A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

Abstract

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.

Appendix: \(L^2\) Projection and Some Technical Results

In this section, we shall present some technical results for the $L^2$ projection operators with respect to the finite element space $V_h$. These results are useful for the error estimates of the WG finite element method.

Lemma 7.1

([39])  (Trace Inequality) Let $\mathcal {T}_h$ be a partition of the domain $\Omega$ into polygons in 2D or polyhedra in 3D. Assume that the partition $\mathcal {T}_h$ satisfies the assumptions (A1), (A2), and (A3) as specified in [39]. Then, there exists a constant $C$ such that for any $T\in \mathcal {T}_h$ and edge/face $e\in \partial T$, we have

$$\begin{aligned} \Vert \theta \Vert ^p_{e}\le Ch_T^{-1}(\Vert \theta \Vert ^p_{T}+h^p_T\Vert \nabla \theta \Vert ^p_{T}), \end{aligned}$$

(7.1)

where $\theta \in H^{1}(T)$ is any function.

Lemma 7.2

([39]) (Inverse Inequality) Let $\mathcal {T}_h$ be a partition of the domain $\Omega$ into polygons or polyhedra. Assume that $\mathcal {T}_h$ satisfies all the assumptions (A1)–(A4) as specified in [39]. Then, there exists a constant $C(n)$ such that

$$\begin{aligned} \Vert \nabla \varphi \Vert _{T}\le C(n)h^{-1}_T\Vert \varphi \Vert _{T},\quad \forall T\in \mathcal {T}_h \end{aligned}$$

(7.2)

for any piecewise polynomial $\varphi$ of degree $n$ on $\mathcal {T}_h$.

1.1 Approximation Properties

The following lemma provides some approximation properties for the projection operators $Q_h$ and $\mathbb {Q}_h$.

Lemma 7.3

([32]) Let $\mathcal {T}_h$ be a finite element partition of $\Omega$ satisfying the shape regularity assumptions. Then, for any $0\le s\le 2$ and $2\le m\le k$ we have

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h}h^{2s}_T\Vert u-Q_0u\Vert ^2_{s,T}\le Ch^{2(m+1)}\Vert u\Vert ^2_{m+1},\end{aligned}$$

(7.3)

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h}h^{2s}_T\Vert \Delta u-\mathbb {Q}_h\Delta u\Vert ^2_{s,T}\le Ch^{2(m-1)}\Vert u\Vert ^2_{m+1}. \end{aligned}$$

(7.4)

Lemma 7.4

Let $2 \le m\le k, \omega \in H^{m+2}(\Omega )$. There exists a constant $C$ such that the following estimates hold true:

$$\begin{aligned} \left( \sum _{T\in \mathcal {T}_h} h_T\Vert \Delta \omega -\mathbb {Q}_h\Delta \omega \Vert ^2_{\partial T}\right) ^{\frac{1}{2}}&\le Ch^{m-1}\Vert \omega \Vert _{m+1},\end{aligned}$$

(7.5)

$$\begin{aligned} \left( \sum _{T\in \mathcal {T}_h}h^3_T\Vert \nabla (\Delta \omega -\mathbb {Q}_h\Delta \omega )\Vert ^2_{\partial T}\right) ^{\frac{1}{2}}&\le Ch^{m-1}(\Vert \omega \Vert _{m+1}+h\delta _{m,2}\Vert \omega \Vert _4),\end{aligned}$$

(7.6)

$$\begin{aligned} \left( \sum _{T\in \mathcal {T}_h}h^{-1}_T\Vert \nabla (Q_0\omega )\cdot \mathbf n _e -Q_b(\nabla \omega \cdot \mathbf n _e)\Vert ^2_{\partial T}\right) ^{\frac{1}{2}}&\le Ch^{m-1}\Vert \omega \Vert _{m+1},\end{aligned}$$

(7.7)

$$\begin{aligned} \left( \sum _{T\in \mathcal {T}_h}h^{-3}_T\Vert Q_b Q_0\omega -Q_b\omega \Vert ^2_{\partial T}\right) ^{\frac{1}{2}}&\le Ch^{m-1}\Vert \omega \Vert _{m+1}, \end{aligned}$$

