A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

Article

Abstract

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.

Keywords

Weak Galerkin Finite element methods Weak gradient The Reissner–Mindlin plate Polygonal partitions

Mathematics Subject Classification

Primary 65N15 65N30 Secondary 35J50

1 Introduction

The Reissner–Mindlin equations with clamped boundary seek a rotation $${\varvec{\theta }}$$ and transverse displacement w satisfying
\begin{aligned} -\nabla \cdot (\mathbb {C}{\epsilon ({\varvec{\theta }})})-\lambda t^{-2}(\nabla w-{\varvec{\theta }})= & {} 0,\quad \text{ in }\;\Omega , \end{aligned}
(1.1)
\begin{aligned} -\nabla \cdot \lambda t^{-2}(\nabla w-{\varvec{\theta }})= & {} g,\quad \text{ in }\;\Omega , \end{aligned}
(1.2)
\begin{aligned} {\varvec{\theta }}=0, \quad w= & {} 0,\quad \text{ on }\;\partial \Omega , \end{aligned}
(1.3)
where the load g is in $$L^2(\Omega )$$, $$\mathbb {C}$$ is the tensor of bending moduli, $$\lambda$$ is the shear correction factor, t is the plate thickness, and $$\epsilon ({\varvec{\theta }})=\frac{1}{2} (\nabla {\varvec{\theta }}+\nabla {\varvec{\theta }}^T)$$. For simplicity, we assume $$\lambda =1$$ in the mathematical development.
A typical weak formulation for (1.1)–(1.3) seeks $$({\varvec{\theta }},w)\in [H_0^1(\Omega )]^2\times H_0^1(\Omega )$$ such that
\begin{aligned} (\mathbb {C}\epsilon ({\varvec{\theta }}),\;{\epsilon ({\varvec{\eta }})})+t^{-2}(\nabla w-{\varvec{\theta }}, \nabla v-{\varvec{\eta }})=(g,\;v) \end{aligned}
(1.4)
for all $$({\varvec{\eta }}, v)\in [H_0^1(\Omega )]^2\times H^1_0(\Omega )$$.

The Reissner–Mindlin model is frequently used by engineers for plates and shells of small to moderate thickness. This model is well known for its “locking” phenomenon so that many numerical approximations behave poorly when the thickness parameter t tends to zero. There have been extensive efforts on seeking locking free finite element schemes. Some special locking free elements have been developed on triangles, rectangles and quadrilaterals [3, 7, 11, 23]. Several numerical methods have been developed for the Reissner–Mindlin model based on the mixed formulations [6, 8, 9, 10, 18, 22]. Recently, discontinuous Galerkin finite element methods have been applied successfully to the Reissner–Mindlin equations [1, 2, 12, 13].

Weak Galerkin (WG) methods refer to a general finite element technique for partial differential equations where the differential operators are approximated by weakly-defined discrete analogues and the smoothness be enhanced by using stabilizers or smoothers. The WG method was first introduced in [20, 21] for second order elliptic equations and has been further developed for other PDEs with the most recent primal-dual formulation [19]. Like the discontinuous Galerkin method, weak Galerkin methods make use of discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygon or polyhedron.

In this paper, we shall develop a weak Galerkin finite element method for the Reissner–Mindlin equations. Compared with the existing numerical schemes, our method has a unique advantage by allowing more general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. This feature is highly desirable in adaptive finite element method because it makes adaptive mesh refinements more effective and local. To the best of our knowledge, all the existing work cited here requires the mesh to be either triangular or quadrilateral in two dimensional space. We have implemented the weak Galerkin scheme and conducted several numerical experiments on polygonal partitions. The numerical results confirm the stability and the theoretical convergence. In addition, our numerical experiments reveal some superconvergence phenomena for both the rotation variable $${\varvec{\theta }}$$ and the transverse displacement w. With respect to fast solving, the weak Galerkin finite element formulation is in the primary form, which yields a positive definite discrete linear system. Piecewise polynomials without any unconventional terms such as “bubbles” are used for both the transverse displacements and the rotations. Error estimates of optimal order with respect to the mesh size h are established in various Sobolev norms.

Although we cannot prove the convergence rate independent of thickness t theoretically, our numerical results show the uniform accuracy of the WG method in plate thickness. A theory on the uniform convergence of the WG method respect to the plate thickness t is open for future studies.

2 Weak Galerkin Finite Element Scheme

Let $${{\mathcal {T}}}_h$$ be a partition of the domain $$\Omega$$ consisting of polygons satisfying a set of conditions specified in [21]. Denote by $${{\mathcal {E}}}_h$$ the set of all edges in $${{\mathcal {T}}}_h$$, and let $${{\mathcal {E}}}_h^0={{\mathcal {E}}}_h\backslash \partial \Omega$$ be the set of all interior edges. For every element $$T\in {\mathcal {T}}_h$$, we denote by $$h_T$$ its diameter and mesh size $$h=\max _{T\in {\mathcal {T}}_h} h_T$$ for $${{\mathcal {T}}}_h$$.

The weak Galerkin methods introduce a new way to define a function v, called weak function, that allows v taking different forms in the interior and on the boundary of the element:
\begin{aligned} v= \left\{ \begin{array}{ll} \displaystyle v_0,&{}\quad \mathrm{in}\; T, \\ \displaystyle v_b,&{}\quad \mathrm{on}\;\partial T. \end{array} \right. \end{aligned}
Since a weak function v is formed by two parts $$v_0$$ and $$v_b$$, we write v as $$v=\{v_0,v_b\}$$ in short without confusion.
For a given integer $$k\ge 1$$, we define two weak Galerkin finite element spaces as follows
\begin{aligned} {\varvec{\Theta }}_h=\left\{ {\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}: {\varvec{\eta }}_0|_T\in [P_k(T)]^2, {\varvec{\eta }}_b|_e\in [P_{k}(e)]^2, e\in {\partial T}, {\varvec{\eta }}_b=0 \text{ on } \partial \Omega \right\} \end{aligned}
(2.1)
and
\begin{aligned} W_h=\{v=\{v_0,v_b\}:\; v_0|_T\in P_k(T), \ v_b|_e\in P_{k-1}(e), \;v_b=0 \text{ on } \partial \Omega \}. \end{aligned}
(2.2)

Definition 2.1

(Weak Symmetric Gradient) For any $${\varvec{\eta }}\in {\varvec{\Theta }}_h$$, a weak symmetric gradient $${\epsilon _w({\varvec{\eta }})}\in [P_{k-1}(T)]^{2\times 2}$$ is defined on each element T such that equation
\begin{aligned} ({\epsilon _w({\varvec{\eta }})}, {\varvec{\tau }})_K = -({\varvec{\eta }}_0,\nabla \cdot {\bar{{\varvec{\tau }}}})_K+ \langle {\varvec{\eta }}_b, {\bar{{\varvec{\tau }}}}\cdot \mathbf{n}\rangle _{\partial K},\quad \forall {\varvec{\tau }}\in [P_{k-1}(K)]^{2\times 2}, \end{aligned}
(2.3)
where $${\bar{{\varvec{\tau }}}}=\frac{1}{2}({\varvec{\tau }}+{\varvec{\tau }}^T)$$.

Definition 2.2

(Weak Gradient) For any $$v\in W_h$$, a weak gradient $$\nabla _{w}v$$ is defined as the unique polynomial $$\nabla _w v \in [P_{k-1}(T)]^2$$ satisfying the following equation
\begin{aligned} (\nabla _w v, \mathbf{q})_T = -(v_0,\nabla \cdot \mathbf{q})_T+ \langle v_b, \mathbf{q}\cdot \mathbf{n}\rangle _{\partial T},\quad \forall \mathbf{q}\in [P_{k-1}(T)]^2. \end{aligned}
(2.4)

For each element $$T\in {\mathcal {T}}_h$$, denote by $$\mathbf{Q}_0$$ and $$Q_0$$ the $$L^2$$ projections from $$[L^2(T)]^2$$ and $$L^2(\Omega )$$ to $$[P_k(T)]^2$$ and $$P_k(T)$$ respectively. Let $$\mathbf{Q}_b$$ and $$Q_b$$ the $$L^2$$ projections from $$[L^2(e)]^2$$ and $$L^2(e)$$ to $$[P_{k}(e)]^2$$ and $$P_{k-1}(e)$$ respectively. Denote by $$\mathbb {Q}_h$$ and $$\mathbf{{\mathcal {Q}}}_h$$ the $$L^2$$ projections onto the local gradient space $$[P_{k-1}(T)]^2$$ and the local symmetric gradient space $$[P_{k-1}(T)]^{2\times 2}$$.

