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.

1 Introduction

This paper will concern with approximating the solution $u$ of the biharmonic equation

$$\begin{aligned} \Delta ^2 u&= f, \quad \mathrm{in} \ \Omega , \end{aligned}$$

(1.1)

with clamped boundary conditions

$$\begin{aligned} u&= g,\quad \mathrm{on}\ \partial \Omega , \end{aligned}$$

(1.2)

$$\begin{aligned} \frac{\partial u}{\partial \mathbf n }&= \phi ,\quad \mathrm{on}\ \partial \Omega , \end{aligned}$$

(1.3)

where $\Delta$ is the Laplacian operator, $\Omega$ is a bounded polygonal or polyhedral domain in $\mathbb {R}^d$ for $d=2, 3$ and $\mathbf n$ denotes the outward unit normal vector along $\partial \Omega$. We assume that $f, g, \phi$ are given, sufficiently smooth functions.

This problem mainly arises in fluid dynamics where the stream functions $u$ of incompressible flows are sought and elasticity theory, in which the deflection of a thin plate of the clamped plate bending problem is sought [26, 34, 36].

Due to the significance of the biharmonic problem, a large number of methods for discretizing (1.1)–(1.3) have been proposed. These methods include dealing with the biharmonic operator directly, such as discretizing (1.1)–(1.3) on a uniform grid using a 13-point or 25-point direct approximation of the fourth order differential operator [9, 24]; mixed methods, that is, splitting the biharmonic equation into two coupled Poisson equations [1, 4–7, 12, 15, 17–20, 25, 27]. Also there are some other approaches to the biharmonic problems, like the conformal mapping methods [11, 35], integral equations [29], orthogonal spline collocation method [8] and the fast multipole methods [23], etc.

Among these methods, finite element methods are one of the most widely used technique, which is based on variational formulations of the equations considered. In fact, the biharmonic equation is also one of the most important applicable problems of the finite element methods, cf. [2, 13, 14, 16, 22, 41]. The Galerkin methods, discretizing the corresponding variational form of (1.1) is given by seeking $u\in H^2(\Omega )$ satisfying

$$\begin{aligned} u|_{\partial \Omega }=g, \qquad \frac{\partial u}{\partial \mathbf n }|_{\partial \Omega }=\phi \end{aligned}$$

such that

$$\begin{aligned} (\Delta u, \Delta v)=(f,v), \quad \forall v\in H_0^2(\Omega ), \end{aligned}$$

(1.4)

where $H_0^2(\Omega )$ is the subspace of $H^2(\Omega )$ consisting of functions with vanishing value and normal derivative on $\partial \Omega$.

Standard finite element methods for solving (1.1)–(1.3) based on the variational form (1.4) with conforming finite element require rather sophisticated finite elements such as the 21-degrees-of-freedom of Argyris (see [3]) or nonconforming elements of Hermite type. Since the complexity in the construction for the finite element with high continuous elements, $H^2$ conforming element are seldom used in practice for the biharmonic problem. To avoid using of $C^1$-elements, besides the mixed methods, an alternative approach, nonconforming and discontinuous Galerkin finite element methods have been developed for solving the biharmonic equation over the last several decades. Morley element [28] is a well known nonconforming element for the biharmonic equation for its simplicity. A $C^0$ interior penalty method was developed in [10, 21]. In [30], a hp-version interior penalty discontinuous Galerkin method was presented for the biharmonic equation.

Recently a new class of finite element methods, called weak Galerkin(WG) finite element methods were developed for the biharmonic equation for its highly flexible and robust properties. The WG method refers to a numerical scheme for partial differential equations in which differential operators are approximated by weak forms as distributions over a set of generalized functions. This thought was first proposed in [38] for a model second order elliptic problem, and this method was further developed in [31, 39, 40]. In [32], a weak Galerkin method for the biharmonic equation was derived by using discontinuous functions of piecewise polynomials on general partitions of polygons or polyhedra of arbitrary shape. After that, in order to reduce the number of unknowns, a $C^0$ WG method [33] was proposed and analyzed. However, due to the continuity limitation, the $C^0$ WG scheme only works for the traditional finite partitions, while not arbitrary polygonal or polyhedral girds as allowed in [32].

In order to realize the aim that reducing the unknown numbers and suit for general partitions of polygons or polyhedra of arbitrary shape at the same time, in this paper we construct a reduction WG scheme based on the use of a discrete weak Laplacian plus a new stabilization that is also parameter free. The goal of this paper is to specify all the details for the reduction WG method for the biharmonic equations and present the numerical analysis by presenting a mathematical convergence theory.

An outline of the paper is as follows. In the remainder of the introduction we shall introduce some preliminaries and notations for Sobolev spaces. In Sect. 2 is devoted to the definitions of weak functions and weak derivatives. The WG finite element schemes for the biharmonic Eqs. (1.1)–(1.3) are presented in Sect. 3. In Sect. 4, we establish an optimal order error estimates for the WG finite element approximation in an $H^2$ equivalent discrete norm. In Sect. 5, we shall drive an error estimate for the WG finite element method in the standard $L^2$ norm. Section 6 contains the numerical results of the WG method. The theoretical results are illustrated by these numerical examples. Finally, we present some technical estimates for quantities related to the local $L^2$ projections into various finite element spaces and some approximation properties which are useful in the convergence analysis in “Appendix”.

Now let us define some notations. Let $D$ be any open bounded domain with Lipschitz continuous boundary in $\mathbb {R}^d, d=2, 3$. We use the standard definition for the Sobloev space $H^s(D)$ and their associated inner products $(\cdot , \cdot )_{s, D}$, norms $\Vert \cdot \Vert _{s, D}$, and seminorms $|\cdot |_{s, D}$ for any $s\ge 0$.

The space $H^0(D)$ coincides with $L^2(D)$, for which the norm and the inner product are denoted by $\Vert \cdot \Vert _D$ and $(\cdot ,\cdot )_D$, respectively. When $D=\Omega$, we shall drop the subscript $D$ in the norm and in the inner product notation.

The space $H(\mathrm{div}; D)$ is defined as the set of vector-valued functions on $D$ which, together with their divergence, are square integrable; i.e.,

$$\begin{aligned} H({\mathrm{div}}; D)=\left\{ \mathbf{v}: \mathbf{v}\in [L^2(D)]^d, \nabla \cdot \mathbf{v}\in L^2(D)\right\} . \end{aligned}$$

The norm in $H(\mathrm{div}; D)$ is defined by

$$\begin{aligned} \Vert \mathbf v \Vert _{H(\mathrm{div}; D)}=\left( \Vert \mathbf v \Vert ^2_D+\Vert \nabla \cdot \mathbf v \Vert ^2_D\right) ^{\frac{1}{2}}. \end{aligned}$$

2 Weak Laplacain and Discrete Weak Laplacian

For the biharmonic equation (1.1), the underlying differential operator is the Laplacian $\Delta$. Thus, we shall first introduce a weak version for the Laplacian operator defined on a class of discontinuous functions as distributions [32].

