New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

Given a nonnegative integer $k$ and a bounded domain ${\rm\Omega}\subset\mathbb{R}^{2}$ with boundary $\partial{\rm\Omega}$, let $W^{k,\infty}({\rm\Omega},\mathbb{R})$, $H^{k}({\rm\Omega},\mathbb{R})$, $\parallel\cdot\parallel_{k,{\rm\Omega}}$ and $|\cdot|_{k,{\rm\Omega}}$ denote the usual Sobolev spaces, norm, and semi-norm, respectively. The Sobolev spaces

$$H_{0}^{1}({\rm\Omega},\mathbb{R})=\{u\in H^{1}({\rm\Omega},\mathbb{R}):u|_{\partial{\rm\Omega}}=0\},\ H_{0}^{2}({\rm\Omega},\mathbb{R})=\{u\in H^{2}({\rm\Omega},\mathbb{R}):u|_{\partial{\rm\Omega}}={\partial u\over\partial\bm{n}}|_{\partial{\rm\Omega}}=0\}.$$

Denote the standard $L^{2}({\rm\Omega},\mathbb{R})$ inner product and $L^{2}(K,\mathbb{R})$ inner product by $(\cdot,\cdot)$ and $(\cdot,\cdot)_{0,K}$, respectively.

Suppose that ${\rm\Omega}\subset\mathbb{R}^{2}$ is a convex polygonal domain and the partition $\mathcal{T}_{h}$ of domain ${\rm\Omega}$ is assumed to be uniform in the sense that any two adjacent triangles form a parallelogram. Let $|K|$ denote the area of element $K$ and $|e|$ the length of edge $e$. Let $h_{K}$ denote the diameter of element $K\in\mathcal{T}_{h}$ and $h=\max_{K\in\mathcal{T}_{h}}h_{K}$. Denote the set of all interior edges and boundary edges of $\mathcal{T}_{h}$ by $\mathcal{E}_{h}^{i}$ and $\mathcal{E}_{h}^{b}$, respectively, and $\mathcal{E}_{h}=\mathcal{E}_{h}^{i}\cup\mathcal{E}_{h}^{b}$.

Let element $K$ have vertices $\mathbb{p}_{i}=(p_{i1},p_{i2}),1\leq i\leq 3$ oriented counterclockwise, and corresponding barycentric coordinates $\{\psi_{i}\}_{i=1}^{3}$. Let $M_{K}=(M_{1},M_{2})$ denote the centroid of the element, $\{e_{i}\}_{i=1}^{3}$ the edges of element $K$, $\{d_{i}\}_{i=1}^{3}$ the perpendicular heights, $\{\theta_{i}\}_{i=1}^{3}$ the internal angles, $\{\mathbb{m}_{i}\}_{i=1}^{3}$ the midpoints of edges $\{e_{i}\}_{i=1}^{3}$, and $\{\mathbb{n}_{i}\}_{i=1}^{3}$ the unit outward normal vectors, $\{\mathbb{t}_{i}\}_{i=1}^{3}$ the unit tangent vectors with counterclockwise orientation (see Fig. 1).

There hold the following relationships $d_{i}|e_{i}|=2|K|$ and

(2.1)

$$\begin{split}\nabla\psi_{i}=-\frac{\mathbb{n}_{i}}{d_{i}},\quad\sin\theta_{i}=\bm{n}_{i-1}\cdot\bm{t}_{i+1}=-\bm{n}_{i+1}\cdot\bm{t}_{i-1},\\ \cos\theta_{i}=-\bm{n}_{i-1}\cdot\bm{n}_{i+1}={|e_{i-1}|^{2}+|e_{i+1}|^{2}-|e_{i}|^{2}\over 2|e_{i-1}||e_{i+1}|}\end{split}$$