Now we introduce some bilinear forms as for $${\varvec{\phi }}=\{{\varvec{\phi }}_0,{\varvec{\phi }}_b\},{\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ and $$u=\{u_0,u_b\},v=\{v_0,v_b\}\in W_h$$:
\begin{aligned} s_1({\varvec{\phi }},{\varvec{\eta }})= & {} \sum _{T\in {{\mathcal {T}}}_h} h^{-1}\langle {\varvec{\phi }}_0-{\varvec{\phi }}_b,{\varvec{\eta }}_0-{\varvec{\eta }}_b\rangle _{\partial T},\\ a({\varvec{\phi }},{\varvec{\eta }})= & {} \sum _{T\in {{\mathcal {T}}}_h}( \mathbb {C}{\epsilon _w({\varvec{\phi }})}, {\epsilon _w({\varvec{\eta }})})_T+s_1({\varvec{\phi }},{\varvec{\eta }}),\\ s_2(u,v)= & {} \sum _{T\in {{\mathcal {T}}}_h}h^{-1}\langle Q_bu_0-u_b, Q_bv_0-v_b\rangle _{\partial T}. \end{aligned}

Weak Galerkin Algorithm 2.1

A numerical approximation for (1.1) and (1.3) can be obtained by seeking $${\varvec{\theta }}_h=\{{\varvec{\theta }}_0,\;{\varvec{\theta }}_b\}\in {\varvec{\Theta }}_h$$ and $$w_h=\{w_0,w_b\}\in W_h$$ satisfying the following equation for $${\varvec{\eta }}=\{{\varvec{\eta }}_0,\;{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ and $$v=\{v_0,v_b\}\in W_h$$,
\begin{aligned} a({\varvec{\theta }}_h,\;{\varvec{\eta }})+t^{-2}(\nabla _ww_h-\mathbb {Q}_h{\varvec{\theta }}_0, \nabla _wv-\mathbb {Q}_h{\varvec{\eta }}_0)+s_2(w_h,v)=(g,\;v_0). \end{aligned}
(2.5)
Let elements $$T_1$$ and $$T_2$$ have e as a common edge. We define the jump and average of $${\varvec{\eta }}_0$$ with $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ on e as
\begin{aligned}{}[{\varvec{\eta }}_0]_e= \left\{ \begin{array}{ll} {\varvec{\eta }}_0|_{{\partial T_1}}-{\varvec{\eta }}_0|_{{\partial T_2}}, &{}\quad e\in {{\mathcal {E}}}_h^0,\\ {\varvec{\eta }}_0, &{}\quad e\in \partial \Omega , \end{array} \right. \quad \{{\varvec{\eta }}_0\}_e= \left\{ \begin{array}{ll} \frac{1}{2}({\varvec{\eta }}_0|_{{\partial T_1}}+{\varvec{\eta }}_0|_{{\partial T_2}}), &{}\quad e\in {{\mathcal {E}}}_h^0,\\ {\varvec{\eta }}_0, &{}\quad e\in \partial \Omega . \end{array} \right. \end{aligned}
The order of $$T_1$$ and $$T_2$$ is non-essential as long as the difference is taken in a consistent way.

3 Well-Posedness

We start this section by defining two norms. For any $${\varvec{\eta }}\in {\varvec{\Theta }}_h$$, define
\begin{aligned} {|||} {\varvec{\eta }}{|||}_\theta ^2=a({\varvec{\eta }},{\varvec{\eta }}). \end{aligned}
(3.1)
For any $$v\in W_h$$, define
\begin{aligned} {|||} v{|||}_w^2=\sum _{T\in {\mathcal {T}}_h}\Vert \nabla _wv\Vert _T^2+s_2(v,v). \end{aligned}
(3.2)
For any function $$\varphi \in H^1(T)$$, the following trace inequality holds true (see [21] 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}
(3.3)
By simple algebraic manipulation, we have the following inequality,
\begin{aligned} s_1({\varvec{\eta }},{\varvec{\eta }})=\sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert ^2_{\partial T}\ge \frac{1}{2} \left( \sum _{e\in {\mathcal {E}}_h} h^{-1}(\Vert \{{\varvec{\eta }}_0\}-{\varvec{\eta }}_b\Vert _e^2+\Vert [{\varvec{\eta }}_0]\Vert _e^2\right) , \end{aligned}
(3.4)
which implies
\begin{aligned} \sum _{e\in {\mathcal {E}}_h} h^{-1}\Vert [{\varvec{\eta }}_0]\Vert _e^2\le Cs_1({\varvec{\eta }},{\varvec{\eta }})\le C{|||}{\varvec{\eta }}{|||}_\theta . \end{aligned}
(3.5)

Lemma 3.1

For any $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$, we have
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert {\epsilon ({\varvec{\eta }}_0)}\Vert _T^2\le & {} C{|||}{\varvec{\eta }}{|||}_\theta ^2. \end{aligned}
(3.6)

Proof

Using the integration by parts, (2.3), (3.3) and the inverse inequality, we have
\begin{aligned} ({\epsilon ({\varvec{\eta }}_0)},{\epsilon ({\varvec{\eta }}_0)})_T= & {} ({\epsilon ({\varvec{\eta }}_0)},\nabla {\varvec{\eta }}_0)_T\\= & {} -(\nabla \cdot {\epsilon ({\varvec{\eta }}_0)}, {\varvec{\eta }}_0)_T+{\langle }{\epsilon ({\varvec{\eta }}_0)}\cdot \mathbf{n}, {\varvec{\eta }}_0{\rangle }_{\partial T}\\= & {} ({\epsilon _w({\varvec{\eta }})},{\epsilon ({\varvec{\eta }}_0)})_T+{\langle }{\epsilon ({\varvec{\eta }}_0)}\cdot \mathbf{n}, {\varvec{\eta }}_0-{\varvec{\eta }}_b{\rangle }_{\partial T}\\\le & {} C(\Vert {\epsilon _w({\varvec{\eta }})}\Vert _T\Vert {\epsilon ({\varvec{\eta }}_0)}\Vert _T+h^{-1/2} \Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert _{\partial T}\Vert \Vert {\epsilon ({\varvec{\eta }}_0)}\Vert _T). \end{aligned}
The above estimate implies that
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert {\epsilon ({\varvec{\eta }}_0)}\Vert ^2_T\le C\left( \sum _{T\in {\mathcal {T}}_h}\Vert {\epsilon _w({\varvec{\eta }})}\Vert _T^2 +\sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert _{\partial T}^2\right) \le C{|||}{\varvec{\eta }}{|||}_\theta ^2. \end{aligned}
We proved the lemma. $$\square$$

Lemma 3.2

For any $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ and $$v\in W_h$$, we have
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \nabla {\varvec{\eta }}_0\Vert _T^2\le & {} C {|||}{\varvec{\eta }}{|||}_\theta ^2, \end{aligned}
(3.7)
\begin{aligned} \Vert {\varvec{\eta }}_0\Vert\le & {} C{|||}{\varvec{\eta }}{|||}_\theta , \end{aligned}
(3.8)
\begin{aligned} \Vert v_0\Vert\le & {} C{|||} v{|||}_w. \end{aligned}
(3.9)

Proof

For $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$, the following estimate can be found for example in [1, 5],
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \nabla {\varvec{\eta }}_0\Vert _T^2\le C\left( \sum _{T\in {\mathcal {T}}_h}\Vert {\epsilon ({\varvec{\eta }}_0)}\Vert _T^2+\sum _{e\in {\mathcal {E}}_h}h^{-1}\Vert [{\varvec{\eta }}_0]\Vert _e^2\right) . \end{aligned}
(3.10)
Using (3.10), (3.6) and (3.5), we obtain,
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \nabla {\varvec{\eta }}_0\Vert _T^2\le & {} C\left( \sum _{T\in {\mathcal {T}}_h}\Vert {\epsilon ({\varvec{\eta }}_0)}\Vert _T^2+\sum _{e\in {\mathcal {E}}_h}h^{-1}\Vert [{\varvec{\eta }}_0]\Vert _e^2\right) \le C{|||}{\varvec{\eta }}{|||}_\theta ^2. \end{aligned}
To prove (3.8), we need the following estimate which can be found in [14] for example:
\begin{aligned} \Vert {\varvec{\eta }}_0\Vert ^2\le C\left( \sum _{T\in {\mathcal {T}}_h}\Vert \nabla {\varvec{\eta }}_0\Vert _T^2 +\sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert _{\partial T}^2\right) . \end{aligned}
Using the estimate above and (3.7), we proved (3.8). Similarly, we can prove (3.9). $$\square$$

Lemma 3.3

$${|||}\cdot {|||}_\theta$$ and $${|||}\cdot {|||}_w$$ defined in (3.1) and (3.2) provide norms in $${\varvec{\Theta }}_h$$ and $$W_h$$ respectively.