Let $K$ be any polygonal or polyhedral domain with boundary $\partial K$. A weak function on the region $K$ refers to a function $v= \{v_0, v_b, \mathbf v _g\}$ such that $v_0\in L^2(K), v_b\in L^{2}(\partial K)$, and $\mathbf v _g\cdot \mathbf n \in L^{2}(\partial K)$, where $\mathbf n$ is the outward unit normal vector along $\partial K$. Denote by $\mathcal {W}(K)$ the space of all weak functions on $K$, that is,

$$\begin{aligned} \mathcal {W}(K)=\left\{ v=\{v_0,v_b, \mathbf v _g\}: v_0\in L^2(K), v_b, \mathbf v _g\cdot \mathbf n \in L^{2}(\partial K)\right\} . \end{aligned}$$

(2.1)

Recall that, for any $v\in \mathcal {W}(K)$, the weak Laplacian of $v=\{v_0, v_b, \mathbf v _g\}$ is defined as a linear functional $\Delta _w v$ in the dual space of $H^2(K)$ whose action on each $\varphi \in H^2(K)$ is given by

$$\begin{aligned} (\Delta _w v, \varphi )_K=(v_0, \Delta \varphi )_K-\langle v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}+\langle \mathbf v _g\cdot \mathbf n ,\varphi \rangle _{\partial K}, \end{aligned}$$

(2.2)

where $(\cdot , \cdot )_K$ stands for the $L^2$-inner product in $L^2(K)$ and $\langle \cdot , \cdot \rangle _{\partial K}$ is the inner product in $L^2(\partial K)$.

The Sobolev space $H^2(K)$ can be embedded into the space $\mathcal {W}(K)$ by an inclusion map $i_\mathcal {W}: H^2(K)\rightarrow \mathcal {W}(K)$ defined as follows

$$\begin{aligned} i_\mathcal {W}(\phi )=\{\phi |_K, \phi |_{\partial K}, (\nabla \phi \cdot \mathbf n )\mathbf n |_{\partial K}\}, \quad \phi \in H^2(K). \end{aligned}$$

With the help of the inclusion map $i_\mathcal {W}$, the Sobolev space $H^2(K)$ can be viewed as a subspace of $\mathcal {W}(K)$ by identifying each $\phi \in H^2(K)$ with $i_\mathcal {W}(\phi )$.

Analogously, a weak function $v= \{v_0,v_b, \mathbf v _g\}\in \mathcal {W}(K)$ is said to be in $H^2(K)$ if it can be identified with a function $\phi \in H^2(K)$ through the above inclusion map. Here the first components $v_0$ can be seen as the value of $v$ in the interior and the second component $v_b$ represents the value of $v$ on $\partial K$. Denote $\nabla v\cdot \mathbf n$ by $v_n$, then the third component $\mathbf v _g$ represents $(\nabla v\cdot \mathbf n )\mathbf n |_{\partial K}=v_n \mathbf n$. Obviously, $\mathbf v _g\cdot \mathbf n = \nabla v\cdot \mathbf n$. Note that if $v\not \in H^2(K)$, then $v_b$ and $\mathbf v _g$ may not necessarily be related to the trace of $v_0$ and $(\nabla v_0\cdot \mathbf n )\mathbf n$ on $\partial K$, respectively.

For $v\in H^2(K)$, from integration by parts we have

$$\begin{aligned} (\Delta _w v, \varphi )_K&= (v, \Delta \varphi )_K-\langle v, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}+\langle \nabla v\cdot \mathbf n ,\varphi \rangle _{\partial K} \\&= (v_0, \Delta \varphi )_K-\langle v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}+\langle \mathbf v _g\cdot \mathbf n ,\varphi \rangle _{\partial K}. \end{aligned}$$

Thus the weak Laplacian is identical with the strong Laplacian, i.e.,

$$\begin{aligned} \Delta _w i_\mathcal {W}(v)=\Delta v \end{aligned}$$

for smooth functions in $H^2(K)$.

For numerical implementation purpose, we define a discrete version of the weak Laplacain operator by approximating $\Delta _w$ in polynomial subspaces of the dual of $H^2(K)$. To this end, for any non-negative integer $r\ge 0$, let $P_r(K)$ be the set of polynomials on $K$ with degree no more than $r$.

Definition 2.1

([32]) A discrete weak Laplacian operator, denoted by $\Delta _{w,r,K}$, is defined as the unique polynomial $\Delta _{w, r, K} v\in P_r(K)$ satisfying

$$\begin{aligned} \qquad (\Delta _{w,r,K} v, \varphi )_K=(v_0, \Delta \varphi )_K-\langle v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}+\langle \mathbf{v}_n\cdot \mathbf n ,\varphi \rangle _{\partial K}, \quad \forall \varphi \in P_r(K).\qquad \end{aligned}$$

(2.3)

From the integration by parts, we have

$$\begin{aligned} (v_0, \Delta \varphi )_K = (\Delta v_0, \varphi )_K+\langle v_0, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}-\langle \nabla v_0\cdot \mathbf n ,\varphi \rangle _{\partial K}. \end{aligned}$$

Substituting the above identity into (2.3) yields

$$\begin{aligned} \qquad (\Delta _{w,r,K} v, \varphi )_K-(\Delta v_0, \varphi )_K=\langle v_0-v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial K}-\langle (\nabla v_0-\mathbf v _g)\cdot \mathbf n ,\varphi \rangle _{\partial K},\qquad \quad \end{aligned}$$

(2.4)

for all $\varphi \in P_r(K)$.

3 Weak Galerkin Finite Element Scheme

Let $\mathcal {T}_h$ be a partition of the domain $\Omega$ into polygons in 2D or polyhedra in 3D. Assume that $\mathcal {T}_h$ is shape regular in the sense as defined in [39]. Denote by $\mathcal {E}_h$ the set of all edges or flat faces in $\mathcal {T}_h$, and let $\mathcal {E}_h^0=\mathcal {E}_h\setminus \partial \Omega$ be the set of all interior edges or flat faces.

Since $v_n$ represents $\nabla v\cdot \mathbf n$, then $v_n$ is naturally dependent on $\mathbf n$. To ensure a single valued function $v_n$ on $e\in \mathcal {E}_h$, we introduce a set of normal directions on $\mathcal {E}_h$ as follows

$$\begin{aligned} \mathcal {N}_h=\{\mathbf{n}_e: \mathbf{n}_e \hbox { is unit and normal to } e, \ e\in \mathcal {E}_h\}. \end{aligned}$$

(3.1)

For any given integer $k\ge 2, T\in \mathcal {T}_h$, denote by $\mathcal {W}_k(T)$ the discrete weak function space given by

$$\begin{aligned} \mathcal {W}_k(T)=\{\{v_0,v_b, v_n \mathbf n _e\}: v_0\in P_k(T), v_b, v_n\in P_{k-1}(e), e\subset \partial T\}. \end{aligned}$$

(3.2)

By patching $\mathcal {W}_k(T)$ over all the elements $T\in \mathcal {T}_h$ through a common value on the interface $\mathcal {E}_h^0$, we arrive at a weak finite element space $V_h$ given by

$$\begin{aligned} V_h=\left\{ \left\{ v_0, v_b, v_n \mathbf n _e \right\} : \{v_0,v_b, v_n \mathbf n _e\}\big |_T\in \mathcal {W}_k(T), \quad \forall T\in \mathcal {T}_h\right\} . \end{aligned}$$

Denote by $V_h^0$ the subspace of $V_h$ constituting discrete weak functions with vanishing traces; i.e.,

$$\begin{aligned} V_h^0=\{\{v_0, v_b, v_n \mathbf n _e \}: \{v_0,v_b, v_n \mathbf n _e\}\in V_h, v_b|_e=0, v_n|_e=0, \quad e\in \partial T\cap \partial \Omega \}. \end{aligned}$$

Denote by $\Lambda _h$ the trace of $V_h$ on $\partial \Omega$ from the component $v_b$. It is obvious that $\Lambda _h$ consists of piecewise polynomials of degree $k-1$. Similarly, denote by $\Upsilon _h$ the trace of $V_h$ from the component of $v_n$ as piecewise polynomials of degree $k-1$. Denote by $\Delta _{w, k-2}$ the discrete weak Laplacian operator on the finite element space $V_h$ computed by using (2.3) on each element $T$ for $k\ge 2$, that is,

$$\begin{aligned} (\Delta _{w, k-2} v)|_T=\Delta _{w, k-2, T}(v|_T)\quad \forall v\in V_h. \end{aligned}$$