among the quantities [28]. For $K\subset\mathbb{R}^{2},\ r\in\mathbb{Z}^{+}$, let $P_{r}(K,\mathbb{R})$ be the space of all polynomials of degree not greater than $r$ on $K$. Denote the piecewise Hessian operator by $\nabla_{h}^{2}$. For any $1\leq i,j,k\leq 2$, denote the third order derivative $\frac{\partial^{3}v}{\partial x_{i}\partial x_{j}\partial x_{k}}$ by $\partial_{ijk}v$. For any $\alpha=(\alpha_{1},\alpha_{2})$, denote

$$|\alpha|=\alpha_{1}+\alpha_{2},\qquad\alpha!=\alpha_{1}!\alpha_{2}!,\qquad D^{\alpha}v=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}v.$$

For ease of presentation, the symbol $A\lesssim B$ will be used to denote that $A\leq CB$, where $C$ is a positive constant.

Consider the biharmonic eigenvalue problem for plate bending, which finds $\lambda$ and $\|u\|_{0,{\rm\Omega}}=1$ such that

(2.2)

$$\Delta^{2}u=\lambda u,\quad\text{ in }\Omega,$$

with the clamped boundary condition

(2.3)

$$u|_{\partial\Omega}={\partial u\over\partial\bm{n}}\big{|}_{\partial{\rm\Omega}}=0,$$

or the simply supported boundary condition

(2.4)

$$u|_{\partial\Omega}={\partial^{2}u\over\partial\bm{n}^{2}}\big{|}_{\partial{\rm\Omega}}=0.$$

The weak formulation for (2.2) is to find $(\lambda,u)\in\mathbb{R}\times V$ such that $\parallel u\parallel_{0,{\rm\Omega}}=1$ and

(2.5)

$\displaystyle a(u,v)=\lambda(u,v)\quad\forall\ v\in V,$

with $\displaystyle a(w,v)=\int_{{\rm\Omega}}\nabla^{2}w:\nabla^{2}v\,dx$ and

$$V=\begin{cases}H_{0}^{2}({\rm\Omega},\mathbb{R})&\mbox{ for clamped boundary plates with }\eqref{bc:cb},\\ H^{2}({\rm\Omega},\mathbb{R})\cap H_{0}^{1}({\rm\Omega},\mathbb{R})&\mbox{ for simply supported plates with }\eqref{bc:ssc}.\end{cases}$$

The bilinear form $a(\cdot,\cdot)$ is symmetric, bounded, and coercive, namely

$$a(w,v)=a(v,w),\quad|a(w,v)|\lesssim\parallel w\parallel_{2,{\rm\Omega}}\parallel v\parallel_{2,{\rm\Omega}},\quad\parallel v\parallel_{2,{\rm\Omega}}^{2}\lesssim a(v,v),\quad\forall w,v\in V.$$

The eigenvalue problem (2.5) has a sequence of eigenvalues

$$0<\lambda^{1}\leq\lambda^{2}\leq\lambda^{3}\leq...\nearrow+\infty,$$

and the corresponding eigenfunctions $u^{1},u^{2},u^{3},...,$ with $(u^{i},u^{j})=\delta_{ij}.$

The nonconforming Morley element space $V_{\rm M}$ over $\mathcal{T}_{h}$ is defined [44, 40] by

$$\begin{split}V_{\rm M}:=&\big{\{}v\in L^{2}({\rm\Omega},\mathbb{R})\big{|}v|_{K}\in P_{2}(K,\mathbb{R})\text{ for any }K\in\mathcal{T}_{h},v\ \text{is continuous at each interior}\\ &\text{ vertex and vanishes on each boundary vertex},\ \int_{e}\big{[}{\partial v\over\partial n}\big{]}\,ds=0\text{ for any }e\in\mathcal{E}_{h}^{i},\\ &\text{ and }\int_{e}{\partial v\over\partial n}\,ds=0\text{ for any }e\in\mathcal{E}_{h}^{b}\big{\}}\end{split}$$