Proof

To prove $${|||}\cdot {|||}_\theta$$ being a norm in $${\varvec{\Theta }}_h$$, one needs to show that $${\varvec{\eta }}=0$$ if $${|||}{\varvec{\eta }}{|||}_\theta =0$$ for any $${\varvec{\eta }}\in {\varvec{\Theta }}_h$$. For a given $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ and $${|||}{\varvec{\eta }}{|||}_\theta =0$$, it follows from (3.8),
\begin{aligned} \Vert {\varvec{\eta }}_0\Vert \le C{|||}{\varvec{\eta }}{|||}_\theta =0. \end{aligned}
Combining the above estimate with $$s_1({\varvec{\eta }},{\varvec{\eta }})=0$$ implies $${\varvec{\eta }}=0$$. Similarly, we can prove that $${|||}\cdot {|||}_w$$ defines a norm in $$W_h$$. $$\square$$

We are ready to justify the well-posedness of the WG formulation (2.5).

Lemma 3.4

The Algorithm 2.1 has a unique solution.

Proof

It suffices to show that zero is the only solution of (2.5) if $$g=0$$. Letting $$({\varvec{\eta }},v)=({\varvec{\theta }}_h,w_h)$$ and $$g=0$$ in (2.5), we have
\begin{aligned} a({\varvec{\theta }}_h,{\varvec{\theta }}_h)+t^{-2}\sum _{T\in {\mathcal {T}}_h}\Vert \nabla _ww_h-\mathbb {Q}_h {\varvec{\theta }}_0\Vert _T^2+\sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert Q_bw_0-w_b\Vert _{{\partial T}}^2=0. \end{aligned}
(3.11)
It follows the fact $$a({\varvec{\theta }}_h,{\varvec{\theta }}_h)=0$$ and (3.7),
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \nabla {\varvec{\theta }}_0\Vert _T^2\le C{|||}{\varvec{\theta }}_h{|||}^2_\theta =Ca({\varvec{\theta }}_h,{\varvec{\theta }}_h)=0, \end{aligned}
which implies that $${\varvec{\theta }}_0$$ is a constant on each $$T\in {\mathcal {T}}_h$$. Combining with $${\varvec{\theta }}_0={\varvec{\theta }}_b$$ on $$e\in {\partial T}$$ and $${\varvec{\theta }}_b=0$$ on $$\partial \Omega$$, we have $${\varvec{\theta }}_h=0$$. It follows from (3.11) and $${\varvec{\theta }}_h=0$$ that
\begin{aligned} 0=\Vert \nabla _ww_h-\mathbb {Q}_h{\varvec{\theta }}_0\Vert ^2=\Vert \nabla _ww_h\Vert ^2. \end{aligned}
Using $$\nabla _w w_h=0$$ and (2.4), we have for any $$\mathbf{q}\in [P_{k-1}(T)]^2$$
\begin{aligned} 0= & {} (\nabla _w w_h,\mathbf{q})_T\\= & {} (\nabla w_0,\mathbf{q})_T-\langle w_0-w_b,\mathbf{q}\cdot \mathbf{n}\rangle _{\partial T}\\= & {} (\nabla w_0,\mathbf{q})_T, \end{aligned}
which implies $$w_0$$ is a constant on $$T\in {\mathcal {T}}_h$$. Since $$w_0=w_b$$ on $${\partial T}$$ and $$w_b=0$$ on $$\partial \Omega$$, we prove $$w_h=0$$. $$\square$$
Let $$({\varvec{\theta }},w)$$ be the solution of (1.1)–(1.3). Define
\begin{aligned} \mathbf{Q}_h{\varvec{\theta }}=\{\mathbf{Q}_0{\varvec{\theta }},\mathbf{Q}_b{\varvec{\theta }}\}\in {\varvec{\Theta }}_h,\quad Q_h w=\{Q_0w,Q_bw\}\in W_h. \end{aligned}

Lemma 3.5

Let $$\mathbf{Q}_h$$, $$Q_h$$, $$\mathbb {Q}_h$$ and $${{\mathcal {Q}}}_h$$ be the $$L^2$$ projection operators as defined. Then, on each element $$T\in {\mathcal {T}}_h$$, we have the following commutative properties
\begin{aligned} \nabla _w (Q_h \varphi )= & {} \mathbb {Q}_h (\nabla \varphi ),\quad \forall \varphi \in H^1(T), \end{aligned}
(3.12)
\begin{aligned} {\epsilon _w(\mathbf{Q}_h{\varvec{\eta }})}= & {} {{\mathcal {Q}}}_h{\epsilon ({\varvec{\eta }})}, \quad \forall {\varvec{\eta }}\in [H^1(T)]^2. \end{aligned}
(3.13)

Proof

Using (2.4), the integration by parts and the definitions of $$Q_h$$ and $$\mathbb {Q}_h$$, we have that for any $$\mathbf{q}\in [P_{k-1}(T)]^2$$
\begin{aligned} (\nabla _w (Q_h \varphi ),\mathbf{q})_T= & {} -(Q_0 \varphi ,\nabla \cdot \mathbf{q})_T +\langle Q_b \varphi ,\mathbf{q}\cdot \mathbf{n}\rangle _{{\partial T}}\\= & {} -(\varphi ,\nabla \cdot \mathbf{q})_T + \langle \varphi ,\mathbf{q}\cdot \mathbf{n}\rangle _{\partial T}\\= & {} (\nabla \varphi ,\mathbf{q})_T=(\mathbb {Q}_h(\nabla \varphi ),\mathbf{q})_T, \end{aligned}
which implies the desired identity (3.12). Similarly, we can prove (3.13). $$\square$$

4 Error Equation

Let $$({\varvec{\theta }},w)$$ and $$({\varvec{\theta }}_h,w_h)$$ be the solution of (1.1)–(1.3) and (2.5) respectively. For simplicity, we introduce shear stress $${\varvec{\gamma }}=t^{-2}(\nabla w-{\varvec{\theta }})$$ and its approximation $${\varvec{\gamma }}_h=t^{-2}(\nabla _w w_h-\mathbb {Q}_h{\varvec{\theta }}_0)$$.

The followings are the error functions between the WG finite element solution and the $$L^2$$ projection of the exact solution.
\begin{aligned} \mathbf{e}_h= & {} \mathbf{Q}_h{\varvec{\theta }}-{\varvec{\theta }}_h=\{\mathbf{Q}_0{\varvec{\theta }}-{\varvec{\theta }}_0,\;\mathbf{Q}_b{\varvec{\theta }}-{\varvec{\theta }}_b\}, \xi _h=Q_hw-w_h=\{Q_0w-w_0,Q_bw-w_b\},\\ {\varvec{\zeta }}_h= & {} \mathbb {Q}_h{\varvec{\gamma }}-{\varvec{\gamma }}_h. \end{aligned}
Define
\begin{aligned} \ell _1({\varvec{\theta }},{\varvec{\eta }})= & {} \sum _{T\in {\mathcal {T}}_h}{\langle }\mathbb {C}({\epsilon ({\varvec{\theta }})}\cdot \mathbf{n}-{{\mathcal {Q}}}_h{\epsilon ({\varvec{\theta }})}\cdot \mathbf{n}),{\varvec{\eta }}_0-{\varvec{\eta }}_b{\rangle }_{\partial T}\\ \ell _2({\varvec{\gamma }},v)= & {} \sum _{T\in {\mathcal {T}}_h}{\langle }{\varvec{\gamma }}\cdot \mathbf{n}-\mathbb {Q}_h{\varvec{\gamma }}\cdot \mathbf{n}, v_0-v_b{\rangle }_{\partial T},\\ \ell _3({\varvec{\gamma }},{\varvec{\eta }})= & {} ({\varvec{\gamma }}-\mathbb {Q}_h{\varvec{\gamma }}, {\varvec{\eta }}_0). \end{aligned}

Lemma 4.1

Let $$(\mathbf{e}_h,\xi _h)\in {\varvec{\Theta }}_h\times W_h$$ be the error of the weak Galerkin finite element solution arising from (2.5). Then, for any $$({\varvec{\eta }},v)\in {\varvec{\Theta }}_h\times W_h$$ we have
\begin{aligned} a(\mathbf{e}_h,{\varvec{\eta }})+({\varvec{\zeta }}_h, \nabla _wv-\mathbb {Q}_h{\varvec{\eta }}_0)+s_2(\xi _h,v) =\,&\ell _1({\varvec{\theta }},{\varvec{\eta }})+\ell _2({\varvec{\gamma }},v)\nonumber \\&+ s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }})+s_2(Q_hw,v)+\ell _3({\varvec{\gamma }},{\varvec{\eta }}). \end{aligned}
(4.1)