(7.8)

$$\begin{aligned} \left( \sum _{T\in \mathcal {T}_h} \Vert \nabla (\Delta \omega )-Q_b(\nabla (\Delta \omega ))\Vert ^2_{\partial T}\right) ^{\frac{1}{2}}&\le Ch^{m-\frac{3}{2}}\Vert \omega \Vert _{m+2}. \end{aligned}$$

(7.9)

Here $\delta _{i,j}$ is the usual Kronecker’s delta with value $1$ when $i=j$ and value 0 otherwise.

Proof

To derive (7.5), we use the trace inequality (7.1) and the estimate (7.4) to obtain

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h} h_T\Vert \Delta \omega -\mathbb {Q}_h\Delta \omega \Vert ^2_{\partial T}\nonumber \\&\quad \le C\sum _{T\in \mathcal {T}_h}\Big (\Vert \Delta \omega -\mathbb {Q}_h\Delta \omega \Vert ^2_T+h^2_T\Vert \nabla \left( \Delta \omega -\mathbb {Q}_h\Delta \omega \right) \Vert ^2_T\Big )\\&\quad \le Ch^{2m-2}\Vert \omega \Vert ^2_{m+1}. \end{aligned}$$

As to (7.6), we use the trace inequality (7.1) and the estimate (7.4) to obtain

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h} h^3_T\Vert \nabla \left( \Delta \omega -\mathbb {Q}_h\Delta \omega \right) \Vert ^2_{\partial T}\\&\quad \le C\sum _{T\in \mathcal {T}_h}\left( h^2_T\Vert \nabla (\Delta \omega -\mathbb {Q}_h\Delta \omega )\Vert ^2_T+h^4_T\Vert \nabla ^2(\Delta \omega -\mathbb {Q}_h\Delta \omega )\Vert ^2_T\right) \\&\quad \le Ch^{2m-2}\Big (\Vert \omega \Vert ^2_{m+1}+h^2\delta _{m,2}\Vert \omega \Vert ^2_4\Big ). \end{aligned}$$

As to (7.7), we have from the definition of $Q_b$, the trace inequality (7.1), and the estimate (7.3) that

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h} h^{-1}_T\Vert \nabla (Q_0\omega )\cdot \mathbf n _e-Q_b(\nabla \omega \cdot \mathbf n _e)\Vert ^2_{\partial T}\\&\quad \le \sum _{T\in \mathcal {T}_h}h^{-1}_T\Vert (\nabla Q_0\omega -\nabla \omega )\cdot \mathbf n _e\Vert ^2_{\partial T}\\&\quad \le C\sum _{T\in \mathcal {T}_h}\left( h^{-2}_T\Vert \nabla Q_0\omega -\nabla \omega \Vert ^2_T+\Vert \nabla Q_0\omega -\nabla \omega \Vert ^2_{1,T}\right) \\&\quad \le Ch^{2m-2}\Vert \omega \Vert ^2_{m+1}. \end{aligned}$$

Notice that $Q_b$ is a linear bounded operator, we use the definition of $Q_b$ and the trace inequality (7.1) to obtain

$$\begin{aligned}&\sum _{T\in T_h}h_T^{-3}\Vert Q_b Q_0\omega -Q_b \omega \Vert _{\partial T}^2 \\&\quad \le \sum _{T\in T_h}\left( h_T^{-4}\Vert Q_0\omega -\omega \Vert _{T}^2+h_T^{-2}\Vert \nabla (Q_0\omega -\omega )\Vert _T^2\right) \\&\quad \le Ch^{2m-2}\Vert \omega \Vert _{m+1}^2. \end{aligned}$$