(3.3)

For simplicity, we shall drop the subscript $k-2$ in the notation $\Delta _{w, k-2}$ for the discrete weak Laplacian operator. We also introduce the following notation

$$\begin{aligned} (\Delta _{w}v,\Delta _{w}w)_h=\sum _{T\in \mathcal {T}_h} (\Delta _{w}v,\Delta _{w}w)_T. \end{aligned}$$

For each element $T\in \mathcal {T}_h$, denote by $Q_0$ the $L^2$ projection onto $P_k(T), k\ge 2$. For each edge/face $e\subset \partial T$, denote by $Q_b$ the $L^2$ projection onto $P_{k-1}(e)$. Now for any $u\in H^2(\Omega )$, we shall combine these two projections together to define a projection into the finite element space $V_h$ such that on the element $T$

$$\begin{aligned} Q_hu=\{Q_0 u, Q_b u, (Q_b (\nabla u\cdot \mathbf n _e))\mathbf n _e\}. \end{aligned}$$

Theorem 3.1

Let $\mathbb {Q}_h$ be the local $L^2$ projection onto $P_{k-2}$. Then the following commutative diagram holds true on each element $T\in {\mathcal {T}}_h$:

$$\begin{aligned} \Delta _w Q_h u=\mathbb {Q}_h \Delta u,\qquad \forall u\in H^2(T). \end{aligned}$$

(3.4)

Proof

For any $\phi \in P_{k-2}(T)$, from the definition of the discrete weak Laplacian and the $L^2$ projection

$$\begin{aligned} (\Delta _w Q_h u, \phi )_T&= (Q_0 u,\Delta \phi )_T-\langle Q_b u, \nabla \phi \cdot \mathbf n \rangle _{\partial T}+\langle Q_b(\nabla u\cdot \mathbf n _e)\mathbf n _e\cdot \mathbf n , \phi \rangle _{\partial T} \\&= (u, \Delta \phi )_T-\langle u, \nabla \phi \cdot \mathbf n \rangle _{\partial T}+\langle \nabla u\cdot \mathbf n , \phi \rangle _{\partial T} \\&= (\Delta u, \phi )_T=(\mathbb {Q}_h \Delta u, \phi ), \end{aligned}$$

which implies (3.4). $\square$

The commutative property (3.4) indicates that the discrete weak Laplacian of the $L^2$ projection of $u$ is a good approximation of the Laplacian of $u$ in the classical sense. This is a good property of the discrete weak Laplacian in application to algorithm and analysis.

For any $u_h=\{u_0, u_b, u_n \mathbf n _e\}$ and $v=\{v_0, v_b, v_n \mathbf n _e \}$ in $V_h$, we introduce a bilinear form as follows

$$\begin{aligned} s(u_h, v)&= \displaystyle \sum _{T\in \mathcal {T}_h}h_T^{-1} \langle \nabla u_0\cdot \mathbf n _e-u_n, \nabla v_0\cdot \mathbf n _e-v_n \rangle _{\partial T} \\&+\sum _{T\in \mathcal {T}_h}h_T^{-3}\langle Q_b u_0-u_b, Q_b v_0-v_b \rangle _{\partial T}. \end{aligned}$$

Weak Galerkin Algorithm 1

Find $u_h = \{u_0, u_b, u_n\mathbf n _e\}\in V_h$ satisfying $u_b = Q_b g$ and $u_n = Q_{b}\phi$ on $\partial \Omega$ and the following equation:

$$\begin{aligned} (\Delta _w u_h, \Delta _w v)_h+s(u_h, v)=(f, v_0), \ \forall v=\{v_0, v_b, v_n\mathbf n _e\}\in V_h^0. \end{aligned}$$

(3.5)

Lemma 3.2

For any $v \in V_h^0$, let $|\!|\!| v |\!|\!|$ be given by

$$\begin{aligned} |\!|\!| v |\!|\!| ^2=(\Delta _w v, \Delta _w v)_h+s(v, v). \end{aligned}$$

(3.6)

Then, $|\!|\!| \cdot |\!|\!|$ defines a norm in the linear space $V_h^0$.

Proof

For simplicity, we shall only prove the positivity property for $|\!|\!| \cdot |\!|\!|$. Assume that $|\!|\!| v |\!|\!| = 0$ for some $v \in V_h^0$. It follows that $\Delta _w v=0$ on each element T, $Q_b v_0=v_b$ and $\nabla v_0\cdot \mathbf n _e = v_n$ on each edge $\partial T$. We claim that $\Delta v_0 = 0$ holds true locally on each element T. To this end, for any $\varphi \in P_{k-2}(T)$ we use $\Delta _w v=0$ and the identify (2.4) to obtain

$$\begin{aligned} 0&= (\Delta _w v, \varphi )_T\\ \nonumber&= (\Delta v_0, \varphi )_T+\langle Q_b v_0-v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial T} +\langle v_n\mathbf n _e \cdot \mathbf n -\nabla v_0\cdot \mathbf n , \varphi \rangle _{\partial T} \\ \nonumber&= (\Delta v_0, \varphi )_T, \end{aligned}$$

(3.7)

where we have used the fact that $Q_b v_0-v_b=0$ and

$$\begin{aligned} v_n \mathbf n _e\cdot \mathbf n - \nabla v_0\cdot \mathbf n =\pm (v_n-\nabla v_0\cdot \mathbf n _e) = 0 \end{aligned}$$

in the last equality. The identity (3.7) implies that $\Delta v_0=0$ holds true locally on each element $T$.

Next, we claim that $\nabla v_0 = 0$ also holds true locally on each element $T$. For this purpose, for any $\phi \in P_{k}(T)$, we utilize the Gauss formula to obtain

$$\begin{aligned} (\nabla v_0, \nabla \phi )_{T}&= -(\Delta v_0, \phi )_T +\langle \nabla v_0\cdot \mathbf n , \phi \rangle _{\partial T} = \langle \nabla v_0\cdot \mathbf n , \phi \rangle _{\partial T}. \end{aligned}$$

(3.8)

By letting $\phi =v_0$ on each element $T$ and summing over all $T$ we obtain

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}(\nabla v_0, \nabla v_0)_{T}=\sum _{T\in \mathcal {T}_h}\langle \nabla v_0\cdot \mathbf n , v_0\rangle _{\partial T}. \end{aligned}$$

(3.9)

For two elements $T_1, T_2\in \mathcal {T}_h$, which share $e\in \mathcal {E}_h\setminus \partial \Omega$ as a common edge, denote $v_0^1, v_0^2$ the values of $v$ in the interior of $T_1, T_2$, respectively. It follows from $Q_b v_0^1=Q_b v_0^2=v_b$ on edge $e$ and the fact $\nabla v_0\cdot {\mathbf {n}}_e=v_n\in P_{k-1}(e)$ that

$$\begin{aligned} \left\langle \nabla v_0^1\cdot \mathbf n _{T_1}, v_0^1\right\rangle _{e}+\left\langle \nabla v_0^2\cdot \mathbf n _{T_2}, v_0^2\right\rangle _{e} =\pm \left\langle v_n, v_0^1-v_0^2\right\rangle _{e}=\pm \left\langle v_n, Q_bv_0^1-Q_bv_0^2\right\rangle _{e}=0, \end{aligned}$$

where $\mathbf n _{T_1}, \mathbf n _{T_2}$ denote the outward unit normal vectors on $e$ according to elements $T_1, T_2$, respectively. This, together with $\nabla v_0\cdot \mathbf n =v_n=0$ on the boundary edge $e\in \mathcal {E}_h\cap \partial \Omega$ implies

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}\langle \nabla v_0\cdot \mathbf n , v_0\rangle _{\partial T}=0. \end{aligned}$$

It follows from Eq. (3.9) that $\Vert \nabla v_0\Vert _T=0$ on each element $T$. Thus, $v_0= const$ locally on each element and is then continuous across each interior edge $e$ as

$$\begin{aligned} v_0|_e = Q_b v_0 = v_b. \end{aligned}$$

The boundary condition of $v_b=0$ then implies that $v\equiv 0$ on $\Omega$, which completes the proof. $\square$

Lemma 3.3

The weak Galerkin finite element scheme (3.5) has a unique solution.