Proof

Testing (1.1) by using $${\varvec{\eta }}_0$$ of $${\varvec{\eta }}=\{{\varvec{\eta }}_0,{\varvec{\eta }}_b\}\in {\varvec{\Theta }}_h$$ we arrive at
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}(\mathbb {C}{\epsilon ({\varvec{\theta }})},{\epsilon ({\varvec{\eta }}_0)})_T-\sum _{T\in {\mathcal {T}}_h} \langle \mathbb {C}{\epsilon ({\varvec{\theta }})}\cdot \mathbf{n},{\varvec{\eta }}_0-{\varvec{\eta }}_b\rangle _{\partial T}-({\varvec{\gamma }},{\varvec{\eta }}_0)=0, \end{aligned}
(4.2)
where we have used the fact that $$\sum _{T\in {\mathcal {T}}_h}\langle \mathbb {C}{\epsilon ({\varvec{\theta }})}\cdot \mathbf{n}, {\varvec{\eta }}_b\rangle _{\partial T}=0$$. To deal with the term $$\sum _{T\in {\mathcal {T}}_h}(\mathbb {C}{\epsilon ({\varvec{\theta }})},{\epsilon ({\varvec{\eta }}_0)})_T$$ in (4.2), we need the following equation. For any $${\varvec{\phi }}\in [H^1(T)]^2$$ and $${\varvec{\eta }}\in {\varvec{\Theta }}_h$$, it follows from (3.13), the definition of the discrete weak symmetric gradient (2.3), and the integration by parts that
\begin{aligned} (\mathbb {C}{\epsilon _w(\mathbf{Q}_h{\varvec{\phi }})},{\epsilon _w({\varvec{\eta }})})_T= & {} (\mathbb {C}{{\mathcal {Q}}}_h {\epsilon ({\varvec{\phi }})},{\epsilon _w({\varvec{\eta }})})_T\nonumber \\= & {} -\,({\varvec{\eta }}_0,\nabla \cdot (\mathbb {C}{{\mathcal {Q}}}_h{\epsilon ({\varvec{\phi }})}))_T +\langle {\varvec{\eta }}_b,\mathbb {C}{{\mathcal {Q}}}_h{\epsilon ({\varvec{\phi }})}\cdot \mathbf{n}\rangle _{\partial T}\nonumber \\= & {} (\nabla {\varvec{\eta }}_0,\mathbb {C}{{\mathcal {Q}}}_h{\epsilon ({\varvec{\phi }})})_T-\langle {\varvec{\eta }}_0-{\varvec{\eta }}_b,\mathbb {C}{{\mathcal {Q}}}_h{\epsilon ({\varvec{\phi }})}\cdot \mathbf{n}\rangle _{\partial T}\nonumber \\= & {} (\mathbb {C}{\epsilon ({\varvec{\phi }})},{\epsilon ({\varvec{\eta }}_0)})_T-{\langle }\mathbb {C}{{\mathcal {Q}}}_h{\epsilon ({\varvec{\phi }})}\cdot \mathbf{n},\ {\varvec{\eta }}_0-{\varvec{\eta }}_b{\rangle }_{\partial T}. \end{aligned}
(4.3)
By letting $${\varvec{\phi }}={\varvec{\theta }}$$ in (4.3), we have from combining (4.3) and (4.2) that
\begin{aligned} \sum _{T\in {\mathcal {T}}_h} (\mathbb {C}{\epsilon _w(\mathbf{Q}_h{\varvec{\theta }})},{\epsilon _w({\varvec{\eta }})})_T-({\varvec{\gamma }},{\varvec{\eta }}_0)= & {} \ell _1({\varvec{\theta }},{\varvec{\eta }}). \end{aligned}
Adding $$s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }})$$ to both sides of the above equation gives
\begin{aligned} a(\mathbf{Q}_h{\varvec{\theta }}, {\varvec{\eta }})-({\varvec{\gamma }},{\varvec{\eta }}_0)=\ell _1({\varvec{\theta }},{\varvec{\eta }}) +s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }}). \end{aligned}
(4.4)
Testing (1.2) by using $$v_0$$ of $$v=\{v_0,v_b\}\in W_h$$ we arrive at
\begin{aligned} -(\nabla \cdot {\varvec{\gamma }},v_0)=\sum _{T\in {\mathcal {T}}_h}({\varvec{\gamma }},\nabla v_0)_T-\sum _{T\in {\mathcal {T}}_h}{\langle }v_0-v_b,{\varvec{\gamma }}\cdot \mathbf{n}{\rangle }_{\partial T}=(g,v_0). \end{aligned}
(4.5)
It follows from (3.12), the definition of the discrete weak gradient (2.4), and the integration by parts that
\begin{aligned} ({\varvec{\gamma }},\nabla v_0)_T= & {} (\mathbb {Q}_h{\varvec{\gamma }},\nabla v_0)_T\nonumber \\= & {} -(v_0,\nabla \cdot (\mathbb {Q}_h{\varvec{\gamma }})_T+\langle v_0, \mathbb {Q}_h{\varvec{\gamma }}\cdot \mathbf{n}\rangle _{\partial T}\nonumber \\= & {} (\mathbb {Q}_h{\varvec{\gamma }},\nabla _wv)_T+\langle v_0-v_b,\mathbb {Q}_h{\varvec{\gamma }}\cdot \mathbf{n}\rangle _{\partial T}. \end{aligned}
Using the equation above, (4.5) becomes
\begin{aligned} ({\varvec{\gamma }},\nabla _wv)=(g,v_0)+\ell _2({\varvec{\gamma }},v). \end{aligned}
(4.6)
Adding $$s_2(Q_hw,v)$$ to both sides of the above equation gives
\begin{aligned} ({\varvec{\gamma }},\nabla _wv)+s_2(Q_hw,v)=(g,v_0)+\ell _2({\varvec{\gamma }},v)+s_2(Q_hw,v). \end{aligned}
(4.7)
Adding (4.4) and (4.7), we have
\begin{aligned} a(\mathbf{Q}_h{\varvec{\theta }}, {\varvec{\eta }})+({\varvec{\gamma }},\nabla _wv-{\varvec{\eta }}_0)+s_2(Q_hw,v)= & {} (g,v_0)+\ell _1({\varvec{\theta }},{\varvec{\eta }})+s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }})\\&+\,\ell _2({\varvec{\gamma }},v)+s_2(Q_hw,v). \end{aligned}
By adding the term $$(\mathbb {Q}_h{\varvec{\gamma }},\nabla _wv-{\varvec{\eta }}_0)$$ on the both side of equation above, we obtain
\begin{aligned} a(\mathbf{Q}_h{\varvec{\theta }}, {\varvec{\eta }})+(\mathbb {Q}_h{\varvec{\gamma }},\nabla _wv-\mathbb {Q}_h{\varvec{\eta }}_0) +s_2(Q_hw,v)=&(g,v_0)+\ell _1({\varvec{\theta }},{\varvec{\eta }})+s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }})\nonumber \\&+\,\ell _2({\varvec{\gamma }},v)+s_2(Q_hw,v)+\ell _3({\varvec{\gamma }},{\varvec{\eta }}). \end{aligned}
(4.8)
Subtracting (2.5) from (4.8) yields the error equation (4.1)
\begin{aligned} a(\mathbf{e}_h, {\varvec{\eta }})+({\varvec{\zeta }}_h,\nabla _wv-\mathbb {Q}_h{\varvec{\eta }}_0)+s_2(\xi _h,v)= & {} \ell _1({\varvec{\theta }},{\varvec{\eta }})+s_1(\mathbf{Q}_h{\varvec{\theta }},{\varvec{\eta }})+\ell _2({\varvec{\gamma }},v)\nonumber \\&+\,s_2(Q_hw,v)+\ell _3({\varvec{\gamma }},{\varvec{\eta }}). \end{aligned}
(4.9)
This completes the proof of the lemma. $$\square$$

5 Error Analysis

The following lemma provides the estimates of the terms on the right hand side of the error equation (4.1).

Lemma 5.1

Assume that $${\mathcal {T}}_h$$ is shape regular. Then for any $${\varvec{\theta }}\in [H^{k+1}(\Omega )]^2$$, $$w\in H^{k+1}(\Omega )$$ and $${\varvec{\gamma }}\in [H^{k}(\Omega )]^2$$, $$({\varvec{\eta }}, v)\in {\varvec{\Theta }}_h\times W_h$$, we have
\begin{aligned} |\ell _1({\varvec{\theta }},{\varvec{\eta }})|\le & {} Ch^k\Vert {\varvec{\theta }}\Vert _{k+1}{|||}{\varvec{\eta }}{|||}_\theta , \end{aligned}
(5.1)
\begin{aligned} |\ell _2({\varvec{\gamma }},v)|\le & {} Ch^k\Vert {\varvec{\gamma }}\Vert _{k}(s_2(v,v))^{1/2}, \end{aligned}
(5.2)
\begin{aligned} |s_1(\mathbf{Q}_h{\varvec{\theta }}, {\varvec{\eta }})|\le & {} Ch^k\Vert {\varvec{\theta }}\Vert _{k+1}{|||} {\varvec{\eta }}{|||}_\theta , \end{aligned}
(5.3)
\begin{aligned} |s_2(Q_hw,v)|\le & {} C h^k\Vert w\Vert _{k+1} (s_2(v,v))^{1/2}, \end{aligned}
(5.4)
\begin{aligned} |\ell _3({\varvec{\gamma }},{\varvec{\eta }})|\le & {} C h^k\Vert {\varvec{\gamma }}\Vert _{k}{|||}{\varvec{\eta }}{|||}_\theta . \end{aligned}
(5.5)