To derive (7.9), we use the trace inequality (7.1) and the estimate (7.4) to obtain

$$\begin{aligned}&\sum _{T\in \mathcal {T}_h} \Vert \nabla (\Delta \omega )-Q_b(\nabla (\Delta \omega ))\Vert ^2_{\partial T}\\&\quad \le C\sum _{T\in \mathcal {T}_h}\left( h^{-1}_T\Vert \nabla (\Delta \omega )-Q_b(\nabla (\Delta \omega ))\Vert ^2_T +h_T\Vert \nabla (\nabla (\Delta \omega )-Q_b(\nabla (\Delta \omega )))\Vert ^2_T\right) \\&\quad \le Ch^{2m-3}\Vert \omega \Vert ^2_{m+2}. \end{aligned}$$

This completes the proof of (7.9), and hence the lemma. $\square$

1.2 Technical Inequalities

The goal here is to present some technical estimates useful for deriving error estimates for the WG finite element scheme (3.5).

Lemma 7.5

There exists a constant $C$ such that, for any $v=\{v_0, v_b,v_n\mathbf n _e\}\in V_h$, the following holds true

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\Vert \Delta v_0\Vert ^2_T\le C|\!|\!| v |\!|\!|^2. \end{aligned}$$

(7.10)

Proof

From the identity (2.4) with $\phi =\Delta v_0$ we have

$$\begin{aligned} \Vert \Delta v_0\Vert ^2_T&= (\Delta _w v, \Delta v_0)_T-\langle Q_b v_0-v_b, \nabla (\Delta v_0)\cdot \mathbf{n} \rangle _{\partial T} +\langle (\nabla v_0-v_n\mathbf{n}_e)\cdot \mathbf{n}, \Delta v_0 \rangle _{\partial T}. \end{aligned}$$

Thus, using the Cauchy–Schwarz inequality, trace inequality, and the inverse inequality we obtain

$$\begin{aligned} \Vert \Delta v_0\Vert ^2_T&\le \Vert \Delta _w v\Vert _T\Vert \Delta v_0\Vert _T+\Vert Q_bv_0-v_b\Vert _{\partial T}\Vert \nabla (\Delta v_0)\cdot \mathbf{n}\Vert _{\partial T}\\&+\,\Vert (\nabla v_0-v_n\mathbf{n}_e)\cdot \mathbf{n}\Vert _{\partial T}\Vert \Delta v_0\Vert _{\partial T}\\&\le C (\Vert \Delta _w v\Vert _T\Vert \Delta v_0\Vert _T+h_T^{-\frac{1}{2}}\Vert Q_bv_0-v_b\Vert _{\partial T}\Vert \nabla (\Delta v_0)\cdot \mathbf{n}\Vert _{T}\\&+\,h_T^{-\frac{1}{2}}\Vert (\nabla v_0-v_n\mathbf{n}_e)\cdot \mathbf{n}\Vert _{\partial T}\Vert \Delta v_0\Vert _{T}) \\&\le C(\Vert \Delta _w v\Vert _T\Vert \Delta v_0\Vert _T+h_T^{-\frac{3}{2}}\Vert Q_bv_0-v_b\Vert _{\partial T}\Vert \Delta v_0\Vert _{T}\\&+\,h_T^{-\frac{1}{2}}\Vert (\nabla v_0-v_n\mathbf{n}_e)\cdot \mathbf{n}\Vert _{\partial T}\Vert \Delta v_0\Vert _{T}).\\ \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \Delta v_0\Vert ^2_T\le C(\Vert \Delta _w v\Vert _T^2+h_T^{-3}\Vert Q_bv_0-v_b\Vert _{\partial T}^2 +h_T^{-1}\Vert (\nabla v_0-v_n\mathbf{n}_e)\cdot \mathbf{n}\Vert _{\partial T}^2), \end{aligned}$$

which verifies the inequality (7.10). $\square$

Lemma 7.6

([37], Lemma 10.4) There exists a constant $C$ such that, for any $v\in V_h^0$, we have the following Poincaré inequality:

$$\begin{aligned} \Vert v_0\Vert ^2 \le C \left( \sum _{T\in {\mathcal {T}}_h}\Vert \nabla v_0\Vert _T^2 +h^{-1} \sum _{T\in \mathcal {T}_h}\Vert Q_bv_0-v_b\Vert ^2_{\partial T}\right) \!. \end{aligned}$$

(7.11)

The following lemma provides an estimate for the term $\sum _{T\in {\mathcal {T}}_h}\Vert \nabla v_0\Vert _T^2$. Note that $v_0$ is a piecewise polynomial of degree $k\ge 2$. Thus, Lemma 7.7 is concerned only with piecewise polynomials; no boundary condition is necessary.