if the clamped boundary condition (2.3) is imposed, and

$$\begin{split}V_{\rm M}:=&\big{\{}v\in L^{2}({\rm\Omega},\mathbb{R})\big{|}v|_{K}\in P_{2}(K,\mathbb{R})\text{ for any }K\in\mathcal{T}_{h},v\ \text{is continuous at each interior}\\ &\text{ vertex and vanishes on each boundary vertex},\ \int_{e}\big{[}{\partial v\over\partial n}\big{]}\,ds=0\text{ for any }e\in\mathcal{E}_{h}^{i}\big{\}}\end{split}$$

if the simply supported boundary condition (2.4) is imposed. The corresponding canonical interpolation operator $\Pi_{\rm M}:V\rightarrow V_{\rm M}$ is defined by

(2.6)

$$\int_{e}{\partial\Pi_{\rm M}v\over\partial n}\,ds={\int}_{e}{\partial v\over\partial n}\,ds,\forall e\in\mathcal{E}_{h},\quad\Pi_{\rm M}v(\bm{p})=v(\bm{p})\quad\text{ for any vertex }\bm{p}.$$

The corresponding finite element approximation of (2.5) is to find $(\lambda_{\rm M},u_{\rm M})\in\mathbb{R}\times V_{\rm M}$ such that $\parallel u_{\rm M}\parallel_{0,{\rm\Omega}}=1$ and

(2.7)

$\displaystyle a_{h}(u_{\rm M},v_{h})=\lambda_{\rm M}(u_{\rm M},v_{h})\quad\forall\ v_{h}\in V_{\rm M},$

with the discrete bilinear form $\displaystyle a_{h}(w_{h},v_{h}):=\sum_{K\in\mathcal{T}_{h}}\int_{K}\nabla_{h}^{2}w_{h}:\nabla_{h}^{2}v_{h}\,dx.$ Denote the approximate solution of (2.7) by $(\lambda_{\rm M}^{i},u_{\rm M}^{i})$ with $0<\lambda_{\rm M}^{1}\leq\lambda_{\rm M}^{2}\leq\cdots\nearrow\lambda_{\rm M}^{N_{V}},$ where $N_{V}={\rm dim}V_{\rm M}$ and $(u_{\rm M}^{i},u_{\rm M}^{j})=\delta_{ij}$, $1\leq i,\ j\leq N_{V}$.

Let $\lambda$ be an eigenvalue of Problem (2.5) with multiplicity $q$ and

$$M(\lambda)=\{w\in V:w\mbox{ is an eigenvector of Problem \eqref{variance} corresponding to}\ \lambda\}.$$

Without loss of generality, assume the index of $\lambda$ are $k_{0}+1,\cdots,k_{0}+q$, that is, $\lambda^{k_{0}}<\lambda=\lambda^{k_{0}+1}=\cdots=\lambda^{k_{0}+q}<\lambda^{k_{0}+q+1}.$ Denote

$\displaystyle M_{h}(\lambda)$

$\displaystyle={\rm span}\{u_{\rm M}^{k_{0}+1},\ u_{\rm M}^{k_{0}+2},\cdots,\ u_{\rm M}^{k_{0}+q}\}\subset V_{\rm M}.$

Suppose that $(\lambda_{\rm M},u_{\rm M})$ is the $i$-th eigenpair of Problem (2.7) by the Morley element, the theory of nonconforming eigenvalue approximations, see for instance, [3, 4, 6, 18, 42] and the references therein, indicates that there exists $u\in M(\lambda)$ with $\lambda=\lambda^{i}$ such that

(2.8)

$$\begin{split}h|\lambda-\lambda_{\rm M}|+h^{j}|u-\Pi_{\rm M}u|_{j,h}+h\parallel u-u_{\rm M}\parallel_{0,{\rm\Omega}}+h^{j}\parallel\nabla_{h}^{j}(u-u_{\rm M})\parallel_{0,{\rm\Omega}}&\lesssim h^{3}\parallel u\parallel_{3,{\rm\Omega}}\end{split}$$

provided that the domain is convex and $M(\lambda)\subset V\cap H^{3}({\rm\Omega})$, where $j=1$, $2$. Whenever there is no ambiguity, $(\lambda,u)$ defined this way is the called the corresponding eigenpair to $(\lambda_{\rm M},u_{\rm M})$ of Problem (2.7) if the estimate (2.8) holds.