Proof

We will prove (5.1), (5.4) and (5.5). The estimates (5.2) and (5.3) can be proved similarly. As to (5.1), it follows from the Cauchy–Schwarz inequality, the trace inequality (3.3) that
\begin{aligned} |\ell _1({\varvec{\theta }},{\varvec{\eta }})|= & {} \left| \sum _{T\in {\mathcal {T}}_h}{\langle }\mathbb {C}({\epsilon ({\varvec{\theta }})}\cdot \mathbf{n}-{{\mathcal {Q}}}_h{\epsilon ({\varvec{\theta }})}\cdot \mathbf{n}),{\varvec{\eta }}_0-{\varvec{\eta }}_b{\rangle }_{\partial T}\right| \nonumber \\\le & {} C \sum _{T\in {\mathcal {T}}_h}\Vert \mathbb {C}({\epsilon ({\varvec{\theta }})}-{{\mathcal {Q}}}_h{\epsilon ({\varvec{\theta }})})\Vert _{{\partial T}} \Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert _{\partial T}\nonumber \\\le & {} C \left( \sum _{T\in {\mathcal {T}}_h}h\Vert \mathbb {C}({\epsilon ({\varvec{\theta }})}-{{\mathcal {Q}}}_h{\epsilon ({\varvec{\theta }})})\Vert _{{\partial T}}^2\right) ^{\frac{1}{2}} \left( \sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert {\varvec{\eta }}_0-{\varvec{\eta }}_b\Vert _{\partial T}^2\right) ^{\frac{1}{2}}\nonumber \\\le & {} Ch^k\Vert {\varvec{\theta }}\Vert _{k+1}\left( \sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert {\varvec{\eta }}_0 -{\varvec{\eta }}_b\Vert _{\partial T}^2\right) ^{\frac{1}{2}}\le Ch^k\Vert {\varvec{\theta }}\Vert _{k+1}{|||}{\varvec{\eta }}{|||}_\theta . \end{aligned}
Using the definition of $$Q_b$$, (3.3), we obtain
\begin{aligned} |s_2(Q_hw, v)|= & {} \left| \sum _{T\in {\mathcal {T}}_h} h^{-1}\langle Q_bQ_0w-Q_bw,\; Q_bv_0-v_b\rangle _{\partial T}\right| \\= & {} \left| \sum _{T\in {\mathcal {T}}_h} h^{-1}\langle Q_0w-w,\; v_0-v_b\rangle _{\partial T}\right| \\\le & {} C\left( \sum _{T\in {\mathcal {T}}_h}\left( h^{-2}\Vert Q_0w-w\Vert _T^2+ \Vert \nabla (Q_0w-w)\Vert _T^2\right) \right) ^{\frac{1}{2}}\cdot \\&\left( \sum _{T\in {\mathcal {T}}_h}h^{-1}\Vert Q_bv_0-v_b\Vert ^2_{{\partial T}}\right) ^{\frac{1}{2}}\\\le & {} Ch^k\Vert w\Vert _{k+1}(s_2(v,v))^{1/2}. \end{aligned}
The estimate (5.5) follows from (3.8) and the definition of $$\mathbb {Q}_h$$ that
\begin{aligned} |\ell _3({\varvec{\gamma }},{\varvec{\eta }})|= & {} |({\varvec{\gamma }}-\mathbb {Q}_h{\varvec{\gamma }}, {\varvec{\eta }}_0)|\le \Vert \mathbb {Q}_h{\varvec{\gamma }}-{\varvec{\gamma }}\Vert \Vert {\varvec{\eta }}_0\Vert \le C h^k\Vert {\varvec{\gamma }}\Vert _{k}{|||}{\varvec{\eta }}{|||}_\theta . \end{aligned}
We have proved the lemma. $$\square$$

The error equation (4.1) can be used to derive the following error estimate for the WG finite element solution.

Theorem 5.2

Let $$({\varvec{\theta }}_h,w_h)\in {\varvec{\Theta }}_h\times W_h$$ be the weak Galerkin finite element solution of the problem (1.1)–(1.3) arising from (2.5). Then, there exists a constant C such that
\begin{aligned} {|||} \mathbf{Q}_h{\varvec{\theta }}-{\varvec{\theta }}_h{|||}_\theta +{|||} Q_hw-w_h{|||}_w+t\Vert \mathbb {Q}_h{\varvec{\gamma }}-{\varvec{\gamma }}_h\Vert \le Ch^{k}(\Vert {\varvec{\theta }}\Vert _{k+1}+\Vert w\Vert _{k+1}+\Vert {\varvec{\gamma }}\Vert _k), \end{aligned}
(5.6)
where $${\varvec{\gamma }}=t^{-2}(\nabla w-{\varvec{\theta }})$$ and $${\varvec{\gamma }}_h=t^{-2}(\nabla _w w_h-\mathbb {Q}_h{\varvec{\theta }}_0)$$.

Proof

It follows the definition of $${\varvec{\gamma }}$$ and $${\varvec{\gamma }}_h$$ and (3.12),
\begin{aligned} t^2{\varvec{\zeta }}_h= & {} t^2(\mathbb {Q}_h{\varvec{\gamma }}-{\varvec{\gamma }}_h)=\mathbb {Q}_h(\nabla w-{\varvec{\theta }}) -(\nabla _ww_h-\mathbb {Q}_h{\varvec{\theta }}_0)\\= & {} \nabla _w(Q_hw-w_h)-(\mathbb {Q}_h{\varvec{\theta }}-\mathbb {Q}_h{\varvec{\theta }}_0) =\nabla _w\xi _h-(\mathbb {Q}_h{\varvec{\theta }}-\mathbb {Q}_h{\varvec{\theta }}_0). \end{aligned}
The above equation implies
\begin{aligned} ({\varvec{\zeta }}_h, \nabla _w\xi _h-\mathbb {Q}_h\mathbf{e}_0)= & {} ({\varvec{\zeta }}_h, \nabla _w\xi _h -\mathbf{e}_0)=({\varvec{\zeta }}_h, \nabla _w\xi _h-(\mathbf{Q}_0{\varvec{\theta }}-{\varvec{\theta }}_0))\\= & {} ({\varvec{\zeta }}_h, \nabla _w\xi _h-(\mathbb {Q}_h{\varvec{\theta }}-\mathbb {Q}_h{\varvec{\theta }}_0))=t^2 ({\varvec{\zeta }}_h,{\varvec{\zeta }}_h). \end{aligned}
Letting $$({\varvec{\eta }},v)=(\mathbf{e}_h,\xi _h)$$ in (4.1) and using Lemma 5.1, we have
\begin{aligned} {|||} \mathbf{e}_h{|||}_\theta ^2+t^2\Vert {\varvec{\zeta }}_h\Vert ^2 +s_2(\xi _h,\xi _h)= & {} \ell _1({\varvec{\theta }},\mathbf{e}_h) +\ell _2({\varvec{\gamma }},\xi _h)+s_1(\mathbf{Q}_h{\varvec{\theta }},\mathbf{e}_h)\\&+\,s_2(Q_hw,\xi _h)+\ell _3({\varvec{\gamma }},\mathbf{e}_h)\\\le & {} Ch^k(\Vert {\varvec{\theta }}\Vert _{k+1}+\Vert w\Vert _{k+1}+\Vert {\varvec{\gamma }}\Vert _k) ({|||}\mathbf{e}_h{|||}_\theta \\&+\,(s_2(\xi _h,\xi _h))^{1/2}), \end{aligned}
which implies
\begin{aligned} {|||} \mathbf{e}_h{|||}_\theta +t\Vert {\varvec{\zeta }}_h\Vert +s_2(\xi _h,\xi _h)^{1/2} \le Ch^k(\Vert {\varvec{\theta }}\Vert _{k+1}+\Vert w\Vert _{k+1}+\Vert {\varvec{\gamma }}\Vert _k). \end{aligned}
(5.7)
Using the definition of $${\varvec{\zeta }}_h$$, (5.7) and (3.8), we have
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \nabla _w\xi _h\Vert _T^2\le & {} C\left( t^2 \Vert {\varvec{\zeta }}_h\Vert _T^2+\sum _{T\in {\mathcal {T}}_h}\Vert \mathbb {Q}_h{\varvec{\theta }}-\mathbb {Q}_h{\varvec{\theta }}_0\Vert _T^2\right) \\\le & {} C\left( t^2\Vert {\varvec{\zeta }}_h\Vert _T^2+\sum _{T\in {\mathcal {T}}_h}\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert _T^2\right) \\\le & {} C\left( t^2\Vert {\varvec{\zeta }}_h\Vert _T^2+\sum _{T\in {\mathcal {T}}_h}\Vert {\varvec{\theta }}-\mathbf{Q}_0{\varvec{\theta }}\Vert _T^2 +\sum _{T\in {\mathcal {T}}_h}\Vert \mathbf{e}_0\Vert _T^2\right) \\\le & {} Ch^k(\Vert {\varvec{\theta }}\Vert _{k+1}+\Vert w\Vert _{k+1}+\Vert {\varvec{\gamma }}\Vert _k). \end{aligned}
This completes the proof. $$\square$$