Lemma 7.7

Let $\varphi$ be any piecewise polynomial of degree $k\ge 2$ on each element $T$. Denote by $\nabla _h\varphi$ and $\Delta _h \varphi$ the gradient and Laplacian of $\varphi$ taken on each element. Then, for any $\varepsilon >0$, there exists a constant $C$ such that

$$\begin{aligned} \nonumber \Vert \nabla _h \varphi \Vert ^2&\le \varepsilon \Vert \varphi \Vert ^2 + C \varepsilon ^{-1} \Vert \Delta _h \varphi \Vert ^2\\&+\, C \varepsilon ^{-1}h^{-1} \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L} + \frac{\partial \varphi _R}{\partial \mathbf n _R}\right) ^2 ds\right) \\&\nonumber + \,Ch^{-1}\left( \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b \varphi _R-Q_b \varphi _L)^2 ds\right) . \end{aligned}$$

(7.12)

Here $\varphi _L$ is the trace of $\varphi$ on $e$ as seen from the “left” or the opposite direction of ${\mathbf {n}}_e$. If $e$ is a boundary edge, then the trace from the outside of $\Omega$ is defined as zero.

Proof

On each element $T$, we have

$$\begin{aligned} \int _T |\nabla \varphi |^2 dT&= - \int _T \varphi \Delta \varphi \ dT +\int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n }\, \varphi \, ds \\&= - \int _T \varphi \Delta \varphi dT +\int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n }\, Q_b \varphi \, ds. \end{aligned}$$

Summing over all $T\in \mathcal {T}_h$, we have

$$\begin{aligned} \Vert \nabla _h \varphi \Vert ^2 = - \int _\Omega \varphi \Delta _h \varphi dT +\sum _{T\in \mathcal {T}_h} \int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n } \, Q_b \varphi \, ds. \end{aligned}$$

(7.13)

Using the identity $a_L b_L+a_R b_R=(a_L+a_R)b_{L}+a_R(b_R-b_L)$ we obtain

$$\begin{aligned} \sum _{T\in \mathcal {T}_h} \int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n } \, Q_b \varphi \, ds&= \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L} \, Q_b \varphi _L+ \frac{\partial \varphi _R}{\partial \mathbf n _R} \, Q_b \varphi _R\, \right) ds \\&= \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L}+ \frac{\partial \varphi _R}{\partial \mathbf n _R}\right) \, Q_b \varphi _L ds \\&+ \sum _{e\in \mathcal {E}_h}\int _{e} \frac{\partial \varphi _R}{\partial \mathbf n _R}\, (Q_b \varphi _R-Q_b \varphi _L) ds. \end{aligned}$$

Thus, from the Cauchy–Schwarz inequality we have

$$\begin{aligned} \left| \sum _{T\in \mathcal {T}_h} \int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n } \, Q_b \varphi \, ds\right|&\le \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L} + \frac{\partial \varphi _R}{\partial \mathbf n _R}\right) ^2 ds\right) ^{\!\!\frac{1}{2}} \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left| Q_b \varphi _L \right| ^2 ds\right) ^{\!\!\frac{1}{2}} \nonumber \\&+ \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left| \frac{\partial \varphi _R}{\partial \mathbf n _R}\right| ^2 ds\right) ^{\!\!\frac{1}{2}} \left( \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b \varphi _R-Q_b \varphi _L)^2 ds\right) ^{\!\!\frac{1}{2}}.\nonumber \\ \end{aligned}$$

(7.14)

