A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

Abstract

A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete \(H^2\) norm are established for the corresponding finite element solutions. Error estimates in the \(L^2\) norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.

Introduction

We consider the biharmonic equation of the form

$$\begin{aligned} \Delta ^2 u=\, & f\quad \text{ in }\;\varOmega , \end{aligned}$$
(1)
$$\begin{aligned} u=\, & 0\quad \text{ on }\;\partial \varOmega , \end{aligned}$$
(2)
$$\begin{aligned} \frac{\partial u}{\partial n}=\, & 0\quad \text{ on }\;\partial \varOmega , \end{aligned}$$
(3)

where \(\varOmega\) is a bounded polytopal domain in \({\mathbb {R}}^d\), \(d=2\) or 3.

The weak formulation of the boundary value problem (2) and (3) is seeking \(u\in H^2_0(\varOmega )\) satisfying

$$\begin{aligned} (\Delta u, \Delta v) = (f, v),\qquad \forall v\in H_0^2(\varOmega ). \end{aligned}$$
(4)

It is known that \(H^2\)-conforming methods require \(C^1\)-continuous piecewise polynomials on simplicial meshes, which impose difficulty in practical computation, cf. [4, 5, 17, 18]. Due to the complexity in the construction of \(C^1\)-continuous elements, \(H^2\)-conforming finite element methods are rarely used in practice for solving the biharmonic equation, especially in 3D [18, 19].

Due to the flexibility of discontinuous finite element methods in element constructions and in mesh generations, many finite element methods have been developed using totally discontinuous polynomials for the biharmonic equation to avoid the construction of \(H^2\)-conforming elements. Here, we are only interested in interior penalty discontinuous Galerkin (IPDG) methods, since the proposed method shares the same finite element spaces with the IPDG method. For the biharmonic equation, IPDG finite element methods have been studied in [1,2,3, 6, 7, 10]. One obvious disadvantage of discontinuous finite element methods is that their formulations are rather complicated, which are often necessary to guarantee the well posedness and the convergence of the methods. For example, the symmetric IPDG method for the biharmonic equation with homogenous boundary conditions has the following formulation [1, 2]:

$$\begin{aligned} &(\Delta u_h,\Delta v)_{{{\mathcal {T}}}_h}+\sum \int _e(\{\nabla \Delta u_h\}\cdot [v]+\{\nabla \Delta v\}\cdot [u_h]){\rm{d}}s\nonumber \\&+\sum \int _e(\{\Delta u_h\}\cdot [\nabla v]+\{\Delta v\}\cdot [\nabla u_h]){\rm{d}}s\nonumber \\&+ \sum \int _e\left( \frac{\sigma }{h^3} [u_h]\cdot [v]+\frac{\tau }{h}[\nabla u_h][\nabla v]\right) {\rm{d}}s=(f,v), \end{aligned}$$
(5)

where \(\sigma\) and \(\tau\) are two parameters that need to be tuned.

In this work, a modified weak Galerkin (MWG) finite element method is introduced for the biharmonic equation. This new MWG method shares the same finite element space with the IPDG method, but with a much simpler, parameter-independent, formulation:

$$\begin{aligned} (\Delta _w u_h,\ \Delta _w v)_{{{\mathcal {T}}}_h}+\sum _{T\in {{\mathcal {T}}}_h}(h^{-3}_T{\langle }[u_h], [v]{\rangle }_{\partial T}+ h^{-1}_T{\langle }[\nabla u_h], [\nabla v]{\rangle }_{\partial T})=(f,\;v). \end{aligned}$$
(6)

The MWG finite element method was first introduced in [13] for second-order elliptic equations. Optimal order error estimates in a discrete \(H^2\) norm for the polynomial degree \(k\ge 2\) and in the \(L^2\) norm for \(k>2\) are established for the corresponding finite element solutions. Numerical results are provided to confirm the theories.

The corresponding work for second-order elliptic equations was done in [14,15,16].

Finite Element Method

Let \({{\mathcal {T}}}_h\) be a partition of the domain \(\varOmega\) consisting of polygons in two dimensions or polyhedra in three dimensions satisfying a set of conditions defined in [11]. Denote by \({{\mathcal {E}}}_h\) the set of all edges or flat faces in \(\mathcal{T}_h\), and let \({{\mathcal {E}}}_h^0={{\mathcal {E}}}_h\backslash \partial \varOmega\) be the set of all interior edges or flat faces.

For simplicity, we adopt the following notations:

$$\begin{aligned} (v,w)_{{{\mathcal {T}}}_h}\,\,= & \sum _{T\in {{\mathcal {T}}}_h}(v,w)_T=\sum _{T\in {{\mathcal {T}}}_h}\int _T vw {\text {d}}\mathbf{x},\\ {\langle }v,w{\rangle }_{\partial {{\mathcal {T}}}_h}\,\,= & \sum _{T\in {{\mathcal {T}}}_h} {\langle }v,w{\rangle }_{\partial T}=\sum _{T\in {{\mathcal {T}}}_h} \int _{\partial T}vw {\text {d}}s. \end{aligned}$$

Let \(P_k(K)\) consist of all the polynomials degree less or equal to k defined on K.

