Partial Relaxation of 𝐶0 Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem

Jun Hu; Rui Ma

doi:10.1515/cmam-2020-0003

Article

Partial Relaxation of 𝐶⁰ Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem

Jun Hu and Rui Ma

Published/Copyright: May 27, 2020

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From the journal Computational Methods in Applied Mathematics Volume 21 Issue 1

Abstract

A conforming triangular mixed element recently proposed by Hu and Zhang for linear elasticity is extended by rearranging the global degrees of freedom. More precisely, adaptive meshes $𝒯_{1}, \dots, 𝒯_{N}$ $𝒯 1 , … , 𝒯 N$ which are successively refined from an initial mesh $𝒯_{0}$ $𝒯 0$ through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex $𝒙_{e}$ $𝒙 e$ of the mesh $𝒯_{ℓ}$ $𝒯 ℓ$ is the midpoint of an edge e of the coarse mesh $𝒯_{ℓ - 1}$ $𝒯 ℓ - 1$ . Such a hierarchical structure is explored to partially relax the $C^{0}$ $C 0$ vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on $𝒯_{ℓ}$ $𝒯 ℓ$ and results in an extended discrete stress space: for such an internal vertex $𝒙_{e}$ $𝒙 e$ located at the coarse edge e with the unit tangential vector $t_{e}$ $t e$ and the unit normal vector $n_{e} = t_{e}^{⊥}$ $n e = t e ⊥$ , the pure tangential component basis function $φ_{𝒙_{e}} (𝒙) t_{e} t_{e}^{T}$ $φ 𝒙 e ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T$ of the original discrete stress space associated to vertex $𝒙_{e}$ $𝒙 e$ is split into two basis functions $φ_{𝒙_{e}}^{+} (𝒙) t_{e} t_{e}^{T}$ $φ 𝒙 e + ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T$ and $φ_{𝒙_{e}}^{-} (𝒙) t_{e} t_{e}^{T}$ $φ 𝒙 e - ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T$ along edge e, where $φ_{𝒙_{e}} (𝒙)$ $φ 𝒙 e ⁢ ( 𝒙 )$ is the nodal basis function of the scalar-valued Lagrange element of order k (k is equal to the polynomial degree of the discrete stress) on $𝒯_{ℓ}$ $𝒯 ℓ$ with $φ_{𝒙_{e}}^{+} (𝒙)$ $φ 𝒙 e + ⁢ ( 𝒙 )$ and $φ_{𝒙_{e}}^{-} (𝒙)$ $φ 𝒙 e - ⁢ ( 𝒙 )$ denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions $φ_{𝒙_{e}} (𝒙) n_{e} n_{e}^{T}$ $φ 𝒙 e ⁢ ( 𝒙 ) ⁢ n e ⁢ n e T$ , $φ_{𝒙_{e}} (𝒙) (n_{e} t_{e}^{T} + t_{e} n_{e}^{T})$ $φ 𝒙 e ⁢ ( 𝒙 ) ⁢ ( n e ⁢ t e T + t e ⁢ n e T )$ are the same as those associated to $𝒙_{e}$ $𝒙 e$ of the original discrete stress space, the number of the global basis functions associated to $𝒙_{e}$ $𝒙 e$ of the extended discrete stress space becomes four rather than three (for the original discrete stress space). As a result, though the extended discrete stress space on $𝒯_{ℓ}$ $𝒯 ℓ$ is still a $H (div)$ $H ⁢ ( div )$ subspace, the pure tangential component along the coarse edge e of discrete stresses in it is not necessarily continuous at such vertices like $𝒙_{e}$ $𝒙 e$ . A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh $𝒯$ $𝒯$ is a subspace of a space on any refinement $\hat{𝒯}$ $𝒯 ^$ of $𝒯$ $𝒯$ , which allows a proof of convergence of a standard adaptive algorithm. The idea is extended to impose a general traction boundary condition on the discrete level. Numerical experiments are provided to illustrate performance on both uniform and adaptive meshes.

Keywords: Linear Elasticity; Nested Mixed Finite Element; Adaptive Algorithm

MSC 2010: 65N30; 74B05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11625101

Award Identifier / Grant number: 11421101

Funding statement: The authors were supported by NSFC projects 11625101 and 11421101.

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Received: 2020-01-10

Accepted: 2020-04-23

Published Online: 2020-05-27

Published in Print: 2021-01-01

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DOI: doi.org/10.1515/cmam-2020-0003

Keywords for this article

Linear Elasticity; Nested Mixed Finite Element; Adaptive Algorithm