Next, we use the trace inequality (7.1) and the inverse inequality (7.2) to obtain

$$\begin{aligned} \int _{e}\left| Q_b \varphi _L \right| ^2 ds&\le \int _{e}\left| \varphi _L \right| ^2 ds\nonumber \\ \nonumber&\le C \left[ h^{-1} \int _{T} \varphi ^2 dT + h \int _{T} |\nabla \varphi |^2 dT\right] \\&\le Ch^{-1} \int _{T} \varphi ^2 dT, \end{aligned}$$

(7.15)

and

$$\begin{aligned} \int _{e}\left| \frac{\partial \varphi _R}{\partial \mathbf n _R}\right| ^2 ds&\le C \left[ h^{-1} \int _{T} |\nabla \varphi |^2 dT + h \int _{T} |\nabla ^2 \varphi |^2 dT\right] \nonumber \\&\le C h^{-1} \int _{T} |\nabla \varphi |^2 dT. \end{aligned}$$

(7.16)

Substituting (7.15) and (7.16) into (7.14) yields

$$\begin{aligned} \left| \sum _{T\in \mathcal {T}_h} \int _{\partial T}\frac{\partial \varphi }{\partial \mathbf n } \, Q_b \varphi \, ds\right|&\le C h^{-\frac{1}{2}}\Vert \varphi \Vert \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L} + \frac{\partial \varphi _R}{\partial \mathbf n _R}\right) ^2 ds\right) ^{\frac{1}{2}} \nonumber \\&+ \,C h^{-\frac{1}{2}}\Vert \nabla _h \varphi \Vert \left( \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b \varphi _R-Q_b \varphi _L)^2 ds\right) ^{\!\!\frac{1}{2}}\!. \end{aligned}$$

(7.17)

Substituting (7.17) into (7.13) gives

$$\begin{aligned} \Vert \nabla _h \varphi \Vert ^2&\le \Vert \Delta _h \varphi \Vert \ \Vert \varphi \Vert +Ch^{-\frac{1}{2}}\Vert \varphi \Vert \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial \varphi _L}{\partial \mathbf n _L} + \frac{\partial \varphi _R}{\partial \mathbf n _R}\right) ^2 ds\right) ^{\frac{1}{2}} \\&+\, C h^{-\frac{1}{2}}\Vert \nabla _h \varphi \Vert \left( \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b \varphi _R-Q_b \varphi _L)^2 ds\right) ^{\frac{1}{2}}, \end{aligned}$$

which, through an use of Young’s inequality, implies the desired estimate (7.12). This completes the proof. $\square$

Lemma 7.8

There exists a constant $C$ such that for any $v=\{v_0,v_b,v_n{\mathbf {n}}_e\}\in V_h^0$ the following Poincaré type inequality holds true

$$\begin{aligned} \Vert \nabla _h v_0\Vert \le C {|||} v {|||}. \end{aligned}$$

(7.18)

In addition, we have the following estimate

$$\begin{aligned} \Vert \nabla _h v_0\Vert \le \lambda h^{-1} \Vert v\Vert + C h {|||} v{|||}, \end{aligned}$$

(7.19)

where $\lambda$ is a positive constant.

Proof

The first component $v_0$ is a piecewise polynomial of degree $k\ge 2$. Using the estimate (7.12) in Lemma 7.7 we have

$$\begin{aligned} \nonumber \Vert \nabla _h v_0\Vert ^2&\le \varepsilon \Vert v\Vert ^2 + C \varepsilon ^{-1} \Vert \Delta _h v_0\Vert ^2\\&+\, C \varepsilon ^{-1} h^{-1} \left( \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial {v_0}_L}{\partial \mathbf n _L} + \frac{\partial {v_0}_R}{\partial \mathbf n _R}\right) ^2 ds\right) \nonumber \\&+\, Ch^{-1}\left( \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b {v_0}_R-Q_b {v_0}_L)^2 ds\right) \!. \end{aligned}$$

(7.20)

