Partial Relaxation of 𝐶0 Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem
Authors: Hu Jun , Ma Rui 
Published: May 2020

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

,
…
,

𝒯
N

{\mathcal{T}_{1},\ldots,\mathcal{T}_{N}}

which are successively refined from an initial mesh

𝒯
0

{\mathcal{T}_{0}}

through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex

𝒙
e

{\boldsymbol{x}_{e}}

of the mesh

𝒯
ℓ

{\mathcal{T}_{\ell}}

is the midpoint of an edge e of the coarse mesh

𝒯

ℓ
-
1

{\mathcal{T}_{\ell-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

{\mathcal{T}_{\ell}}

and results in an extended discrete stress space: for such an internal vertex

𝒙
e

{\boldsymbol{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}^{\perp}}

, the pure tangential component basis function

φ

𝒙
e

⁢

(
𝒙
)

⁢

t
e

⁢

t
e
T

{\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}}

of the original discrete stress space associated to vertex

𝒙
e

{\boldsymbol{x}_{e}}

is split into two basis functions

φ

𝒙
e

+

⁢

(
𝒙
)

⁢

t
e

⁢

t
e
T

{\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}}

and

φ

𝒙
e

-

⁢

(
𝒙
)

⁢

t
e

⁢

t
e
T

{\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}}

along edge e, where

φ

𝒙
e

⁢

(
𝒙
)

{\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{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

{\mathcal{T}_{\ell}}

with

φ

𝒙
e

+

⁢

(
𝒙
)

{\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})}

and

φ

𝒙
e

-

⁢

(
𝒙
)

{\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{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

{\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}}

,

φ

𝒙
e

⁢

(
𝒙
)

⁢

(

n
e

⁢

t
e
T

+

t
e

⁢

n
e
T

)

{\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})(n_{e}t_{e}^{T}+t_{e}n_{e}^{T})}

are the same as those associated to

𝒙
e

{\boldsymbol{x}_{e}}

of the original discrete stress space, the number of the global basis functions associated to

𝒙
e

{\boldsymbol{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

{\mathcal{T}_{\ell}}

is still a

H
⁢

(
div
)

{H(\operatorname{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

{\boldsymbol{x}_{e}}

.
A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh

𝒯

{\mathcal{T}}

is a subspace of a space on any refinement

𝒯
^

{\hat{\mathcal{T}}}

of

𝒯

{\mathcal{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.