Corollary 5.3

Let $$({\varvec{\theta }}_h,w_h)\in {\varvec{\Theta }}_h\times W_h$$ be the weak Galerkin finite element solution of the problem (1.1)–(1.3) arising from (2.5). Then, there exists a constant C such that
\begin{aligned} \sum _{T\in {\mathcal {T}}_h}\Vert \mathbf{Q}_0{\varvec{\theta }}-{\varvec{\theta }}_0\Vert _{1,T}+\Vert Q_0w-w_0\Vert \le Ch^{k}(\Vert {\varvec{\theta }}\Vert _{k+1}+\Vert w\Vert _{k+1}+\Vert {\varvec{\gamma }}\Vert _k), \end{aligned}
(5.8)
where $${\varvec{\gamma }}=t^{-2}(\nabla w-{\varvec{\theta }})$$.

Proof

The estimate (5.8) is a direct result of (5.6) and Lemma 3.2. $$\square$$

6 Numerical Experiments

The goal of this section is to study the weak Galerkin algorithm 2.1 through several numerical examples for the Reissner–Mindlin equations (1.1) and (1.2) with tensor of bending moduli given by
\begin{aligned} \mathbb {C}\varvec{\tau }=\frac{E}{12(1-\nu ^2)}[(1-\nu )\varvec{\tau }+\nu \text{ tr } (\varvec{\tau })I],\quad \forall \varvec{\tau }=\{\tau _{ij}\}_{2\times 2}, \ \tau _{12}=\tau _{21} \end{aligned}
and the shear correction factor $$\lambda =\frac{5E}{12(1+\nu )}$$. Here E is the Young’s modulus and $$\nu$$ is the Poisson’s ratio. The accuracy for the WG approximate solutions will be measured in four norms defined as follows:
\begin{aligned}&\text{ Discrete } H^1\text{-norm: } {|||} \varvec{\theta }{|||}_{\theta }:=\bigg (a(\varvec{\theta }, \varvec{\theta })\bigg )^{1/2},\\&\text{ Element-based } L^2\text{-norm: } \Vert \varvec{\theta }_0\Vert :=\bigg (\sum _{T\in \mathcal {T}_h}\int _T|\varvec{\theta }_0|^2dx\bigg )^{1/2},\\&\text{ Discrete } H^1\text{-norm: } {|||} w{|||}_{w} :=\bigg (\sum _{T\in \mathcal {T}_h}\Vert \nabla _w w\Vert _T^2+s_2(w,w)\bigg )^{1/2}\\&\text{ Element-based } L^2\text{-norm: } \Vert w_0\Vert :=\bigg (\sum _{T\in \mathcal {T}_h}\int _T|w_0|^2dx\bigg )^{1/2}, \end{aligned}
where $$\varvec{\theta }=\{\varvec{\theta }_0,\varvec{\theta }_b\}\in \Theta _h$$ and $$w=\{w_0,w_b\}\in W_h.$$

To demonstrate the performance of the WG algorithm 2.1 on quadrilateral partitions, we consider a model Reissner–Mindlin equation for a material with Poisson’s ratio $$\nu =0.3$$ and Young’s modulus $$E=1.092\times 10^3\,\hbox {N}/\hbox {m}^2$$. The domain is given by the unit square $$\Omega =(0,1)^2$$. The exact solution is chosen as
\begin{aligned} \varvec{\theta }(x,y)= & {} \begin{pmatrix} y^3(y-1)^3x^2(x-1)^2(2x-1)\\ x^3(x-1)^3y^2(y-1)^2(2y-1) \end{pmatrix},\\ w(x,y)= & {} \frac{1}{3}x^3(x-1)^3y^3(y-1)^3\\&-\,\frac{2t^2}{5(1-\nu )}\left[ y^3(y-1)^3x(x-1)(5x^2-5x+1)\right. \\&+\left. x^3(x-1)^3y(y-1)(5y^2-5y+1)\right] , \end{aligned}
and the body load can be calculated as
\begin{aligned} g(x,y)= & {} \frac{E}{12(1\!-\!\nu ^2)}\left[ 12y(y\!-\!1)(5x^2\!-\!5x+1) (2y^2(y-1)^2+x(x-1)(5y^2-5y+1))\right. \\&+\left. 12x(x-1)(5y^2-5y+1) (2x^2(x-1)^2+y(y-1)(5x^2-5x+1))\right] . \end{aligned}

6.1.1 On Uniform Rectangular Partitions

The uniform rectangular mesh is given by partitioning the unit square domain into $$n\times n$$ sub-rectangles, shown as in Fig. 1a with $$n=4$$. The mesh size is given by $$h=1/n$$.

The lowest order WG element (i.e., $$k=1$$) was employed in this numerical experiment. This setting gives numerical solutions $$\varvec{\theta }_h=\{\varvec{\theta }_0, \varvec{\theta }_b\}$$ and $$w_h=\{w_0, w_b\}$$ such that $$\varvec{\theta }_0|_T\in [P_1(T)]^2$$, $$\varvec{\theta }_b|_{\partial T}\in [P_1(e)]^2$$, $$w_0|_T\in P_1(T)$$, and $$w_b|_{\partial T}\in P_0(e)$$ locally on each element T. The numerical tests were conducted for the thickness $$t=1,\ 10^{-3},\ 10^{-6}$$. Table 1 illustrates the relative errors and convergence rates in the discrete $$H^1$$ and $$L^2$$ norms. For the case of $$t=1$$, it can be seen that the convergence for $$\varvec{\theta }_h$$ and $$w_h$$ has the optimal order of $$O(h^2)$$ and O(h) in $$L^2$$ and the discrete $$H^1$$ norms, respectively. For the cases of $$t=10^{-3}$$ and $$t=10^{-6}$$, a super-convergence was observed in the discrete $$H^1$$ norm. The computational performance for the approximation of $${\varvec{\gamma }}=t^{-2}(\nabla w-\varvec{\theta })$$ as given by $${\varvec{\gamma }}_h =t^{-2}(\nabla _w w_h-\mathbb {Q}_h\varvec{\theta }_0)$$ in the $$L^2$$ norm is shown in Table 2. A second order of convergence can be seen for $${\varvec{\gamma }}_h$$ on uniform rectangular partitions, which is indeed a superconvergence phenomena.
Table 1

Numerical results on uniform rectangular partitions for the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

1/4

7.6291e−01

6.2392e−01

6.6788e−01

5.8677e−01

1/8

3.4910e−01

1.13

2.1897e−01

1.51

2.9056e−01

1.20

1.5986e−01

1.88

1/16

1.6832e−01

1.05

6.2644e−02

1.81

1.1760e−01

1.30

4.0598e−02

1.98

1/32

8.3319e−02

1.01

1.6355e−02

1.94

5.1418e−02

1.19

1.0181e−02

2.00

1/64

4.1553e−02

1.00

4.1378e−03

1.99

2.4457e−02

1.07

2.5471e−03

2.00

1/128

2.0786e−02

1.00

1.0343e−03

2.00

1.2226e−02

1.00

6.3679e−04

2.00

$$t=10^{-3}$$

1/4

7.1360e−01

7.4757e−01

7.3852e−01

7.2895e−01

1/8

4.9875e−01

0.52

5.1634e−01

0.53

5.0673e−01

0.54

2.1155e−01

1.78

1/16

2.6864e−01

0.89

2.7468e−01

0.91

2.7074e−01

0.90

6.3711e−02

1.73

1/32

9.7213e−02

1.47

9.8404e−02

1.48

9.7624e−02

1.47

1.8383e−02

1.79

1/64

2.9224e−02

1.73

2.9404e−02

1.74

2.9285e−02

1.74

4.9772e−03

1.89

1/128

9.7413e−03

1.59

7.3510e−03

2.00

8.8742e−03

1.72

1.2443e−03

2.00

$$t=10^{-6}$$

1/4

7.1328e−01

7.4757e−01

7.3851e−01

7.2896e−01

1/8

4.9864e−01

0.52

5.1635e−01

0.53

5.0673e−01

0.54

2.1155e−01

1.78

1/16

2.6859e−01

0.89

2.7469e−01

0.91

2.7074e−01

0.90

6.3711e−02

1.73

1/32

9.7175e−02

1.47

9.8414e−02

1.48

9.7631e−02

1.47

1.8383e−02

1.79

1/64

2.9203e−02

1.73

2.9425e−02

1.74

2.9304e−02

1.74

4.9778e−03

1.88

1/128

9.7343e−03

1.59

7.3562e−03

2.00

8.8800e−03

1.72

1.2445e−03

2.00

Table 2

Numerical results for $${\varvec{\gamma }}_h=t^{-2}(\nabla _w w_h-\mathbb {Q}_h\varvec{\theta }_0)$$ with the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on uniform partitions