We define a finite element space \(V_h\) for \(k\ge 2\) as follows:

$$\begin{aligned} V_h=\left\{ v\in L^2(\varOmega ):\ v|_{T}\in P_{k}(T),\;\; T\in {{\mathcal {T}}}_h \right\} . \end{aligned}$$
(7)

Let \(T_1\) and \(T_2\) be two polygons/polyhedrons sharing e, and \(\mathbf{n}_1\) and \(\mathbf{n}_2\) be the unit outward normal vectors of \(T_1\) and \(T_2\) on e, respectively. Let v and \(\mathbf{q}\) be a scalar and a vector valued functions, respectively. The jumps [v] and \([\mathbf{q}]\) are defined as

$$\begin{aligned} {[{\it{v}}]}=v|_{T_1}\mathbf{n}_1+ v|_{T_2}\mathbf{n}_2, \quad [\mathbf{q}]=\mathbf{q}|_{T_1}\cdot \mathbf{n}_1+ \mathbf{q}|_{T_2}\cdot \mathbf{n}_2, \end{aligned}$$
(8)

and the averages \(\{v\}\) and \(\{\mathbf{q}\}\) are defined as

$$\begin{aligned} \{v\}=\frac{1}{2}(v|_{T_1}+v|_{T_2}), \quad \{\mathbf{q}\}=\frac{1}{2} (\mathbf{q}|_{T_1}+ \mathbf{q}|_{T_2}). \end{aligned}$$
(9)

If e is on \(\partial \varOmega\), then

$$\begin{aligned} \{v\}=0,\quad \{\mathbf{q}\}=0,\quad [v]= v\mathbf{n},\quad [\mathbf{q}]=\mathbf{q}\cdot \mathbf{n}. \end{aligned}$$
(10)

The MWG finite element method for the biharmonic equation (1)–(3) is defined as follows.

Definition 1

A numerical approximation for (1)–(3) can be obtained by seeking \(u_h\in V_h\) satisfying the following equation:

$$\begin{aligned} a(u_h,v)\equiv (\Delta _w u_h,\ \Delta _w v)_{{{\mathcal {T}}}_h}+s(u_h,v)=(f,\;v), \quad \forall v\in V_h, \end{aligned}$$
(11)

where

$$\begin{aligned} s(u_h,v)=\sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T{\langle }[\nabla u_h], [\nabla v]{\rangle }_{\partial T}+\sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T{\langle }[u_h], [v]{\rangle }_{\partial T}. \end{aligned}$$

The \(\Delta _w\) in (11) is called the weak Laplacian. A weak derivative was introduced in [11, 12] for weak functions in weak Galerkin methods and was modified in [9, 13] for MWG methods. A weak Laplacian \(\Delta _{w}\) is defined for any \(v\in V_h \cup H^2(\varOmega )\) as the unique polynomial \(\Delta _{w}v \in \prod _{T\in {{\mathcal {T}}}_h} P_{k-2}(T)\) that satisfies the following equation:

$$\begin{aligned} (\Delta _{w} v, \ \varphi )_T = ( v, \ \Delta \varphi )_T-{\langle }\{v\},\ \nabla \varphi \cdot \mathbf{n}{\rangle }_{\partial T}+{\langle }\{\nabla v\}\cdot \mathbf{n}, \ \varphi {\rangle }_{\partial T}\end{aligned}$$
(12)

for all \(\varphi \in P_{k-2}(T)\).

Lemma 1

Let\(\phi \in H_0^2(\varOmega )\). Then, on any\(T\in {{\mathcal {T}}}_h\),

$$\begin{aligned} \Delta _{w} \phi = \mathbb {Q}_h (\Delta \phi ), \end{aligned}$$
(13)

where\(\mathbb {Q}_h\)is a locally defined\(L^2\)projection operator onto\(P_{k-2}(T)\), on each element\(T\in {{\mathcal {T}}}_h\).

Proof

By the definition of \(\Delta _w\) in (12), one has for any \(\tau \in P_{k-2}(T)\),

$$\begin{aligned} (\Delta _{w} \phi ,\ \tau )_T\,\,= \, & (\phi ,\ \Delta \tau )_T + \langle \{\nabla \phi \}\cdot \mathbf{n}, \;\tau \rangle _{{\partial T}}-\langle \{\phi \},\ \nabla \tau \cdot \mathbf{n}\rangle _{{\partial T}}\\\,\,= \, & (\phi , \Delta \tau )_T + \langle \nabla \phi \cdot \mathbf{n},\ \tau \rangle _{\partial T}-\langle \phi ,\ \nabla \tau \cdot \mathbf{n}\rangle _{{\partial T}}\\\,\,= \, & (\Delta \phi ,\ \tau )_T=(\mathbb {Q}_h\Delta \phi ,\ \tau )_T, \end{aligned}$$

which implies that

$$\begin{aligned} \Delta _{w} \phi = \mathbb {Q}_h (\Delta \phi ). \end{aligned}$$
(14)

We complete the proof.

Well Posedness and Error Equation

In this section, we will prove the well posedness of the method (11) and derive an equation that the error \(e_h=u-u_h\) is satisfied.

First, we define a semi-norm \(||| \cdot |||\) as

$$\begin{aligned} |||v|||^2=a(v,v)=(\Delta _wv,\Delta _wv)_{{{\mathcal {T}}}_h}+s(v,v). \end{aligned}$$
(15)

Then we introduce a discrete \(H^2\) norm as follows:

$$\begin{aligned} \Vert v\Vert _{2,h} = \left( \sum _{T\in {{\mathcal {T}}}_h}\left( \Vert \Delta v\Vert _T^2+h_T^{-3} \Vert [v]\Vert ^2_{\partial T}+h_T^{-1}\Vert [\nabla v]\Vert ^2_{\partial T}\right) \right) ^{\frac{1}{2}}. \end{aligned}$$
(16)

Lemma 2

There exists a positive constantCsuch that for any\(v\in V_h\), we have

$$\begin{aligned} C\Vert v\Vert _{2,h}\le |||v|||. \end{aligned}$$
(17)