Proof

Assume $u_h^{(1)}$ and $u_h^{(2)}$ are two solutions of the WG finite element scheme (3.5). It is obvious that the difference $\rho _h = u_h^{(1)}- u_h^{(2)}$ is a finite element function in $V_h^0$ satisfying

$$\begin{aligned} (\Delta _w \rho _h, \Delta _w v)_h+s(\rho _h, v)=0, \ \forall v\in V_h^0. \end{aligned}$$

(3.10)

By letting $v=\rho _h$ in above Eq. (3.10) we obtain the following indentity

$$\begin{aligned} (\Delta _w \rho _h, \Delta _w \rho _h)_h+s(\rho _h, \rho _h)=0. \end{aligned}$$

It follows from Lemma 3.2 that $\rho _h\equiv 0$, which shows that $u_h^{(1)}=u_h^{(2)}$. This completes the proof. $\square$

4 An Error Estimate

The goal of this section is to establish an error estimate for the WG-FEM solution $u_h$ arising from (3.5).

First of all, let us derive an error equation for the WG finite element solution obtained from (3.5). This error equation is critical in convergence analysis.

Lemma 4.1

Let $u$ and $u_h \in V_h$ be the solution of (1.1)–(1.3) and (3.5), respectively. Denote by

$$\begin{aligned} e_h=Q_hu-u_h \end{aligned}$$

the error function between the $L^2$ projection of $u$ and its weak Galerkin finite element solution. Then the error function $e_h$ satisfies the following equation

$$\begin{aligned} (\Delta _{\omega } e_h,\Delta _{\omega } v)_h + s(e_h,v)=\ell _u(v) \end{aligned}$$

(4.1)

for all $v\in V^0_h$. Here

$$\begin{aligned} \ell _u(v)&= \sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u, \nabla v_0\cdot \mathbf {n} -v_n\mathbf {n}_e\cdot \mathbf {n}\rangle _{\partial T}\\ \nonumber&-\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, v_0-v_b\rangle _{\partial T} +s(Q_hu,v). \end{aligned}$$

(4.2)

Proof

Using (2.4) with $\varphi =\Delta _{\omega }Q_hu = \mathbb {Q}_h\Delta u$ we obtain

$$\begin{aligned}&(\Delta _{\omega }Q_hu,\Delta _{\omega }v)_T\\&\quad = (\Delta v_0,\mathbb {Q}_h\Delta u)_T + \langle v_0-v_b,\nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf {n}\rangle _{\partial T} -\langle (\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n},\mathbb {Q}_h\Delta u\rangle _{\partial T}\\&\quad = (\Delta u,\Delta v_0)_T + \langle v_0-v_b,\nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf {n}\rangle _{\partial T}-\langle (\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n},\mathbb {Q}_h\Delta u\rangle _{\partial T} , \end{aligned}$$

which implies that

$$\begin{aligned} (\Delta u,\Delta v_0)_T&= (\Delta _{\omega }Q_hu,\Delta _\omega v)_T-\langle v_0-v_b,\nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf {n}\rangle _{\partial T}\\&+\, \langle (\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n},\mathbb {Q}_h\Delta u\rangle _{\partial T}.\nonumber \end{aligned}$$

(4.3)

Next, it follows from the integration by parts that

$$\begin{aligned} (\Delta u, \Delta v_0)_T =(\Delta ^2u,v_0)_T +\langle \Delta u,\nabla v_0\cdot \mathbf {n}\rangle _{\partial T}-\langle \nabla (\Delta u)\cdot \mathbf {n},v_0\rangle _{\partial T}. \end{aligned}$$

By summing over all $T$ and then using the identity $(\Delta ^2u,v_0)=(f, v_0)$ we arrive at

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}(\Delta u,\Delta v_0)_T&= (f, v_0)+\sum _{T\in \mathcal {T}_h}\langle \Delta u, \nabla v_0\cdot \mathbf {n}-v_n\mathbf {n}_e\cdot \mathbf {n}\rangle _{\partial T} \\&-\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u)\cdot \mathbf n , v_0-v_b\rangle _{\partial T}, \end{aligned}$$

where we have used the fact that $v_n$ and $v_b$ vanish on the boundary of the domain. Combining the above equation with (4.3) yields

$$\begin{aligned} (\Delta _{\omega }Q_hu,\Delta _{\omega }v)_h&= (f, v_0) +\sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n}\rangle _{\partial T}\\&-\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n},v_0-v_b\rangle _{\partial T}. \end{aligned}$$

Adding $s(Q_hu, v)$ to both sides of the above equation gives

$$\begin{aligned}&(\Delta _{\omega }Q_hu,\Delta _{\omega }v)_h + s(Q_hu,v) \nonumber \\&\quad = (f, v_0) +\sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n}\rangle _{\partial T}\nonumber \\&\qquad -\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, v_0-v_b\rangle _{\partial T}+s(Q_hu,v). \end{aligned}$$

(4.4)

Subtracting (3.5) from (4.4) leads to the following error equation

$$\begin{aligned} (\Delta _{\omega }e_h,\Delta _{\omega }v)_h+s(e_h,v)&= \sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u,(\nabla v_0-v_n\mathbf {n}_e)\cdot \mathbf {n}\rangle _{\partial T}\\&-\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, v_0-v_b\rangle _{\partial T}+s(Q_hu,v) \end{aligned}$$

for all $v\in V^0_h$. This completes the derivation of (4.1). $\square$

The following Theorem presents an optimal order error estimate for the error function $e_h$ in the trip-bar norm. We believe this tripe-bar norm provides a discrete analogue of the usual $H^2$-norm.

Theorem 4.2

Let $u_h\in V_h$ be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order $k\ge 2$. Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that $u\in H^{k+2}(\Omega )$. Then, there exists a constant $C$ such that

$$\begin{aligned} {|||} u_h-Q_hu {|||} \le Ch^{k-1}\ \Vert u\Vert _{k+2}. \end{aligned}$$

(4.5)

The above estimate is of optimal order in terms of the meshsize $h$, but not in the regularity assumption on the exact solution of the biharmonic equation.