h

$$t=1$$

$$t=10^{-3}$$

$$t=10^{-6}$$

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

1/4

2.9893e−01

2.4987e−01

2.4987e−01

1/8

8.4458e−02

1.8235

7.6348e−02

1.7105

7.6348e−02

1.7105

1/16

2.1402e−02

1.9805

2.3907e−02

1.6752

2.3907e−02

1.6752

1/32

5.3678e−03

1.9953

6.5596e−03

1.8658

6.5596e−03

1.8658

1/64

1.3430e−03

1.9989

1.6819e−03

1.9635

1.6819e−03

1.9635

1/128

3.3581e−04

1.9997

4.2342e−04

1.9899

4.2342e−04

1.9899

We also tested the quadratic WG element (i.e., $$k=2$$) on the uniform rectangular partitions. The corresponding numerical setting gives $$\varvec{\theta }_0|_T\in [P_2(T)]^2$$, $$\varvec{\theta }_b|_{\partial T}\in [P_2(e)]^2$$, $$w_0|_T\in P_2(T)$$, and $$w_b|_{\partial T}\in P_1(e)$$ locally on each element T. The relative errors are shown in Table 3 for different values of the thickness parameter t. It can be seen that the quadratic WG element gives numerical approximations for $$\varvec{\theta }$$ and w which are of $$O(h^2)$$ accuracy in the discrete $$H^1$$-norm and are of $$O(h^3)$$ accuracy in the $$L^2$$-norm. The numerical results are consistent with the convergence theory developed in previous Sections.
Table 3

Numerical results for the WG element $$P_2(T)P_2(e)P_2(T)P_1(e)$$ on uniform rectangular partitions

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

1/4

1.3903e−01

8.2646e−02

1.0151e−01

1.4911e−01

1/8

2.9084e−02

2.26

8.7719e−03

3.24

2.9822e−02

1.77

1.9566e−02

2.93

1/16

6.8900e−03

2.08

9.2268e−04

3.25

8.3788e−03

1.83

2.4048e−03

3.02

1/32

1.7137e−03

2.01

1.0538e−04

3.13

2.2353e−03

1.91

2.9728e−04

3.02

1/64

4.2843e−04

2.00

1.2851e−05

3.04

5.8514e−04

1.93

3.6701e−05

3.02

1/128

1.0711e−04

2.00

1.6064e−06

3.00

1.4629e−04

2.00

4.5876e−06

3.00

$$t=10^{-3}$$

1/4

6.8473e−02

1.2679e−01

8.4180e−02

8.4921e−02

1/8

2.3049e−02

1.57

2.4582e−02

2.37

2.4863e−02

1.76

1.1982e−02

2.83

1/16

5.8623e−03

1.98

3.0927e−03

2.99

6.2557e−03

1.99

1.5977e−03

2.91

1/32

1.4506e−03

2.01

3.9409e−04

2.97

1.6539e−03

1.92

1.8722e−04

3.09

1/64

3.6614e−04

1.99

4.8012e−05

3.04

3.9848e−04

2.05

2.4402e−05

2.94

1/128

9.0135e−05

2.00

6.0015e−06

3.00

9.7121e−05

2.04

2.9253e−06

3.06

$$t=10^{-6}$$

1/4

6.8473e−02

1.2679e−01

8.4180e−02

8.4921e−02

1/8

2.3049e−02

1.57

2.4582e−02

2.37

2.4863e−02

1.76

1.1982e−02

2.83

1/16

5.8723e−03

1.97

3.0937e−03

2.99

6.2567e−03

1.99

1.5977e−03

2.91

1/32

1.4606e−03

2.01

3.9419e−04

2.97

1.6549e−03

1.92

1.8822e−04

3.09

1/64

3.6714e−04

1.99

4.8022e−05

3.04

3.9948e−04

2.06

2.4502e−05

2.94

1/128

9.0035e−05

2.03

6.0015e−06

3.00

9.7221e−05

2.04

2.9353e−06

3.06

6.1.2 On Deformed Rectangular Partitions

In this numerical experiment, we first generate an initial deformed rectangular partition Fig. 1b by perturbing the uniform rectangular partition shown as in Fig. 1a. A sequence of deformed rectangular partitions are then obtained from this initial partition by connecting the midpoints of any two opposite edges in the previous level of partition.

The linear WG element (i.e., $$k=1$$) was employed in this numerical experiment. The corresponding numerical setting yields $$\varvec{\theta }_0|_T\in [P_1(T)]^2$$, $$\varvec{\theta }_b|_{\partial T}\in [P_1(e)]^2$$, $$w_0|_T\in P_1(T)$$, and $$w_b|_{\partial T}\in P_0(e)$$ on each element T. Numerical results were produced for the thickness of $$t=1,\ 10^{-3},\ 10^{-6}$$. The relative errors and the convergence rates are shown in Table 4, indicating a convergence of order $$O(h^2)$$ for the approximations of $$\varvec{\theta }$$ and w in the $$L^2$$-norm. Furthermore, the convergence in $$L^2$$ is of optimal order in the discrete $$H^1$$-norm. Surprisingly, a superconvergence was observed for $$\varvec{\theta }$$ and w in the discrete $$H^1$$-norm with the thickness $$t=10^{-3},\ 10^{-6}$$. For the approximation of $${\varvec{\gamma }}=t^{-2}(\nabla w-\varvec{\theta })$$ as given by $${\varvec{\gamma }}_h =t^{-2}(\nabla _w w_h-\mathbb {Q}_h\varvec{\theta }_0)$$, Table 5 illustrates an optimal order of convergence in the $$L^2$$ norm, which is in consistency with the theory developed in previous sections.
Table 4

Numerical results on deformed rectangular partitions for the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

3.8194e−01

7.9548e−01

7.2503e−01

7.3853e−01

6.4858e−01

1.9097e−01

3.6930e−01

1.11

2.6747e−01

1.44

3.7680e−01

0.97

1.7719e−01

1.87

9.5484e−02

1.7649e−01

1.07

8.3091e−02

1.69

1.6971e−01

1.15

4.4602e−02

1.99

4.7742e−02

8.7041e−02

1.02

2.3191e−02

1.84

7.4775e−02

1.18

1.1150e−02

2.00

2.3871e−02

4.3365e−02

1.01

6.0587e−03

1.94

3.4896e−02

1.10

2.7865e−03

2.00

1.1936e−02

2.1663e−02

1.00

1.5368e−03

1.98

1.7010e−02

1.04

6.9652e−04

2.00

$$t=10^{-3}$$

3.8194e−01

7.7487e−01

8.1814e−01

7.9991e−01

7.4633e−01

1.9097e−01

5.1450e−01

0.59

5.3211e−01

0.62

5.2199e−01

0.62

2.1635e−01

1.79

9.5484e−02

2.6979e−01

0.93

2.7596e−01

0.95

2.7205e−01

0.94

6.4040e−02

1.76

4.7742e−02

9.9082e−02

1.45

1.0048e−01

1.46

9.9650e−02

1.45

1.8349e−02

1.80

2.3871e−02

3.1128e−02

1.67

3.1386e−02

1.68

3.1252e−02

1.67

4.9833e−03

1.88

1.1936e−02

1.0376e−02

1.59

7.8465e−03

2.00

1.0081e−02

1.63

1.2458e−03

2.00

$$t=10^{-6}$$

3.8194e−01

7.7457e−01

8.1814e−01

7.9990e−01

7.4633e−01

1.9097e−01

5.1439e−01

0.59

5.3211e−01

0.62

5.2199e−01

0.62

2.1635e−01

1.79

9.5484e−02

2.6973e−01

0.93

2.7596e−01

0.95

2.7205e−01

0.94

6.4039e−02

1.76

4.7742e−02

9.9035e−02

1.45

1.0048e−01

1.46

9.9647e−02

1.45

1.8349e−02

1.80

2.3871e−02

3.1089e−02

1.67

3.1388e−02

1.68

3.1251e−02

1.67

4.9827e−03

1.88

1.1936e−02

1.0363e−02

1.59

7.8470e−03

2.00

1.0081e−02

1.63

1.2457e−03

2.00

Table 5

Numerical results for $${\varvec{\gamma }}_h=t^{-2}(\nabla _w w_h-\mathbb {Q}_h\varvec{\theta }_0)$$ with the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on non-uniform partitions

h

$$t=1$$

$$t=10^{-3}$$

$$t=10^{-6}$$

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

$$\frac{\Vert {\varvec{\zeta }}_h\Vert }{\Vert \mathbb {Q}_h{\varvec{\gamma }}\Vert }$$