Proof

It is easy to see that the following equations hold true for \(\{v\}\) and \(\{\nabla v\}\) defined in (9) and (10) on T with \(e\subset {\partial T}\):

$$\left\{\begin{array} {l} \Vert v-\{v\}\Vert _e=\Vert [v]\Vert _e\quad \mathrm{if} \;e\subset \partial \varOmega ,\\ \Vert v-\{v\}\Vert _e=\frac{1}{2}\Vert [v]\Vert _e\;\; \mathrm{if} \;e\in {{\mathcal {E}}}_h^0, \end{array}\right.$$
(18)

and

$$\left\{\begin{array} {l} \Vert (\nabla v-\{\nabla v\})\cdot \mathbf{n}\Vert _e=\Vert [\nabla v]\Vert _e\quad \mathrm{if} \;e\subset \partial \varOmega ,\\ \Vert(\nabla v-\{\nabla v\})\cdot \mathbf{n}\Vert _e=\frac{1}{2}\Vert [\nabla v]\Vert _e\;\; \mathrm{if} \;e\in {{\mathcal {E}}}_h^0. \end{array}\right.$$
(19)

For any \(v\in V_h\), it follows from the definition of the weak Laplacian (12) and integration by parts that

$$\begin{aligned} (\Delta _{w} v, \ \varphi )_T& \,= \, ( v, \ \Delta \varphi )_T-{\langle }\{v\},\ \nabla \varphi \cdot \mathbf{n}{\rangle }_{\partial T}+{\langle }\{\nabla v\}\cdot \mathbf{n}, \ \varphi {\rangle }_{\partial T}\nonumber \\&\,=\,-(\nabla v, \ \nabla \varphi )_T+{\langle }v-\{v\},\ \nabla \varphi \cdot \mathbf{n}{\rangle }_{\partial T}+{\langle }\{\nabla v\}\cdot \mathbf{n}, \ \varphi {\rangle }_{\partial T}\nonumber \\&\,=\,(\Delta v, \ \varphi )_T+{\langle }v-\{v\},\ \nabla \varphi \cdot \mathbf{n}{\rangle }_{\partial T}\nonumber \\&\quad +{\langle }(\{\nabla v\}-\nabla v)\cdot \mathbf{n}, \ \varphi {\rangle }_{\partial T}. \end{aligned}$$
(20)

By letting \(\varphi =\Delta v\) in (20) and using (18)–(19), we arrive at

$$\begin{aligned} \Vert \Delta v\Vert ^2_T\,\,= & (\Delta v, \ \Delta _w v)_T-{\langle }v-\{v\}, \nabla (\Delta v)\cdot \mathbf{n}{\rangle }_{\partial T}\\&- {\langle }(\{\nabla v\}-\nabla v)\cdot \mathbf{n}, \ \Delta v{\rangle }_{\partial T}\\\le &\,\, \Vert \Delta v\Vert _T \Vert \Delta _w v\Vert _T+ \Vert v-\{v\}\Vert _{\partial T}\Vert \nabla (\Delta v)\Vert _{\partial T}\\&+\Vert (\{\nabla v\}-\nabla v)\cdot \mathbf{n}\Vert _{\partial T}\Vert \Delta v\Vert _{\partial T}\\\le &\,\, C(\Vert \Delta _w v\Vert _T + h_T^{-3/2}\Vert [v]\Vert _{\partial T}+h_T^{-1/2}\Vert [\nabla v]\Vert _{\partial T}) \Vert \Delta v\Vert _T, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \Delta v\Vert _T\le &\,\, C(\Vert \Delta _w v\Vert _T + h_T^{-3/2}\Vert [v]\Vert _{\partial T}+h_T^{-1/2}\Vert [\nabla v]\Vert _{\partial T}). \end{aligned}$$

Taking the summation of the above inequality over \(T\in {{\mathcal {T}}}_h\) gives (17).

Lemma 3

The finite element scheme (11) has a unique solution.

Proof

It suffices to show that the solution of (11) is trivial if \(f=0\). It follows from (11) that

$$\begin{aligned} |||u_h|||^2=(\Delta _w u_h,\Delta _w u_h)_{{{\mathcal {T}}}_h}+s(u_h,u_h)=0. \end{aligned}$$

The above equation and (17) imply \(\Vert u_h\Vert _{2,h}=0\), i.e.,

$$\begin{aligned} \sum _{T\in {{\mathcal {T}}}_h} \Vert \Delta u_h\Vert _T^2=0,\;\; \sum _{T\in {{\mathcal {T}}}_h} h_T^{-3}\Vert [u_h]\Vert ^2_{\partial T}=0,\;\ \sum _{T\in {{\mathcal {T}}}_h} h_T^{-1}\Vert [\nabla u_h]\Vert ^2_{\partial T}=0. \end{aligned}$$
(21)

Thus, we have that \(u_h\) is a smooth harmonic function on \(\varOmega\) and \(u_h=0\) on \(\partial \varOmega\). Thus, we have \(u_h=0\), which completes the proof.

Next, we derive an error equation for \(e_h=u-u_h\).