For the Morley element, there holds the following commuting property [14, 18]

(2.9)

$$\int_{K}\nabla^{2}(w-\Pi_{\rm M}w):\nabla^{2}v_{h}\,dx=0\quad\forall\ w\in V,v_{h}\in V_{\rm M}.$$

This, together with the technique in [20, 21], guarantees the following expansion

(2.10)

$$\lambda-\lambda_{\rm M}=\|\nabla_{h}^{2}(u-u_{\rm M})\|_{0,{\rm\Omega}}^{2}-2\lambda(u-\Pi_{\rm M}u,u)+\mathcal{O}(h^{4}|u|_{3,{\rm\Omega}}^{2}).$$

The asymptotic expansion in this paper is based on this crucial identity (2.10).

For any source term $f$, the plate bending problem seeks $u^{f}\in V$ such that

(2.11)

$\displaystyle\Delta^{2}u^{f}=f$

with boundary condition (2.3) or (2.4). Define

$$D=\{v\in H^{1}_{0}({\rm\Omega},\mathbb{R}):v|_{K}\in H^{2}(K,\mathbb{R})\}$$

and the space for an auxiliary variable $\sigma^{f}:=\nabla^{2}u^{f}$ by

$$S=\{\tau\in L^{2}({\rm\Omega},\mathcal{S}):\tau|_{K}\in H^{1}(K,\mathcal{S}),\text{ and }M_{\bm{n}\bm{n}}(\tau)\text{ is continuous across interior edges}\}$$

with $\mathcal{S}:=\text{symmetric }\mathbb{R}^{2\times 2}$ if the clamped boundary condition (2.3) is imposed and

$$\begin{split}S=\{\tau\in L^{2}({\rm\Omega},\mathcal{S}):&\ \tau|_{K}\in H^{1}(K,\mathcal{S}),\text{ and }M_{\bm{n}\bm{n}}(\tau)\text{ is continuous across interior edges,}\\ &\ M_{\bm{n}\bm{n}}(\tau)=0\text{ on boundary edges}\}\end{split}$$

if the simply supported boundary condition (2.4) is imposed.

Given $K\in\mathcal{T}_{h}$ and $\tau\in H^{1}(K,\mathcal{S})$, let

$$M_{\bm{n}\bm{n}}(\tau)=\bm{n}^{T}\tau\bm{n},\quad M_{\bm{n}\bm{t}}(\tau)=\bm{t}^{T}\tau\bm{n}$$

with the unit outnormal $\bm{n}$ and unit tangential direction $\bm{t}$ with counterclockwise orientation of $\partial K$. Since $\sigma^{f}\bm{n}:\nabla v=M_{\bm{n}\bm{n}}(\sigma^{f})\frac{\partial v}{\partial\bm{n}}+M_{\bm{n}\bm{t}}(\sigma^{f})\frac{\partial v}{\partial\bm{t}}$, the integration by parts gives

(2.12)

$$(f,v)=(\sigma^{f},\nabla^{2}v)+\sum_{K\in\mathcal{T}_{h}}\int_{\partial K}(\nabla\cdot\sigma^{f})\cdot\bm{n}v-M_{\bm{n}\bm{n}}(\sigma^{f}){\partial v\over\partial\bm{n}}-M_{\bm{n}\bm{t}}(\sigma^{f})\frac{\partial v}{\partial\bm{t}}\,ds.$$

Note that $v\in D$ is continuous on interior edges and zero on the boundary $\partial\Omega$, so is the tangential derivative of $v$. Thus,