Proof

By letting $v =e_h$ in the error Eq. (4.1), we have

$$\begin{aligned} {|||} e_h {|||}^2 = \ell (e_h), \end{aligned}$$

(4.6)

where

$$\begin{aligned} \ell (e_h)&= \sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla e_0-e_n\mathbf {n}_e)\cdot \mathbf {n}\rangle _{\partial T} \\ \nonumber&-\sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, e_0-e_b\rangle _{\partial T} \\ \nonumber&+\sum _{T\in \mathcal {T}_h}h_T^{-1}\langle \nabla Q_0u\cdot \mathbf {n}_e-Q_b (\nabla u\cdot \mathbf {n}_e), \nabla e_0 \cdot \mathbf {n}_e-e_n\rangle _{\partial T} \\ \nonumber&+\sum _{T\in \mathcal {T}_h}h_T^{-3}\langle Q_b Q_0u-Q_bu,Q_b e_0-e_b\rangle _{\partial T}. \end{aligned}$$

(4.7)

The rest of the proof shall estimate each of the terms on the right-hand side of (4.7). For the first term, we use the Cauchy–Schwarz inequality and the estimates (7.5) and (7.6) in Lemma 7.4 (see “Appendix”) with $m = k$ to obtain

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h}\langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla e_0-e_n\mathbf {n}_e)\cdot \mathbf {n}\rangle _{\partial T}\right| \nonumber \\&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T\Vert \Delta u-\mathbb {Q}_h\Delta u\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}} \left( \sum _{T\in \mathcal {T}_h}h_T^{-1}\Vert \nabla e_0\cdot \mathbf {n}_e-e_n\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}}\nonumber \\&\quad \le Ch^{k-1}\Vert u\Vert _{k+1} |\!|\!| e_h |\!|\!|. \end{aligned}$$

(4.8)

For the second term, using Lemmas 7.4, 7.6 and 7.9 we obtain

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, e_0-e_b\rangle _{\partial T}\right| \nonumber \\&\quad \le \left| \sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u) \cdot \mathbf {n}, Q_be_0-e_b\rangle _{\partial T}\right| \nonumber \\&\qquad +\,\,\left| \sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf {n}, e_0-Q_be_0\rangle _{\partial T}\right| \nonumber \\&\quad = \left| \sum _{T\in \mathcal {T}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u) \cdot \mathbf {n}, Q_be_0-e_b\rangle _{\partial T}\right| \nonumber \\&\qquad +\left| \sum _{T\in \mathcal {T}_h}\langle (\nabla ( \Delta u)-Q_b(\nabla (\Delta u)))\cdot \mathbf {n}, e_0-Q_be_0\rangle _{\partial T}\right| \nonumber \\&\quad \le \left( \sum _{T\in \mathcal {T}_h}h^3_T\Vert \nabla ( \Delta u-\mathbb {Q}_h\Delta u)\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}}\cdot \left( \sum _{T\in \mathcal {T}_h}h_T^{-3}\Vert Q_be_0-e_b\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}} \nonumber \\&\qquad +\left( \sum _{T\in \mathcal {T}_h}\Vert \nabla ( \Delta u)-Q_b(\nabla (\Delta u))\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}}\cdot \left( \sum _{T\in \mathcal {T}_h}\Vert e_0-Q_be_0\Vert ^2_{\partial T}\right) ^{\!\!\frac{1}{2}} \nonumber \\&\quad \le Ch^{k-1}\ \Vert u\Vert _{k+2} {|||} e_h{|||}, \end{aligned}$$

(4.9)

where the $H^{k+2}$-norm of $u$ is used because the estimate in Lemma 7.9 is not optimal in terms of the mesh parameter $h$.

The third and fourth terms can be estimated by using the Cauchy–Schwarz inequality and the estimates (7.7) and (7.8) in Lemma 7.4 as follows

$$\begin{aligned} \left| \sum _{T\in \mathcal {T}_h}h^{-1}_T\langle \nabla Q_0u\cdot \mathbf {n}_e-Q_b (\nabla u\cdot \mathbf {n}_e), \nabla e_0 \cdot \mathbf {n}_e-e_n\rangle _{\partial T}\right| \le Ch^{k-1}\Vert u\Vert _{k+1} |\!|\!| e_h |\!|\!|\qquad \qquad \end{aligned}$$

(4.10)

and

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h}h^{-3}_T\langle Q_b Q_0u-Q_bu, Q_b e_0-e_b\rangle _{\partial T}\right| \le Ch^{k-1}\Vert u\Vert _{k+1} |\!|\!| e_h |\!|\!|. \end{aligned}$$

(4.11)

Substituting (4.8)–(4.11) into (4.6) gives

$$\begin{aligned} |\!|\!| e_h |\!|\!| ^2\le Ch^{k-1}\ \Vert u\Vert _{k+2} |\!|\!| e_h |\!|\!| , \end{aligned}$$

which implies (4.5) and hence completes the proof. $\square$

5 Error Estimates in $L^2$

In this section, we shall establish some error estimates for all three components of the error function $e_h$ in the standard $L^2$ norm.

First of all, let us derive an error estimate for the first component of the error function $e_h$ by applying the usual duality argument in the finite element analysis. To this end, we consider the problem of seeking $\varphi$ such that

$$\begin{aligned} \Delta ^2 \varphi&= e_0, \quad \mathrm{in} \ \Omega , \\ \nonumber \varphi&= 0,\quad \ \mathrm{on}\ \partial \Omega , \\ \nonumber \frac{\partial \varphi }{\partial \mathbf n }&= 0,\quad \ \mathrm{on}\ \partial \Omega . \end{aligned}$$

(5.1)

Assume that the dual problem has the $H^4$ regularity property in the sense that the solution function $\varphi \in H^4$ and there exists a constant $C$ such that

$$\begin{aligned} \Vert \varphi \Vert _4\le C\Vert e_0\Vert . \end{aligned}$$

(5.2)

Theorem 5.1

Let $u_h\in V_h$ be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order $k\ge 2$. Let $k_0=\min \{3, k\}$. Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that $u\in H^{k+2}(\Omega )$ and the dual problem (5.1) has the $H^4$ regularity. Then, there exists a constant $C$ such that

$$\begin{aligned} \Vert u_0-Q_0 u \Vert \le Ch^{k+k_0-2}\Vert u\Vert _{k+1}, \end{aligned}$$

(5.3)

which means we have a sub-optimal order of convergence for $k=2$ and optimal order of convergence for $k\ge 3$.

Proof

Testing (5.1) by error function $e_0$ and then using the integration by parts gives

$$\begin{aligned} \Vert e_0\Vert ^2&= (\Delta ^2 \varphi , e_0) \\&= \sum _{T\in \mathcal {T}_h} (\Delta \varphi , \Delta e_0)_T+\sum _{T\in \mathcal {T}_h} \langle \nabla (\Delta \varphi )\cdot \mathbf n , e_0\rangle _{\partial T}-\sum _{T\in \mathcal {T}_h} \langle \Delta \varphi , \nabla e_0\cdot \mathbf n \rangle _{\partial T} \\&= \sum _{T\in \mathcal {T}_h} (\Delta \varphi , \Delta e_0)_T+\sum _{T\in \mathcal {T}_h} \langle \nabla (\Delta \varphi )\cdot \mathbf n , e_0-e_b\rangle _{\partial T} \\&-\sum _{T\in \mathcal {T}_h} \langle \Delta \varphi , (\nabla e_0-e_n\mathbf n _e)\cdot \mathbf n \rangle _{\partial T}, \end{aligned}$$

where we have used the fact that $e_n$ and $e_b$ vanishes on the boundary of the domain $\Omega$. By letting $u=\varphi$ and $v_0=e_h$ in (4.3), we can rewrite the above equation as follows

$$\begin{aligned} \Vert e_0\Vert ^2&= (\Delta _w Q_h\varphi ,\Delta _w e_h)_h+\sum _{T\in \mathcal {T}_h} \langle (\nabla (\Delta \varphi )-\nabla (\mathbb {Q}_h\Delta \varphi )\cdot \mathbf n , e_0-e_b\rangle _{\partial T} \\&-\sum _{T\in \mathcal {T}_h} \langle \Delta \varphi -\mathbb {Q}_h\Delta \varphi , (\nabla e_0-e_n\mathbf n _e)\cdot \mathbf n \rangle _{\partial T}. \end{aligned}$$