Lemma 4

For any\(v\in V_h\), we have

$$\begin{aligned} a(e_h,v)=(\Delta _we_h,\Delta _wv)_{{{\mathcal {T}}}_h}+s(e_h,v)=\ell _1(u,v)+\ell _2(u,v), \end{aligned}$$
(22)

where

$$\begin{aligned} \ell _1(u,v)\,\,= & \langle \nabla (\mathbb {Q}_h\Delta u-\Delta u)\cdot \mathbf{n}, v-\{v\}\rangle _{{\partial T}_h},\\ \ell _2(u,v)\,\,= & \langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla v-\{\nabla v\})\cdot \mathbf{n}\rangle _{{\partial T}_h}. \end{aligned}$$

Proof

Testing (1) by \(v\in V_h\), we arrive at, by using the fact that \(\sum \limits_{T\in {{\mathcal {T}}}_h}\)\(\langle \nabla (\Delta u) \cdot \mathbf{n}, \{v\}\rangle _{\partial T}=0\) and \(\sum \limits_{T\in {{\mathcal {T}}}_h}\langle \Delta u, \{\nabla v\}\cdot \mathbf{n}\rangle _{\partial T}=0\) and integration by parts,

$$\begin{aligned} (f,v)&=(\Delta ^2u, v)_{{{\mathcal {T}}}_h}\nonumber \\&=(\Delta u,\Delta v)_{{{\mathcal {T}}}_h} -\langle \Delta u, \nabla v\cdot \mathbf{n}\rangle _{{\partial T}_h} + \langle \nabla (\Delta u)\cdot \mathbf{n}, v\rangle _{{\partial T}_h}\nonumber \\&=(\Delta u,\Delta v)_{{{\mathcal {T}}}_h} -\langle \Delta u, (\nabla v-\{\nabla v\})\cdot \mathbf{n}\rangle _{{\partial T}_h}\nonumber \\&\quad + \langle \nabla (\Delta u)\cdot \mathbf{n}, v-\{v\}\rangle _{{\partial T}_h}. \end{aligned}$$
(23)

Using (13), integration by parts and the definition of the weak Laplacian (12), we have

$$\begin{aligned} (\Delta u, \Delta v)_{{{\mathcal {T}}}_h}&=(\mathbb {Q}_h\Delta u, \Delta v)_{{{\mathcal {T}}}_h} \\&= (v, \Delta (\mathbb {Q}_h\Delta u))_T + \langle \nabla v\cdot \mathbf{n},\ \mathbb {Q}_h\Delta u\rangle _{{\partial T}}-\langle v, \nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf{n}\rangle _{{\partial T}}\nonumber \\&=(\Delta _w v,\ \mathbb {Q}_h\Delta u)_T-\langle v-\{v\}, \nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf{n}\rangle _{{\partial T}} \\&\quad +\langle (\nabla v-\{\nabla v\})\cdot \mathbf{n}, \mathbb {Q}_h\Delta u\rangle _{{\partial T}}\nonumber \\&=(\Delta _w u,\ \Delta _w v)_T- \langle v-\{v\}, \nabla (\mathbb {Q}_h\Delta u)\cdot \mathbf{n}\rangle _{{\partial T}}\\&\quad +\langle (\nabla v-\{\nabla v\})\cdot \mathbf{n}, \mathbb {Q}_h\Delta u\rangle _{{\partial T}}. \end{aligned}$$

Combining the above two equations gives

$$\begin{aligned} (f,v)&=(\Delta ^2u, v)_{{{\mathcal {T}}}_h}\nonumber \\&=(\Delta _w u,\ \Delta _w v)_{{{\mathcal {T}}}_h}-\langle v-\{v\}, \nabla (\mathbb {Q}_h\Delta u-\Delta u)\cdot \mathbf{n}\rangle _{{\partial T}_h}\nonumber \\&\quad - \langle (\nabla v-\{\nabla v\})\cdot \mathbf{n}, \Delta u-\mathbb {Q}_h\Delta u \rangle _{{\partial T}}. \end{aligned}$$
(24)

It follows from the above equation and the fact \(s(u,v)=0\) that

$$\begin{aligned} a(u,v)=(\Delta _w u,\ \Delta _w v)_{{{\mathcal {T}}}_h}+s(u,v)=(f,v)+\ell _1(u,v)+\ell _2(u,v). \end{aligned}$$

The error equation follows from subtracting (11) from the above equation,

$$\begin{aligned} a(e_h,v)=(\Delta _w e_h ,\ \Delta _w v)_{{{\mathcal {T}}}_h}+s(e_h,v)=\ell _1(u,v)+\ell _2(u,v). \end{aligned}$$

We have proved the lemma.

An Error Estimate in \(H^2\)

In this section, we will obtain the optimal order error estimate for \(e_h\) in \(|||\cdot |||\) norm. Let \(Q _h\) be the element-wise defined \(L^2\) projection onto \(P_{k}(T)\) on each element T.

First, we need the following trace inequality. For any function \(\varphi \in H^1(T)\), the following trace inequality holds true (see [11] for details):

$$\begin{aligned} \Vert \varphi \Vert _{e}^2 \le C \left( h_T^{-1} \Vert \varphi \Vert _T^2 + h_T \Vert \nabla \varphi \Vert _{T}^2\right) . \end{aligned}$$
(25)

Lemma 5

Let\(k\ge 2\)and\(w\in H^{\max \{k+1,4\}}(\varOmega )\). There exists a constantCsuch that the following estimates hold true:

$$\begin{aligned} \left( \sum _{T\in {{\mathcal {T}}}_h} h_T\Vert \Delta w-\mathbb {Q}_h\Delta w\Vert _{\partial T}^2\right) ^{\frac{1}{2}}&\le C h^{k-1}\Vert w\Vert _{k+1}, \end{aligned}$$
(26)
$$\begin{aligned} \left( \sum _{T\in {{\mathcal {T}}}_h} h_T^3\Vert \nabla (\Delta w-\mathbb {Q}_h\Delta w)\Vert _{\partial T}^2\right) ^{\frac{1}{2}}&\le Ch^{k-1}(\Vert w\Vert _{k+1} + \delta _{k,2}\Vert w\Vert _4). \end{aligned}$$
(27)

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

The above lemma can be proved by using the trace inequality (25) and the definition of \(\mathbb {Q}_h\). The proof can also be found in [8].