(2.13)

$$(f,v)=(\sigma^{f},\nabla^{2}v)-\sum_{K\in\mathcal{T}_{h}}\int_{\partial K}M_{\bm{n}\bm{n}}(\sigma^{f}){\partial v\over\partial\bm{n}}\,ds.$$

The mixed formulation of source problem (2.11), which was analyzed in [30], seeks $(\sigma^{f},u^{f})\in S\times D$ such that

(2.14)		$\displaystyle(\sigma^{f},\tau)+\sum_{K\in\mathcal{T}_{h}}-(\tau,\nabla^{2}u^{f})_{0,K}+\int_{\partial K}M_{\bm{n}\bm{n}}(\tau){\partial u^{f}\over\partial\bm{n}}\,ds$	$\displaystyle=0,\$	$\displaystyle\forall\tau\in S,$
		$\displaystyle\sum_{K\in\mathcal{T}_{h}}-(\sigma^{f},\nabla^{2}v)_{0,K}+\int_{\partial K}M_{\bm{n}\bm{n}}(\sigma^{f}){\partial v\over\partial\bm{n}}\,ds$	$\displaystyle=-(f,v),\$	$\displaystyle\forall v\in D.$

Define the discrete spaces [2, 10]

$$U(\mathcal{T}_{h}):=\big{\{}v\in D:v|_{K}\in P_{1}(K,\mathbb{R})\text{ for any }K\in\mathcal{T}_{h}\big{\}},$$

$$\Sigma(\mathcal{T}_{h}):=\big{\{}\tau\in S:\tau|_{K}\in P_{0}(K,\mathcal{S})\text{ for any }K\in\mathcal{T}_{h}\big{\}}.$$

The first order HHJ element [10, 30] of Problem (2.14) seeks $(\sigma_{\rm HHJ}^{f},u_{\rm HHJ}^{f})\in\Sigma(\mathcal{T}_{h})\times U(\mathcal{T}_{h})$ such that

(2.15)		$\displaystyle(\sigma_{\rm HHJ}^{f},\tau_{h})+\sum_{e\in\mathcal{E}_{h}}\int_{e}M_{\bm{n}\bm{n}}(\tau_{h})[{\partial u_{\rm HHJ}^{f}\over\partial\bm{n}}]\,ds$	$\displaystyle=0,\$	$\displaystyle\forall\tau_{h}\in\Sigma(\mathcal{T}_{h}),$
		$\displaystyle\sum_{e\in\mathcal{E}_{h}}\int_{e}M_{\bm{n}\bm{n}}(\sigma_{\rm HHJ}^{f})[{\partial v_{h}\over\partial\bm{n}}]\,ds$	$\displaystyle=-(f,v_{h}),\$	$\displaystyle\forall v_{h}\in U(\mathcal{T}_{h}).$

It follows from the theory of mixed finite element methods [13] that

(2.16)

$\displaystyle\parallel u^{f}-u_{\rm HHJ}^{f}\parallel_{0,{\rm\Omega}}+h\parallel\sigma^{f}-\sigma_{\rm HHJ}^{f}\parallel_{0,{\rm\Omega}}+h|u^{f}-u_{\rm HHJ}^{f}|_{1,{\rm\Omega}}\lesssim h^{2}\parallel u^{f}\parallel_{3,{\rm\Omega}},$

provided that $u^{f}\in V\cap H^{3}({\rm\Omega},\mathbb{R})$. In this paper, we consider two different source terms $f=\lambda u$ and $f=\lambda_{\rm M}u_{\rm M}$. Let $(\sigma_{\rm HHJ}^{\lambda u},u_{\rm HHJ}^{\lambda u})$ and $(\sigma_{\rm HHJ},u_{\rm HHJ})\in\Sigma(\mathcal{T}_{h})\times U(\mathcal{T}_{h})$ be the solutions of Problem (2.15) with source terms $f=\lambda u$ and $f=\lambda_{\rm M}u_{\rm M}$, respectively. Then,

(2.17)

$$\sigma_{\rm HHJ}=\sigma_{\rm HHJ}^{\lambda_{\rm M}u_{\rm M}},\quad u_{\rm HHJ}=u_{\rm HHJ}^{\lambda_{\rm M}u_{\rm M}}.$$

In the rest of this paper, denote $\sigma=\nabla^{2}u$ where $u$ is the eigenfunction of Problem (2.2).

