower order elements for the 2D problem and the 3D problem
  8. The mixed element of degree $k=2$ for the 2D problem
  9. The definition of a macro-element
  10. The macro-element bubble functions
  11. Discrete stability
  12. Error estimates
  13. The mixed element of degree $k=3$ for the 3D problem
  14. The definition of a macro-element
  15. The macro-element bubble functions
  16. ...and 13 more sections

Key Result

Lemma 3.2

There exists an interpolation operator $I_h:H^1(\Omega;\mathbb{S})\rightarrow\Sigma_{2,h}\cap H^1(\Omega;\mathbb{S})$ such that for any macro-element edge $E$, Here $\bm{n}_E$ denotes the unit normal vector of the edge $E$, $M^{+}$ and $M^-$ denote the two macro-elements sharing $E$. Besides, it holds

Figures (9)

  • Figure 1: The macro-element for the mixed element of degree $k=2$ in 2D.
  • Figure 2: The macro-element for the mixed element of degree $k=3$ in 3D.
  • Figure 3: Interior Lagrange nodes of the macro-element face.
  • Figure 4: The macro-element for the mixed element of degree $k=2$ in 3D.
  • Figure 5: Interior Lagrange nodes of the macro-element face.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 27 more