Lemma 6

Let\(w\in H^{\max \{k+1,4\}}(\varOmega )\), and\(v\in V_h\). There exists a constantCsuch that

$$\begin{aligned} |\ell _1(w, v)|\le &\,\, C h^{k-1}(\Vert w\Vert _{k+1} + \delta _{k,2} \Vert w\Vert _4)|||v|||, \end{aligned}$$
(28)
$$\begin{aligned} |\ell _2(w, v)|\le &\,\, Ch^{k-1}|w|_{k+1}|||v|||. \end{aligned}$$
(29)

Proof

Using the Cauchy–Schwartz inequality, (26) and (27), we have

$$\begin{aligned} \ell _1(w,v)\,\,= & \left| \sum _{T\in {{\mathcal {T}}}_h}\langle \nabla (\Delta w-\mathbb {Q}_h\Delta w)\cdot \mathbf{n}, v-\{v\}\rangle _{\partial T}\right| \nonumber \\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^3\Vert \nabla (\Delta w-\mathbb {Q}_h\Delta w)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert v-\{v\}\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\nonumber \\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^3\Vert \nabla (\Delta w-\mathbb {Q}_h\Delta w)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [v]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\nonumber \\\le &\,\, C h^{k-1}(\Vert w\Vert _{k+1} + \delta _{k,2} \Vert w\Vert _4) |||v|||, \end{aligned}$$
(30)

and

$$\begin{aligned} \ell _2(w,v)\,\,= \, & \left| \sum _{T\in {{\mathcal {T}}}_h} \langle \Delta w-\mathbb {Q}_h\Delta w, (\nabla v-\{\nabla v\})\cdot \mathbf{n}\rangle _{\partial T}\right| \nonumber \\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h} h_T\Vert \Delta w-\mathbb {Q}_h\Delta w\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h} h_T^{-1} \Vert [\nabla v]\Vert _{\partial T}^2\right) ^{\frac{1}{2}}\nonumber \\\le &\,\, C h^{k-1}\Vert w\Vert _{k+1} |||v|||. \end{aligned}$$
(31)

We complete the proof.

Lemma 7

Let\(w\in H^{\max \{k+1,4\}}(\varOmega )\). Then,

$$\begin{aligned} |||w-Q_hw|||\le Ch^{k-1}|w|_{k+1}. \end{aligned}$$
(32)

Proof

For any \(T\in {{\mathcal {T}}}_h\), it follows from (12), integration by parts, (25) and inverse inequality that

$$\begin{aligned} \Vert \Delta _w(w-Q_hw)\Vert _T^2& \,= \,(\Delta _w(w-Q_hw), \Delta _w(w-Q_hw))_{T}\\& \,= \,(w-Q_hw, \Delta (\Delta _w(w-Q_hw)))_{T} \\&\quad -{\langle }\{w-Q_hw\}, \nabla (\Delta _w(w-Q_hw))\cdot \mathbf{n}{\rangle }_{\partial T}\\&\quad +{\langle }\{\nabla w-\nabla Q_hw\}\cdot \mathbf{n}, \Delta _w(w-Q_hw){\rangle }_{{\partial T}}\\&\le C(h_T^{-2}\Vert w-Q_hw\Vert _T+h_T^{-3/2}\Vert w-Q_hw\Vert _{\partial T}\\&\quad +h_T^{-1/2}\Vert \nabla w-\nabla Q_hw\Vert _{\partial T})\Vert \Delta _w(w-Q_hw)\Vert _T\\&\le Ch^{k-1}|w|_{k+1, T}\Vert \Delta _w(w-Q_hw)\Vert _T. \end{aligned}$$

It implies

$$\begin{aligned} \Vert \Delta _w(w-Q_hw)\Vert _T\le &\,\, Ch^{k-1}|w|_{k+1, T}. \end{aligned}$$

Taking the summation over \(T\in {{\mathcal {T}}}_h\), we have

$$\begin{aligned} \left( \sum _{T\in {{\mathcal {T}}}_h}\Vert \Delta _w(w-Q_hw)\Vert ^2_T\right) ^{1/2}\le &\,\, Ch^{k-1}|w|_{k+1}. \end{aligned}$$

By the trace inequality (25) and the definition of \(Q_h\), one has

$$\begin{aligned} \left( \sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T \Vert [\nabla (w-Q_hw)]\Vert ^2_{\partial T}\right) ^{1/2}\le &\,\, Ch^{k-1}|w|_{k+1} \end{aligned}$$

and

$$\begin{aligned} \left( \sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T \Vert [w-Q_hw]\Vert ^2_{\partial T}\right) ^{1/2}\le &\,\, Ch^{k-1}|w|_{k+1}. \end{aligned}$$

Combining all the estimates above, we have proved the lemma.