Next, by letting $v=Q_h \varphi$, from the error equation (4.1), we have

$$\begin{aligned} (\Delta _w Q_h\varphi ,\Delta _w e_h)_h&= \sum _{T\in \mathcal {T}_h} \langle (\Delta u- \mathbb {Q}_h \Delta u, (\nabla Q_0\varphi )\cdot \mathbf n -Q_b (\nabla \varphi \cdot \mathbf n _e)\mathbf n _e\cdot \mathbf n \rangle _{\partial T} \\&-\sum _{T\in \mathcal {T}_h} \langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf n , Q_0\varphi -Q_b\varphi \rangle _{\partial T} \\&-\,s(e_h, Q_h \varphi )+s(Q_h u, Q_h \varphi ). \end{aligned}$$

Combining the two equations above gives

$$\begin{aligned} \Vert e_0\Vert ^2&= \sum _{T\in \mathcal {T}_h} \langle (\nabla (\Delta \varphi )-\nabla (\mathbb {Q}_h\Delta \varphi )\cdot \mathbf n , e_0-e_b\rangle _{\partial T} \nonumber \\ \nonumber&-\sum _{T\in \mathcal {T}_h} \langle \Delta \varphi -\mathbb {Q}_h\Delta \varphi , (\nabla e_0\cdot \mathbf n _e -e_n)\cdot \mathbf n \rangle _{\partial T} \\ \nonumber&+\sum _{T\in \mathcal {T}_h} \langle (\Delta u- \mathbb {Q}_h \Delta u, (\nabla Q_0\varphi )\cdot \mathbf n -Q_b (\nabla \varphi \cdot \mathbf n _e)\mathbf n _e\cdot \mathbf n \rangle _{\partial T} \\ \nonumber&-\sum _{T\in \mathcal {T}_h} \langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf n , Q_0\varphi -Q_b\varphi \rangle _{\partial T} \\&-\,s(e_h, Q_h \varphi )+s(Q_h u, Q_h \varphi ). \end{aligned}$$

(5.4)

From the Cauchy–Schwarz inequality and Lemma 7.4, we can estimate the six terms on the right-hand side of the identity above as follows.

For the first term, it follows from Lemmas 7.4, 7.9 and the fact $k_0=\min \{k, 3\}\le 3$ that

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h} \langle (\nabla (\Delta \varphi )-\nabla (\mathbb {Q}_h\Delta \varphi ))\cdot \mathbf n , e_0-e_b\rangle _{\partial T}\right| \nonumber \\ \nonumber&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T^3\Vert \nabla (\Delta \varphi )-\nabla (\mathbb {Q}_h\Delta \varphi )\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in \mathcal {T}_h}h_T^{-3}\Vert Q_b e_0-e_b\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \\ \nonumber&\qquad +\,\left( \sum _{T\in \mathcal {T}_h}\Vert \nabla (\Delta \varphi )-Q_b \nabla (\Delta \varphi )\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in \mathcal {T}_h}\Vert e_0-Q_b e_0\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \\ \nonumber&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T^3\Vert \nabla (\Delta \varphi )-\nabla (\mathbb {Q}_h\Delta \varphi )\Vert _{\partial T}^2\right) ^{\frac{1}{2}} |\!|\!| e_h |\!|\!| \\ \nonumber&\qquad +\,\,C\lambda \left( \sum _{T\in \mathcal {T}_h}\Vert \nabla (\Delta \varphi )-Q_b \nabla (\Delta \varphi )\Vert _{\partial T}^2\right) ^{\frac{1}{2}}\cdot h^{-\frac{1}{2}}\Vert e_0\Vert \\ \nonumber&\qquad +\,\,C\left( \sum _{T\in \mathcal {T}_h}\Vert \nabla (\Delta \varphi )-Q_b \nabla (\Delta \varphi )\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \cdot h^{\frac{3}{2}}|\!|\!| e_h |\!|\!| \\ \nonumber&\quad \le Ch^{k_0-1}(\Vert \varphi \Vert _{k_0+1}+h\delta _{k_0,2}\Vert \varphi \Vert _4) |\!|\!| e_h |\!|\!| + C\lambda h^{\frac{1}{2}}\Vert \varphi \Vert _4\cdot h^{-\frac{1}{2}}\Vert e_0\Vert \\ \nonumber&\qquad +\,\,Ch^{k_0-\frac{5}{2}}(\Vert \varphi \Vert _{k_0+1}+h\delta _{k_0,2}\Vert \varphi \Vert _4) \cdot h^{\frac{3}{2}}|\!|\!| e_h |\!|\!| \\&\quad \le C h^{k_0-1}\Vert \varphi \Vert _4 |\!|\!| e_h|\!|\!| + C \lambda \Vert \varphi \Vert _4 \Vert e_0\Vert . \end{aligned}$$

(5.5)

For the second term, it follows from (7.3) with $m=k_0$ that

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h} \langle \Delta \varphi -\mathbb {Q}_h\Delta \varphi , (\nabla e_0\cdot \mathbf n _e-e_n)\cdot \mathbf n \rangle _{\partial T}\right| \nonumber \\ \nonumber&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T\Vert \Delta \varphi -\mathbb {Q}_h\Delta \varphi \Vert ^2_{\partial T} \right) ^{\frac{1}{2}}\left( \sum _{T\in \mathcal {T}_h}h_T^{-1}\Vert \nabla e_0\cdot \mathbf n _e-e_n\Vert ^2_{\partial T}\right) ^{\frac{1}{2}} \\&\quad \le Ch^{k_0-1}\Vert \varphi \Vert _{k_0+1} |\!|\!| e_h |\!|\!| \le C h^{k_0-1}\Vert \varphi \Vert _4 |\!|\!| e_h |\!|\!|. \end{aligned}$$

(5.6)

As to the third term, it follows from Cauchy–Schwarz inequality and Lemma 7.4 that

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h} \langle \Delta u- \mathbb {Q}_h \Delta u, (\nabla Q_0\varphi )\cdot \mathbf n -Q_b (\nabla \varphi \cdot \mathbf n _e)\mathbf n _e\cdot \mathbf n \rangle _{\partial T}\right| \nonumber \\ \nonumber&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T\Vert \Delta u- \mathbb {Q}_h \Delta u\Vert ^2_{\partial T} \right) ^{\frac{1}{2}}\left( \sum _{T\in \mathcal {T}_h}h_T^{-1}\Vert (\nabla Q_0\varphi )\cdot \mathbf n -Q_b (\nabla \varphi \cdot \mathbf n _e)\Vert ^2_{\partial T}\right) ^{\frac{1}{2}} \\&\quad \le Ch^{k-1}\Vert u\Vert _{k+1}h^{k_0-1}\Vert \varphi \Vert _{k_0+1} \le Ch^{k+k_0-2}\Vert u\Vert _{k+1}\Vert \varphi \Vert _{4}. \end{aligned}$$

(5.7)

For the forth term, by using Lemma 7.3, we have

$$\begin{aligned}&\left| \sum _{T\in \mathcal {T}_h} \langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf n , Q_0\varphi -Q_b\varphi \rangle _{\partial T}\right| \nonumber \\ \nonumber&\quad \le \left( \sum _{T\in \mathcal {T}_h}h_T^3\Vert \nabla (\Delta u-\mathbb {Q}_h\Delta u)\Vert ^2_{\partial T} \right) ^{\frac{1}{2}}\left( \sum _{T\in \mathcal {T}_h}h_T^{-3}\Vert Q_0\varphi -\varphi \Vert ^2_{\partial T}\right) ^{\frac{1}{2}} \nonumber \\&\quad \le Ch^{k-1}(\Vert u\Vert _{k+1}+h\delta _{k,2}\Vert u\Vert _4)h^{t_0-1}\Vert \varphi \Vert _{k_0+1}\nonumber \\&\quad \le Ch^{k-1}(\Vert u\Vert _{k+k_0-2}+h\delta _{k,2}\Vert u\Vert _4)\Vert \varphi \Vert _{4}. \end{aligned}$$

(5.8)