Define the interpolation operator $\Pi_{\rm HHJ}:S\rightarrow\Sigma(\mathcal{T}_{h})$ by

(2.18)

$$\int_{e}M_{\bm{n}\bm{n}}(\Pi_{\rm HHJ}\tau)\,ds=\int_{e}M_{\bm{n}\bm{n}}(\tau)\,ds\text{\quad for any }e\in\mathcal{E}_{h},\mathbb{\tau}\in S.$$

There exists the following identity in [23] that

(2.19)

$$(\sigma_{\rm HHJ}^{\lambda u}-\Pi_{\rm HHJ}\sigma,\sigma_{\rm HHJ}^{\lambda u}-\sigma)=0.$$

For the plate bending problem with the clamped boundary condition (2.3), it was analyzed in [22] that the HHJ element admits an important superconvergence property on uniform triangulations as presented below. For Problem (2.2) with the simply supported condition (2.4), $M_{\bm{n}\bm{n}}(\sigma_{\rm HHJ}^{\lambda u}-\Pi_{\rm HHJ}\sigma)=0$ for all edges $e\in\mathcal{E}_{h}^{b}$. A simple extension of the analysis in [24] proves the superconvergence property of the HHJ element when (2.4) is imposed.

Let $\Pi_{\rm D}v$ be the linear interpolation of any function $v\in V+V_{\rm M}$. Given a function $f\in L^{2}(\Omega,\mathbb{R})$. Let $u_{\rm M}^{f}\in V_{\rm M}$ be the Morley solution of the source problem

(2.21)

$$a_{M}(u_{\rm M}^{f},v_{h})=(f,v_{h}),\quad\forall v_{h}\in V_{\rm M},$$

and $\widetilde{u}_{\rm M}^{f}\in V_{\rm M}$ be the modified Morley solution of the source problem

(2.22)

$$a_{M}(\widetilde{u}_{\rm M}^{f},v_{h})=(f,\Pi_{\rm D}v_{h}),\quad\forall v_{h}\in V_{\rm M}.$$

As analyzed in [1], the HHJ solution of source problem (2.15) and the modified Morley solution of source problem (2.22) are equivalent in the following sense

(2.23)

$$\sigma_{\rm HHJ}^{f}=\nabla_{h}^{2}\widetilde{u}_{\rm M}^{f},\quad u_{\rm HHJ}^{f}=\Pi_{\rm D}\widetilde{u}_{\rm M}^{f}.$$

This equivalence leads to the following lemma.

The classic asymptotic analysis does not work for the Morley element because of the lack of a crucial superclose property. Inspired by [21], we use the equivalence between the mixed HHJ element and the modified Morley element in Lemma 2.2 and the superconvergence property of the mixed HHJ element in Lemma 2.1 to establish an asymptotic expansion of eigenvalues by the Morley element.

To begin with, we list the discrete problems and the corresponding solutions considered in this paper:

1. (P1)

   eigenvalue problem (2.7) by the Morley element: $u_{\rm M}$;

2. (P2)

   source problem (2.21) by the Morley element: $u_{\rm M}^{\lambda u}$;

3. (P3)

   source problem (2.22) by the modified Morley element: $\widetilde{u}_{\rm M}^{\lambda u}$ and $\widetilde{u}_{\rm M}$;

4. (P4)

   source problem (2.15) by the HHJ element: $(\sigma_{\rm HHJ}^{\lambda u},u_{\rm HHJ}^{\lambda u})$, $(\sigma_{\rm HHJ},u_{\rm HHJ})$.