Theorem 1

Let\(u_h\in V_h\)be the finite element solution arising from (11). Assume that the exact solution\(u\in H^{\max \{k+1,4\}}(\varOmega )\). Then, there exists a constantCsuch that

$$\begin{aligned} |||u-u_h|||\le Ch^{k-1}\left( \Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _{4}\right) . \end{aligned}$$
(33)

Proof

Let \(\epsilon _h=Q_hu-u_h\in V_h\). Then, it is straightforward to obtain

$$\begin{aligned} |||e_h|||^2\,\,= \, & a(e_h,e_h)\nonumber \\\,\,= \, & a(e_h,\epsilon _h)+a(e_h,u-Q_hu). \end{aligned}$$
(34)

Letting \(v=\epsilon _h\) in (22) and using (28)–(29) and (32), we have

$$\begin{aligned} |a(e_h,\epsilon _h)|\le &\,\, |\ell _1(u,\epsilon _h)|+|\ell _2(u,\epsilon _h)|\nonumber \\\le &\,\, Ch^{k-1}(\Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _{4})|||\, \epsilon _h|||\nonumber \\\le &\,\, Ch^{k-1}(\Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _{4})(|||u-Q_hu|||+|||u-u_h|||)\nonumber \\\le &\,\, Ch^{2(k-1)}(\Vert u\Vert ^2_{k+1}+ \delta _{k,2}\Vert u\Vert ^2_{4})+\frac{1}{4} |||e_h|||^2. \end{aligned}$$
(35)

To bound the second term on the right-hand side of (34), we have by the Cauchy-Schwarz inequality and (32),

$$\begin{aligned} |a(e_h,u-Q_hu)|\le &\,\, C|||u-Q_hu|||\, |||e_h|||\nonumber \\\le &\,\, Ch^{2(k-1)}\Vert u\Vert ^2_{k+1}+\frac{1}{4}|||e_h|||^2. \end{aligned}$$
(36)

Combining the estimates (35) and (36) with (34), we arrive at

$$\begin{aligned} |||e_h|||\le Ch^{k-1}\left( \Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _{4}\right) , \end{aligned}$$

which completes the proof.

Error Estimates in the \(L^2\) Norm

In this section, we will obtain an error bound for the finite element solution \(u_h\) in the \(L^2\) norm.

The dual problem considered has the following form:

$$\begin{aligned} \Delta ^2w\,\,= \, & e_h\quad \text{ in }\;\varOmega , \end{aligned}$$
(37)
$$\begin{aligned} w\,\,= \, & 0\quad \text{ on }\;\partial \varOmega , \end{aligned}$$
(38)
$$\begin{aligned} \nabla w\cdot \mathbf{n}\,\,= \, & 0\quad \text{ on }\;\partial \varOmega . \end{aligned}$$
(39)

Assume that the \(H^{4}\) regularity holds,

$$\begin{aligned} \Vert w\Vert _4\le C\Vert e_h\Vert . \end{aligned}$$
(40)

Theorem 2

Let\(u_h\in V_h\)be the finite element solution of (11). Assume that the exact solution\(u\in H^{k+1}(\varOmega )\)and (40) hold true. Then, there exists a constantCsuch that

$$\begin{aligned} \Vert u-u_h\Vert \le Ch^{k+1-\delta _{k,2}}(\Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _4). \end{aligned}$$
(41)

Proof

Testing (37) by \(e_h\) and using the fact that \(\sum \limits_{T\in {{\mathcal {T}}}_h}\langle \nabla (\Delta w)\cdot \mathbf{n}, \{e_h\}\rangle _{\partial T}=0\) and \(\sum \limits_{T\in {{\mathcal {T}}}_h}\langle \Delta w, \{\nabla e_h\}\cdot \mathbf{n}\rangle _{\partial T}=0\) and integration by parts, we arrive at

$$\begin{aligned} \Vert e_h\Vert ^2\,\,= \, & (\Delta ^2 w, e_h)_{{{\mathcal {T}}}_h}\nonumber \\\,\,= \, & (\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h} -\langle \Delta w, \nabla e_h\cdot \mathbf{n}\rangle _{{\partial T}_h} + \langle \nabla (\Delta w)\cdot \mathbf{n}, e_h\rangle _{{\partial T}_h}\nonumber \\\,\,= \, & (\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h} -\langle \Delta w, (\nabla e_h-\{\nabla e_h\})\cdot \mathbf{n}\rangle _{{\partial T}_h}\nonumber \\&+ \langle \nabla (\Delta w)\cdot \mathbf{n}, e_h-\{e_h\}\rangle _{{\partial T}_h}\nonumber \\\,\,= \, & (\mathbb {Q}_h\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h}+(\Delta w-\mathbb {Q}_h\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h}\nonumber \\&-\langle \Delta w, (\nabla e_h-\{\nabla e_h\})\cdot \mathbf{n}\rangle _{{\partial T}_h} + \langle \nabla (\Delta w)\cdot \mathbf{n}, e_h-\{e_h\}\rangle _{{\partial T}_h}. \end{aligned}$$

It follows from integration by parts, the definition of the weak Laplacian (12) and (13),

$$\begin{aligned} (\mathbb {Q}_h\Delta w, \Delta e_h)_{{{\mathcal {T}}}_h}\,\,= \, & (e_h, \Delta (\mathbb {Q}_h\Delta w))_T + \langle \nabla e_h\cdot \mathbf{n},\ \mathbb {Q}_h\Delta w\rangle _{{\partial T}} \\&-\langle e_h, \nabla (\mathbb {Q}_h\Delta w)\cdot \mathbf{n}\rangle _{{\partial T}}\nonumber \\\,\,= \, & (\Delta _w e_h,\ \mathbb {Q}_h\Delta w)_T-\langle e_h-\{e_h\}, \nabla (\mathbb {Q}_h\Delta w)\cdot \mathbf{n}\rangle _{{\partial T}}\\&+\langle (\nabla e_h-\{\nabla e_h\})\cdot \mathbf{n}, \mathbb {Q}_h\Delta w\rangle _{{\partial T}}\\\,\,= \, & (\Delta _w w,\ \Delta _w e_h)_T- \langle e_h-\{e_h\}, \nabla (\mathbb {Q}_h\Delta w)\cdot \mathbf{n}\rangle _{{\partial T}}\\& +\langle (\nabla e_h-\{\nabla e_h\})\cdot \mathbf{n}, \mathbb {Q}_h\Delta w\rangle _{{\partial T}}. \end{aligned}$$