As to the fifth term, we also use the Cauchy–Schwarz inequality and Lemma 7.4 to obtain

$$\begin{aligned} |s(e_h, Q_h \varphi )|&\le \left| \sum _{T\in \mathcal {T}_h}h_T^{-1} \langle \nabla e_0\cdot \mathbf n _e-e_n, \nabla Q_0\varphi \cdot \mathbf n _e-Q_b (\nabla \varphi \cdot \mathbf n _e)\rangle _{\partial T}\right| \nonumber \\ \nonumber&+ \left| \sum _{T\in \mathcal {T}_h}h_T^{-3} \langle Q_b e_0-e_b, Q_b Q_0\varphi - Q_b\varphi \rangle _{\partial T}\right| \\&\le Ch^{k_0-1}\Vert \varphi \Vert _4 |\!|\!| e_h |\!|\!|. \end{aligned}$$

(5.9)

The last term can be estimated as follows

$$\begin{aligned} |s(Q_h u, Q_h \varphi )|&\le \left| \sum _{T\in \mathcal {T}_h}h_T^{-1} \langle (\nabla Q_0u\cdot \mathbf n _e-Q_b (\nabla u\cdot \mathbf n _e), (\nabla Q_0\varphi \cdot \mathbf n _e-Q_b (\nabla \varphi \cdot \mathbf n _e)\rangle _{\partial T}\right| \nonumber \\ \nonumber&+ \left| \sum _{T\in \mathcal {T}_h}h_T^{-3} \langle Q_b Q_0 u- Q_b u, Q_b Q_0\varphi - Q_b\varphi \rangle _{\partial T}\right| \\ \nonumber&\le Ch^{k-1}\Vert u\Vert _{k+1} h^{k_0-1} \Vert \varphi \Vert _{k_0+1} \\&\le Ch^{k+k_0-2}\Vert u\Vert _{k+1} \Vert \varphi \Vert _{4}. \end{aligned}$$

(5.10)

Substituting all the six estimates into (5.4) we obtain

$$\begin{aligned} \Vert e_0\Vert ^2\le&Ch^{k+k_0-2}(\Vert u\Vert _{k+1}+h\delta _{k,2}\Vert u\Vert _4)\Vert \varphi \Vert _{4} \\&+\,C h^{k_0-1}\Vert \varphi \Vert _4 |\!|\!| e_h |\!|\!|+C \lambda \Vert \varphi \Vert _4 \Vert e_0\Vert . \end{aligned}$$

Using the regularity estimate (5.2) and choosing constant $\lambda$ such that $C \lambda \Vert \varphi \Vert _4<\frac{1}{2}\Vert e_0\Vert$, we arrive at

$$\begin{aligned} \Vert e_0\Vert&\le C h^{k_0-1} |\!|\!| e_h |\!|\!| +Ch^{k+k_0-2}(\Vert u\Vert _{k+1}+h\delta _{k,2}\Vert u\Vert _4) \\&\le Ch^{k+k_0-2}\Vert u\Vert _{k+2}. \end{aligned}$$

Together with the $H^2$ error estimate (4.5) we have the desired $L^2$ error estimate (5.3). $\square$

In order to study the error estimates on edges, we shall introduce the edge-based $L^2$ norm here. To keep the consistency of order, the edge-based $L^2$ norm is different from the standard $L^2$ norm.

Definition 5.2

For any function $v$ defined on the edges $\mathcal {E}_h$,

$$\begin{aligned} \Vert v\Vert ^2_{\mathcal {E}_h}&= \sum _{e\in \mathcal {E}_h}h_e \Vert v\Vert ^2_{L^2(e)}, \end{aligned}$$

where $h_e$ is the measure of edge $e\in \mathcal {E}_h$.

Next, we shall derive the estimates for the second and third components of the error function $e_h$.

Theorem 5.3

Let $u_h\in V_h$ be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order $k\ge 2$. Let $k_0=\min \{k,3\}$. Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that $u\in H^{k+2}(\omega )$ and the dual problem (5.1) has the $H^4$ regularity property. Then, there exists a constant $C$ such that

$$\begin{aligned} \Vert u_b-Q_b u \Vert _{\mathcal {E}_h}&\le Ch^{k+k_0-2}\Vert u\Vert _{k+2}, \end{aligned}$$

(5.11)

$$\begin{aligned} \Vert u_n-Q_b (\nabla u_0\cdot \mathbf{n}_e)\Vert _{\mathcal {E}_h}&\le Ch^{k+k_0-3}\Vert u\Vert _{k+2}. \end{aligned}$$

(5.12)

Proof

It is obvious that

$$\begin{aligned} \Vert e_b\Vert ^2_{L^2(e)}\le 2(\Vert Q_be_0\Vert ^2_{L^2(e)}+\Vert Q_be_0-e_b\Vert ^2_{L^2(e)}). \end{aligned}$$

Summing over all edges, we have

$$\begin{aligned} \quad \Vert u_b-Q_b u \Vert ^2_{\mathcal {E}_h}&= \sum _{e\in \mathcal {E}_h}h_e \Vert u_b-Q_b u\Vert ^2_{L^2(e)}\nonumber \\ \nonumber&\le 2\left( \sum _{e\in \mathcal {E}_h}h_e\Vert Q_be_0\Vert ^2_{L^2(e)} +\sum _{e\in \mathcal {E}_h}h_e\Vert Q_be_0-e_b\Vert ^2_{L^2(e)}\right) \\&\le C\left( \sum _{T\in \mathcal {T}_h}h_T\Vert Q_be_0\Vert ^2_{L^2(\partial T)} +\sum _{T\in \mathcal {T}_h}h_T\Vert Q_be_0-e_b\Vert ^2_{L^2(\partial T)}\right) .\qquad \qquad \end{aligned}$$

(5.13)

We shall discuss the two terms separately. For the first part, by applying the trace inequality (7.1), the inverse inequality (7.2) and the error estimate for $e_0$ in Theorem 5.1, we have

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}h_T\Vert Q_be_0\Vert ^2_{L^2(\partial T)}&\le \sum _{T\in \mathcal {T}_h}h_T\Vert e_0\Vert ^2_{L^2(\partial T)}\nonumber \\ \nonumber&\le C\sum _{T\in \mathcal {T}_h}\left( \Vert e_0\Vert ^2_{L^2(T)}+h_T^2 \Vert \nabla e_0\Vert ^2_{L^2(T)}\right) \\ \nonumber&\le C\sum _{T\in \mathcal {T}_h}\Vert e_0\Vert ^2_{L^2(T)}\nonumber \\&\le Ch^{2k+2k_0-4}\Vert u\Vert _{k+2}^2. \end{aligned}$$

(5.14)

For the second part, we use the trip-bar norm to handle the second part.

$$\begin{aligned} \sum _{T\in \mathcal {T}_h}h_T\Vert Q_be_0-e_b\Vert ^2_{L^2(\partial T)}&\le h^4\sum _{T\in \mathcal {T}_h}h_T^{-3}\Vert Q_be_0-e_b\Vert ^2_{L^2(\partial T)}\le h^4|\!|\!|e_h|\!|\!|^2\\ \nonumber&\le Ch^{2k+2k_0-4}\Vert u\Vert _{k+2}^2. \end{aligned}$$

(5.15)

Combining the above two estimates gives the desired error estimate (5.11).