By inserting $v_n{\mathbf {n}}_e\cdot {\mathbf {n}}$ in each integrand we obtain

$$\begin{aligned} \sum _{e\in \mathcal {E}_h}\int _{e}\left( \frac{\partial {v_0}_L}{\partial \mathbf n _L} + \frac{\partial {v_0}_R}{\partial \mathbf n _R}\right) ^2 ds \le C \sum _{T\in {\mathcal {T}}_h} \Vert \nabla v_0 \cdot {\mathbf {n}}_e - v_n\Vert _{\partial T}^2. \end{aligned}$$

Similarly, by inserting $v_b$

$$\begin{aligned} \sum _{e\in \mathcal {E}_h}\int _{e} (Q_b {v_0}_R-Q_b {v_0}_L)^2 ds \le C \sum _{T\in {\mathcal {T}}_h} \Vert Q_b v_0 - v_b\Vert ^2_{\partial T}. \end{aligned}$$

Substituting the above two inequalities into (7.20) yields

$$\begin{aligned} \Vert \nabla _h v_0\Vert ^2&\le \varepsilon \Vert v\Vert ^2 + C \varepsilon ^{-1} \Vert \Delta _h v_0\Vert ^2+ Ch^{-1}\sum _{T\in {\mathcal {T}}_h} \Vert Q_b v_0 - v_b\Vert ^2_{\partial T}\nonumber \\&+\, C \varepsilon ^{-1} h^{-1}\sum _{T\in {\mathcal {T}}_h} \Vert \nabla v_0 \cdot {\mathbf {n}}_e - v_n\Vert _{\partial T}^2. \end{aligned}$$

(7.21)

Using the Poincaré inequality (7.11) and the estimate (7.10) we arrive at

$$\begin{aligned} \Vert \nabla _h v_0\Vert ^2\le \varepsilon C \Vert \nabla _h v\Vert ^2 + C \varepsilon ^{-1} {|||} v {|||}^2, \end{aligned}$$

which leads to the inequality (7.18) for sufficiently small $\varepsilon$.

Finally, by setting $\varepsilon = \lambda h^{-2}$ in (7.21) we arrive at

$$\begin{aligned} \Vert \nabla _h v_0\Vert ^2\le \lambda h^{-2} \Vert v\Vert ^2 + C h^2 {|||} v{|||}^2, \end{aligned}$$

where $\lambda$ is a positive constant. This verifies the inequality (7.19), and hence completes the proof of the lemma. $\square$

Lemma 7.9

There exists a constant $C$ such that for any $v=\{v_0,v_b,v_n{\mathbf {n}}_e\}\in V_h^0$ one has

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\int _{\partial T}(v_0-Q_b v_0)^2 ds \le C h {|||} v {|||}^2 \end{aligned}$$

(7.22)

and

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\int _{\partial T}(v_0-Q_b v_0)^2 ds \le C \lambda h^{-1}\Vert v\Vert ^2 + C h^3 {|||} v {|||}^2. \end{aligned}$$

(7.23)

Proof

From the trace inequality (7.1) and the inverse inequality (7.2), we have

$$\begin{aligned} \int _{\partial T}(v_0-Q_b v_0)^2 ds \le C h \int _T |\nabla v_0|^2 dT. \end{aligned}$$

Summing over all $T\in \mathcal {T}_h$ yields

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\int _{\partial T}(v_0-Q_b v_0)^2 ds \le C h\sum _{T\in \mathcal {T}_h} \int _T |\nabla v_0|^2 dT, \end{aligned}$$

(7.24)

which, combined with (7.18) and (7.19), completes the proof of the lemma. $\square$

Remark 7.1

The estimate (7.22) in Lemma 7.9 is sufficient for us to derive an optimal order error estimate for the WG finite element solution arising from (3.5). But the estimate (7.22) is sub-optimal in terms of the mesh parameter $h$. We conjecture that the following inequality holds true

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\int _{\partial T}(v_0-Q_b v_0)^2 ds \le C h^3 \ {|||} v {|||}^2. \end{aligned}$$

(7.25)

However, with the current mathematical approach, we are unable to verify the validity of (7.25). This estimate is then left to interested readers or researchers as an open problem.