Combining the two equations above implies that

$$\begin{aligned} \Vert e_h\Vert ^2\,\,= \, & (\Delta _w w,\ \Delta _w e_h)_{{{\mathcal {T}}}_h}+(\Delta w-\mathbb {Q}_h\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h}\\&+\ell _1(w,e_h)+\ell _2(w,e_h). \end{aligned}$$

By simple manipulation and (22), we have

$$\begin{aligned} (\Delta _w w,\ \Delta _w e_h)_{{{\mathcal {T}}}_h}& \,= \,(\Delta _w Q_hw,\ \Delta _w e_h)_{{{\mathcal {T}}}_h}+(\Delta _w (w-Q_hw),\ \Delta _w e_h)_{{{\mathcal {T}}}_h}\\& \,= \,\ell _1(u,Q_hw)+\ell _2(u, Q_hw)-s(e_h,Q_hw)\\&\quad +(\Delta _w (w-Q_h w),\ \Delta _w e_h)_{{{\mathcal {T}}}_h}. \end{aligned}$$

Combining the two equations above implies that

$$\begin{aligned} \Vert e_h\Vert ^2\,\,=\, & \ell _1(u,Q_hw)+\ell _2(u, Q_hw)-s(e_h,Q_hw)+(\Delta _w e_h,\ \Delta _w (w-Q_hw))_{{{\mathcal {T}}}_h}\\&+(\Delta w-\mathbb {Q}_h\Delta w,\Delta e_h)_{{{\mathcal {T}}}_h} +\ell _1(w,\epsilon _h)+\ell _2(w,\epsilon _h)\\\,\,=\, & I_1+I_2+I_3+I_4+I_5+I_6+I_7. \end{aligned}$$

Next, we will estimate all the terms on the right-hand side of the above equation. Using the Cauchy–Schwartz inequality, (26)–(27), (18) and (30), we have

$$\begin{aligned} I_1&\,=\,\ell _1(u,Q_hw)=\left| \sum _{T\in {{\mathcal {T}}}_h}\langle \nabla (\Delta u-\mathbb {Q}_h\Delta u)\cdot \mathbf{n}, Q_hw-\{Q_hw\}\rangle _{\partial T}\right| \\&\le \left( \sum _{T\in {{\mathcal {T}}}_h}h^3_T\Vert \nabla (\Delta u-\mathbb {Q}_h\Delta u)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [Q_hw]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\\&\le \left( \sum _{T\in {{\mathcal {T}}}_h}h^3_T\Vert \nabla (\Delta u-\mathbb {Q}_h\Delta u)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [Q_hw-w]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\\&\le C h^{k+1-\delta _{k,2}}\left( \Vert u\Vert _{k+1}+ \delta _{k,2}\Vert u\Vert _{4}\right) \Vert w\Vert _4. \end{aligned}$$

Similarly, by the Cauchy–Schwartz inequality, (26)–(27), (19) and (31), we have

$$\begin{aligned} I_2\,\,= \, & \ell _2(u,Q_hw)=\left| \sum _{T\in {{\mathcal {T}}}_h} \langle \Delta u-\mathbb {Q}_h\Delta u, (\nabla Q_hw-\{\nabla Q_hw\})\cdot \mathbf{n})\rangle _{\partial T}\right| \\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h} h^{-1}_T\Vert \Delta u-\mathbb {Q}_h\Delta u\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h} h_T \Vert [\nabla Q_hw-\nabla w]\Vert _{\partial T}^2\right) ^{\frac{1}{2}}\\\le &\,\, C h^{k+1-\delta _{k,2}}\Vert u\Vert _{k+2} \Vert w\Vert _4. \end{aligned}$$

It follows from the trace inequality (25), (17), the definition of \(Q_h\) and (33),

$$\begin{aligned} I_3\,\,= \, & s(e_h,Q_hw)=\sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T{\langle }[\nabla e_h], [\nabla Q_hw]{\rangle }_{\partial T}+\sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T{\langle }[e_h], [Q_hw]{\rangle }_{\partial T}\\\,\,= & \sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T{\langle }[\nabla e_h], [\nabla ( Q_hw-w)]{\rangle }_{\partial T}+\sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T{\langle }[e_h], [Q_hw-w]{\rangle }_{\partial T}\\\le &\,\, C\left( \left( \sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T\Vert [\nabla e_h]\Vert _{\partial T}^2\right) ^{1/2}\left( \sum _{T\in {{\mathcal {T}}}_h}h_T\Vert \nabla (Q_hw-w)\Vert _{\partial T}^2\right) ^{1/2}\right. \\&\left. \quad +\left( \sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T\Vert [e_h]\Vert _{\partial T}^2\right) ^{1/2}\left( \sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T\Vert Q_hw-w\Vert _{\partial T}^2\right) ^{1/2}\right) \\\le &\,\, C|||e_h|||\left( \left( \sum _{T\in {{\mathcal {T}}}_h}h^{-1}_T\Vert \nabla (Q_hw-w)\Vert _{\partial T}^2\right) ^{1/2}+\left( \sum _{T\in {{\mathcal {T}}}_h}h^{-3}_T\Vert Q_hw-w\Vert _{\partial T}^2\right) ^{1/2}\right) \\\le &\,\, C h^{k+1-\delta _{k,2}}\Vert u\Vert _{k+2} \Vert w\Vert _4. \end{aligned}$$

The estimates (33) and (32) give

$$\begin{aligned} I_4\le C h^{k+1}\Vert u\Vert _{k+1} \Vert w\Vert _4. \end{aligned}$$