Similarly, we establish the error estimates for $e_n$.

$$\begin{aligned} \Vert e_n\Vert ^2_{\mathcal {E}_h}&= \sum _{e\in \mathcal {E}_h}h_e\Vert e_n\Vert ^2_{L^2(e)}\nonumber \\ \nonumber&\le C\left( \sum _{T\in \mathcal {T}_h}h_T\Vert \nabla e_0\cdot \mathbf{n}_e\Vert _{\partial T}+\sum _{T\in \mathcal {T}_h}h_T\Vert \nabla e_0\cdot \mathbf{n}_e-e_n\Vert _{\partial T}\right) \\ \nonumber&\le C\left( \sum _{T\in \mathcal {T}_h}h_T\Vert \nabla e_0\Vert _{\partial T}+ h^2\sum _{T\in \mathcal {T}_h}h_T^{-1}\Vert \nabla e_0\cdot \mathbf{n}_e-e_n\Vert _{\partial T}\right) \\ \nonumber&\le C\left( \sum _{T\in \mathcal {T}_h}\Vert \nabla e_0\Vert _{T}+ h^2|\!|\!|e_h|\!|\!|\right) \\ \nonumber&\le C\left( \sum _{T\in \mathcal {T}_h}h_T^{-2}\Vert e_0\Vert _{T}+ h^2|\!|\!|e_h|\!|\!|\right) \\&\le C\left( h^{2k+2k_0-6}+h^{2k}\right) \Vert u\Vert _{k+2}^2. \end{aligned}$$

(5.16)

Thus, we have

$$\begin{aligned} \Vert e_n\Vert _{\mathcal {E}_h}\le Ch^{k+k_0-3}\Vert u\Vert _{k+2}, \end{aligned}$$

which completes the proof. $\square$

6 Numerical Results

In this section, we would like to report some numerical results for the weak Galerkin finite element method proposed and analyzed in previous sections. Here we use the following finite element space

$$\begin{aligned} \tilde{V}_h=\{v=\{v_0, v_b, v_n \mathbf n _e \}, v_0\in P_2(T), v_b, v_n\in P_{1}(e), T\in \mathcal {T}_h, e\subset \mathcal {E}_h\}. \end{aligned}$$

For any given $v=\{v_0, v_b, v_n \mathbf n _e \}\in \tilde{V}_h$ and $\varphi \in P_0(T)$, we compute the discrete weak Laplacian $\Delta _w v$ on each element $T$ as a function in $P_0(T)$ as follows

$$\begin{aligned} \qquad (\Delta _{w} v, \varphi )_T=(v_0, \Delta \varphi )_T-\langle v_b, \nabla \varphi \cdot \mathbf n \rangle _{\partial T}+\langle v_n\mathbf n _e\cdot \mathbf n ,\varphi \rangle _{\partial T}, \end{aligned}$$

which could be simplified as

$$\begin{aligned} \qquad (\Delta _{w} v, \varphi )_T=\langle v_n\mathbf n _e\cdot \mathbf n ,\varphi \rangle _{\partial T}. \end{aligned}$$

The error for the weak Galerkin solution is measured in six norms defined as follows:

$$\begin{aligned} \begin{array}{c} \displaystyle |\!|\!| e_h |\!|\!|^2=\sum _{T\in \mathcal {T}_h} \left( \int _T|\Delta _w v_h|^2 dT+ h_T^{-1}\int _{\partial T}|(\nabla v_0)\cdot \mathbf n _e-v_n |^2 ds\right. \\ \displaystyle \left. +\, h_T^{-3}\int _{\partial T}(Q_bv_0-v_b)^2 ds\right) \qquad \hbox {(A discrete } \ H^2\ \hbox {norm)}\nonumber \\ \displaystyle \Vert Q_0 v-v_0\Vert ^2=\sum _{T\in \mathcal {T}_h} \int _T|Q_0 v- v_0|^2 dT \qquad \hbox {(Element based} \ L^2\ \hbox {norm)} \\ \displaystyle \Vert Q_b v-v_b\Vert ^2_{\mathcal {E}_h}=\sum _{e\in \mathcal {E}_h} h_e\int _e|Q_b v- v_b|^2 ds \qquad \hbox {(Edge based} \ L^2\ \hbox {norm for} \ v_b) \\ \displaystyle \Vert Q_b v-v_n\Vert ^2_{\mathcal {E}_h}=\sum _{e\in \mathcal {E}_h} h_e\int _e|Q_b v- v_n|^2 ds \qquad \hbox {(Edge based} \ L^2\ \hbox {norm for} \ v_n) \\ \displaystyle \Vert Q_b v-v_b\Vert _{\infty }=\max _{e\in \mathcal {E}_h}\{|Q_b v- v_b|\} \qquad \hbox {(Edge based} \ L^\infty \ \hbox {norm for} \ v_b ) \\ \displaystyle \Vert Q_b v-v_n\Vert _{\infty }=\max _{e\in \mathcal {E}_h}\{|Q_b (\nabla u_0\cdot \mathbf{n}_e)- v_n|\} \qquad \hbox {(Edge based} \ L^\infty \ \hbox {norm for} \ v_n ) \end{array} \end{aligned}$$

Example 6.1

Consider the biharmonic problem (1.1)–(1.3) in the square domain $\Omega =(0,1)^2$. It has the analytic solution $u(x)=x^2(1-x)^2y^2(1-y)^2$, and the right hand side function $f$ in (1.1) is computed to match the exact solution. The mesh size is denoted by $h=1/n$. Table 1 shows that the convergence rates for the WG-FEM solution in the $H^2$ and $L^2$ norms are of order $O(h)$ and $O(h^2)$ when $k=2$, respectively.

Table 2 shows that the errors and orders of Example 6.1 in $L^2$ and $L^\infty$ for $e_b$. The numerical results are in consistency with theory for these two cases.

Table 3 shows that the errors and orders of Example 6.1 in $L^2$ and $L^\infty$ for $e_n$. The numerical results are in consistency with theory for these two cases.

Table 1 Errors and orders of Example 6.1 in $H^2$ and $L^2$ with $k=2$

Table 2 Errors and orders of Example 6.1 in $L^2$ and $L^\infty$ for $e_b$ with $k=2$

Table 3 Errors and orders of Example 6.1 in $L^2$ and $L^\infty$ for $e_n$ with $k=2$

In Tables 4, 5 and 6 we investigate the same problem for $k = 3$. Table 4 shows that the convergence rates for the WG-FEM solution in the $H^2$ and $L^2$ norms are of order $O(h^2)$ and $O(h^4)$. Tables 5 and 6 show the errors and orders in $L^2$ and $L^\infty$ for $e_b$ and $e_n$, which are also consistent with theoretical conclusions.

Table 4 Errors and orders of example 6.1 in $H^2$ and $L^2$ with $k=3$

Table 5 Errors and orders of example 6.1 in $L^2$ and $L^\infty$ for $e_b$ with $k=3$

Table 6 Errors and orders of example 6.1 in $L^2$ and $L^\infty$ for $e_n$ with $k=3$

Example 6.2

Consider the biharmonic problem (1.1)–(1.3) in the square domain $\Omega =(0,1)^2$. It has the analytic solution $u(x)=\sin (\pi x)\sin (\pi y)$, and the right hand side function $f$ in (1.1) is computed accordingly.

The numerical results are presented in Tables 7, 8, 9, 10, 11 and 12 which confirm the theory developed in previous sections.

Table 7 Errors and orders of Example 6.2 in $H^2$ and $L^2$ with $k=2$

Table 8 Errors and orders of Example 6.2 in $L^2$ and $L^\infty$ for $e_b$ with $k=2$

Table 9 Errors and orders of Example 6.2 in $L^2$ and $L^\infty$ for $e_n$ with $k=2$

Table 10 Errors and orders of example 6.2 in $H^2$ and $L^2$ with $k=3$

Table 11 Errors and orders of example 6.2 in $L^2$ and $L^\infty$ for $e_b$ with $k=3$

Table 12 Errors and orders of example 6.2 in $L^2$ and $L^\infty$ for $e_n$ with $k=3$

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.