The Morley solution $u_{\rm M}$ in Problem (P1) is also a solution of Problem (P2) with $f=\lambda_{\rm M}u_{\rm M}$. The following Lemma 3.1 analyzes some superclose property of the Morley element and the mixed HHJ element, including the relation between Problems (P1) and (P2), and that between Problems (P2) and (P3). Lemma 2.2 presents the equivalence between the solutions of Problems (P3) and (P4). Thus, the superconvergence in Lemma 2.1 of the HHJ element for Problem (P4) can be used to analyze the expansion of eigenvalues of Problem (P1).

For simplicity of presentation, we introduce the following notations

(3.4)

$$\begin{array}[]{llll}I_{1}&=(\sigma-\sigma_{\rm HHJ}^{\lambda u},\sigma_{\rm HHJ}^{\lambda u}-\sigma_{\rm HHJ}),&I_{2}&=(\sigma-\sigma_{\rm HHJ},\nabla_{h}^{2}(\widetilde{u}_{\rm M}-u_{\rm M})),\\ I_{3}&=\lambda(u-\Pi_{\rm M}u,u).&&\end{array}$$

The asymptotic expansion of eigenvalues by the Morley element is based on the decomposition in the following theorem.

In the rest of this section, we will conduct an asymptotic analysis of each term on the right-hand side of the expansion (3.5) in the above theorem.

Define the three basis functions for the HHJ element

$\displaystyle\phi_{\rm HHJ}^{i}(\bm{x})=-{1\over 2\sin\theta_{i-1}\sin\theta_{i+1}}\big{(}\mathbb{t}_{i-1}\mathbb{t}_{i+1}^{T}+\mathbb{t}_{i+1}\mathbb{t}_{i-1}^{T}\big{)}\in P_{0}(K,\mathcal{S}),\qquad 1\leq i\leq 3.$

By (2.1), it is easy to verify that

(3.10)

$\displaystyle{1\over|e_{j}|}\int_{e_{j}}\bm{n}_{j}^{T}\phi_{\rm HHJ}^{i}\bm{n}_{j}\,ds=\delta_{ij}.$

Define four short-hand notations for the HHJ element

(3.11)

$$\begin{split}\phi_{1}(\bm{x})&={1\over 6|K|^{1/2}}(x_{1}-M_{1})^{3},\quad\phi_{2}(\bm{x})={1\over 2|K|^{1/2}}(x_{1}-M_{1})^{2}(x_{2}-M_{2})\\ \phi_{3}(\bm{x})&={1\over 2|K|^{1/2}}(x_{1}-M_{1})(x_{2}-M_{2})^{2},\quad\phi_{4}(\bm{x})={1\over 6|K|^{1/2}}(x_{2}-M_{2})^{3}.\end{split}$$

Note that $\{\phi_{i}\}_{i=1}^{4}$ are linear independent and

(3.12)

$$P_{3}(K,\mathbb{R})=P_{2}(K,\mathbb{R})\cup{\rm span}\{\phi_{i}:1\leq i\leq 4\},$$

(3.13)

$$\|\partial_{111}\phi_{i}\|_{0,K}=\delta_{1i},\ \|\partial_{112}\phi_{i}\|_{0,K}=\delta_{2i},\ \|\partial_{122}\phi_{i}\|_{0,K}=\delta_{3i},\ \|\partial_{222}\phi_{i}\|_{0,K}=\delta_{4i}.$$

Define

(3.14)

$$\gamma_{\rm HHJ}^{ij}=\frac{1}{|K|}\int_{K}\big{(}(I-\Pi_{\rm HHJ})\nabla^{2}\phi_{i}\big{)}^{T}(I-\Pi_{\rm HHJ})\nabla^{2}\phi_{j}\,dx,\quad 1\leq i,j\leq 4.$$