To estimate \(I_5\), we need to bound \(\Vert \Delta e_h\Vert _T\). By (17), (32), (33) and the definition of \(Q_h\), we have

$$\begin{aligned} \sum _{T\in {{\mathcal {T}}}_h}\Vert \Delta e_h\Vert ^2_{T}\le &\,\, \sum _{T\in {{\mathcal {T}}}_h}\Vert \Delta \epsilon _h\Vert ^2_{T}+\sum _{T\in {{\mathcal {T}}}_h}\Vert \Delta (u-Q_hu)\Vert ^2_{T}\\\le &\,\, C(h^{k-1}\Vert u\Vert _{k+1}+|||\epsilon _h|||^2)\\\le &\,\, C(h^{k-1}\Vert u\Vert _{k+1}+|||e_h|||^2+|||Q_hu-u|||^2)\\\le &\,\, Ch^{k-1}\Vert u\Vert _{k+1}. \end{aligned}$$

The above estimate and the definition of \(\mathbb {Q}_h\) imply that

$$\begin{aligned} I_5\le &\,\, C h^{k+1}\Vert u\Vert _{k+1} \Vert w\Vert _4. \end{aligned}$$

Using the Cauchy–Schwartz inequality, (18), (26), (17), (32) and (33), we have

$$\begin{aligned} I_6\,\,= \, & \ell _1(w,e_h)=\left| \sum _{T\in {{\mathcal {T}}}_h}\langle \nabla (\Delta w-\mathbb {Q}_h\Delta w)\cdot \mathbf{n}, e_h-\{e_h\}\rangle _{\partial T}\right| \\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^3\Vert \nabla (\Delta w-\mathbb {Q}_h\Delta w)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert e_h-\{e_h\}\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\\\le &\,\, \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^3\Vert \nabla (\Delta w-\mathbb {Q}_h\Delta w)\Vert _{\partial T}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [e_h]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\\\le &\,\, C h^2\Vert w\Vert _4\left( |||\epsilon _h|||+\left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [Q_hu-u]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\right) \\\le &\,\, C h^2\Vert w\Vert _4\left( |||e_h|||+|||Q_hu-u|||+\left( \sum _{T\in {{\mathcal {T}}}_h}h_T^{-3}\Vert [Q_hu-u]\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\right) \\\le &\,\, C h^{k+1}\Vert u\Vert _{k+1} \Vert w\Vert _4. \end{aligned}$$

Similarly, we obtain

$$\begin{aligned} I_7\le C h^{k+1}\Vert u\Vert _{k+1} \Vert w\Vert _4. \end{aligned}$$

Combining all the estimates above yields

$$\begin{aligned} \Vert e_h\Vert ^2 \le C h^{k+1-\delta _{k,2}}\Vert u\Vert _{k+1} \Vert w\Vert _4. \end{aligned}$$

It follows from the above inequality and the regularity assumption (40),

$$\begin{aligned} \Vert e_h\Vert \le C h^{k+1-\delta _{k,2}}\Vert u\Vert _{k+1}. \end{aligned}$$

We have completed the proof.

Numerical Experiments

We solve the following biharmonic equation by the MWG finite element methods:

$$\begin{aligned} \Delta ^2 u =f , \quad (x,y)\in \varOmega =(0,1)^2 \end{aligned}$$
(42)

with the boundary conditions \(u=g\) and \(\nabla u\cdot \mathbf {n}=\nabla g\cdot \mathbf {n}\) on \(\partial \varOmega\), where g is defined below in (43). We choose \(f=\Delta ^2 g\) in (42) so that the exact solution is

$$\begin{aligned} u=g=\rm{e}^{x+y}. \end{aligned}$$
(43)
Fig. 1
figure1

The first three levels of triangular grids

In the first computation, the first three levels of triangular grids are plotted in Fig. 1. The error and the order of convergence for the MWG method are listed in Table 1. Here on triangular grids, we compute the weak Laplacian \(\Delta _w v\) of \(P_k\) finite element functions by \(P_{k-2}\) polynomials. The numerical results confirm the convergence theory.

Table 1 Error profiles and convergence rates on triangular grids (Fig. 1)
Fig. 2
figure2

The first three levels of polygonal grids

In the next computation, we use a family of polygonal grids (with quadrilaterals, pentagons and of hexagons) shown in Fig. 2. We use the polynomial \(P_{k-2}\) for computing the weak Laplacian on such polygonal meshes. The rate of convergence is listed in Table 2. The convergence history confirms the theory.

Table 2 Error profiles and convergence rates on polygonal grids (Fig. 2)

We do not have a theorem on the optimal order convergence for the \(H^1\)-seminorm for the \(P_2\) MWG finite element. It seems that on polygonal meshes, the \(P_2\) MWG finite element does not have an order 2 convergence in the \(H^1\)-seminorm, from Table 2. This is why the \(H^1\) error on triangular meshes is much smaller than that on polygonal meshes, cf. Tables 1 and 2.

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Correspondence to Shangyou Zhang.

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M. Cui was supported in part by the National Natural Science Foundation of China (Grant No. 11571026) and the Beijing Municipal Natural Science Foundation of China (Grant No. 1192003). Xiu Ye was supported in part by the National Science Foundation Grant DMS-1620016.

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Cui, M., Ye, X. & Zhang, S. A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes. Commun. Appl. Math. Comput. (2020). https://doi.org/10.1007/s42967-020-00071-9

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Keywords

  • Finite element methods
  • Weak Laplacian
  • Biharmonic equations
  • Polytopal meshes

Mathematics Subject Classification

  • 65N15
  • 65N30
  • 76D07