Rate

3.8194e−01

3.9209e−01

8.4076e−01

8.4076e−01

1.9097e−01

1.1138e−01

1.8157

6.0107e−01

0.4842

6.0107e−01

0.4842

9.5484e−02

2.8073e−02

1.9882

2.6703e−01

1.1705

2.6703e−01

1.1705

4.7742e−02

7.0628e−03

1.9909

1.3743e−01

0.9583

1.3743e−01

0.9583

2.3871e−02

1.7725e−03

1.9945

7.1946e−02

0.9337

7.1946e−02

0.9337

1.1936e−02

4.4432e−04

1.9962

3.7903e−02

0.9247

3.8003e−02

0.9209

6.2 Triangular Partitions

Consider the Reissner–Mindlin problem in the unit disk domain $$\Omega =\{(x,y): \ x^2+y^2<1\}$$ with the loaded force $$g=1$$ and a material with Poisson’s ratio $$\nu =0.3$$ and Young’s modulus $$E=1 N/m^2$$. The exact solution for this model problem is given by
\begin{aligned} \varvec{\theta }(x,y)= & {} \begin{pmatrix} x(x^2+y^2-1)\\ y(x^2+y^2-1) \end{pmatrix}/16D,\\ w(x,y)= & {} \frac{(x^2+y^2)^2}{64D}-(x^2+y^2) \bigg (\frac{t^2}{4\lambda }+\frac{1}{32D}\bigg ) +\frac{t^2}{4\lambda }+\frac{1}{64D}, \end{aligned}
where $$D=E/[12(1-\nu ^2)]$$.
The finite element triangulations were obtained from an initial triangulation shown as in Fig. 2 through successive refinements of the elements from previous levels by connecting the middle points of each edge. The meshsize, denote by h, is the largest length of the edges for each partition. The linear WG element was used in this numerical study, and the relative error profiles for each numerical solution are shown in Table 6. Optimal order of convergence can be seen from this table for both $$\varvec{\theta }$$ and w in the discrete $$H^1$$-norm and the usual $$L^2$$ norm.
Table 6

Relative error and rate of convergence for the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on triangular partitions

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

2.3587e−01

5.3606e−01

6.1991e−02

2.1983e−01

7.9362e−03

1.2175e−01

2.9060e−01

0.93

1.5787e−02

2.07

1.1040e−01

1.04

2.0150e−03

2.07

6.1806e−02

1.4864e−01

0.99

3.9761e−03

2.03

5.5259e−02

1.02

5.0625e−04

2.04

3.1133e−02

7.4761e−02

1.00

9.9654e−04

2.02

2.7637e−02

1.01

1.2675e−04

2.02

1.5624e−02

3.7436e−02

1.00

2.4933e−04

2.01

1.3819e−02

1.01

3.1702e−05

2.01

$$t=10^{-3}$$

2.3587e−01

3.6175e−01

6.1925e−02

5.3706e−02

6.2920e−02

1.2175e−01

1.8460e−01

1.02

1.5756e−02

2.07

2.7700e−02

1.00

1.5905e−02

2.08

6.1806e−02

9.2922e−02

1.01

3.9684e−03

2.03

1.3950e−02

1.01

3.9884e−03

2.04

3.1133e−02

4.6551e−02

1.01

1.0078e−03

2.00

6.9871e−03

1.01

1.0182e−03

1.99

1.5624e−02

2.3274e−02

1.01

2.5193e−04

2.01

3.4933e−03

1.01

2.5450e−04

2.01

$$t=10^{-6}$$

2.3587e−01

3.6175e−01

6.1926e−02

5.3709e−02

6.2921e−02

1.2175e−01

1.8460e−01

1.02

1.5755e−02

2.07

2.7702e−02

1.00

1.5904e−02

2.08

6.1806e−02

9.2922e−02

1.01

3.9707e−03

2.03

1.3951e−02

1.01

3.9923e−03

2.04

3.1133e−02

4.6550e−02

1.01

9.9592e−04

2.02

6.9876e−03

1.01

9.9964e−04

2.02

1.5624e−02

2.3273e−02

1.01

2.4896e−04

2.01

3.4936e−03

1.01

2.4991e−04

2.01

6.3 Hexagonal Partitions

In the test for hexagonal partitions, the model Reissner–Mindlin problem is the same as the one in Sect. 6.1 on the unit square domain $$\Omega =(0,1)\times (0,1)$$. We first consider the uniform hexagonal partition shown as in Fig. 3. Observe that the partition consists of a mixture of uniform hexagons plus some quadrilaterals and pentagons along the boundary of the domain. The initial partition, shown as in Fig. 3a, is called Mesh Level 1. Mesh Level 2 is a finer partition than Mesh Level 1, and is shown in Fig. 3b. The weak Galerkin algorithm 2.1 of the lowest order (i.e., linear element) was employed in our numerical computation, and the relative error profiles are reported in Table 7. Optimal order of convergence can be seen from this table for the WG approximations of $$\varvec{\theta }$$ and w.
Table 7

Numerical results of the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on uniform hexagonal partitions shown as in Fig. 3

Mesh Level

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

Level 1

5.9248e−04

8.5958e−05

9.8780e−04

1.2919e−04

Level 2

2.9538e−04

1.00

3.6473e−05

1.24

2.7291e−04

1.86

3.0259e−05

2.09

Level 3

9.9685e−05

1.57

1.1297e−05

1.69

7.3817e−05

1.89

6.8386e−06

2.15

Level 4

2.9901e−05

1.74

3.0516e−06

1.89

2.2010e−05

1.75

1.5876e−06

2.11

Level 5

9.0588e−06

1.72

7.8682e−07

1.96

7.1028e−06

1.63

3.7967e−07

2.06

Level 6

2.8670e−06

1.66

1.9940e−07

1.98

2.3968e−06

1.57

9.2639e−08

2.04

$$t=10^{-3}$$

Level 1

7.3549e−04

1.0399e−04

7.9517e−04

1.4308e−04

Level 2

4.5562e−04

0.69

5.1332e−05

1.02

2.8006e−04

1.51

4.4845e−05

1.67

Level 3

2.6380e−04

0.79

2.1678e−05

1.24

9.8020e−05

1.51

1.2482e−05

1.85

Level 4

1.4624e−04

0.85

8.0344e−06

1.43

3.5095e−05

1.48

3.3528e−06

1.90

Level 5

7.2864e−05

1.01

2.4121e−06

1.74

1.2656e−05

1.47

8.8379e−07

1.92

Level 6

3.3653e−05

1.11

6.9658e−07

1.80

4.5608e−06

1.47

2.2947e−07

1.95

$$t=10^{-6}$$

Level 1

7.3546e−04

1.0398e−04

7.9517e−04

1.4308e−04

Level 2

4.5563e−04

0.69

5.1330e−05

1.02

2.8007e−04

1.51

4.4846e−05

1.67

Level 3

2.6377e−04

0.79

2.1680e−05

1.24

9.8018e−05

1.51

1.2481e−05

1.85

Level 4

1.4629e−04

0.85

8.0344e−06

1.43

3.5096e−05

1.48

3.3528e−06

1.90

Level 5

7.2895e−05

1.00

2.4121e−06

1.74

1.2654e−05

1.47

8.8339e−07

1.92

Level 6

3.3653e−05

1.11

6.9658e−07

1.80

4.5608e−06

1.47

2.2947e−07

1.95

Next, the uniform hexagonal partitions were slightly perturbed to give distorted hexagonal partitions shown as in Fig. 4. The distorted/deformed hexagonal partitions are denoted as $$\mathcal {T}_h^7$$, and the WG algorithm 2.1 of the lowest order was again employed for computing numerical approximations of $$\varvec{\theta }$$ and w. The performance of the weak Galerkin algorithm on $$\mathcal {T}_h^7$$ is reported in Table 8. Readers are invited to draw their own conclusions on the accuracy and stability of the method.
Table 8

Numerical results for the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on deformed hexagonal partitions $$\mathcal {T}_h^7$$ shown as in Fig. 4

Mesh Level

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

Level 1

5.9357e−04

8.9927e−05

9.7799e−04

1.2687e−04

Level 2

3.0338e−04

0.97

3.6158e−05

1.31

2.9320e−04

1.74

3.0353e−05

2.06

Level 3

1.0828e−04

1.49

1.1140e−05

1.70

1.1209e−04

1.39

7.1754e−06

2.08

Level 4

4.0323e−05

1.43

3.0690e−06

1.86

4.7389e−05

1.24

1.5931e−06

2.17

Level 5

1.6839e−05

1.26

8.0085e−07

1.94

2.2601e−05

1.07

3.8539e−07

2.05

Level 6

8.4195e−06

1.00

2.0021e−07

2.00

1.1300e−05

1.00

9.6347e−08

2.00

$$t=10^{-3}$$

Level 1

7.7602e−04

1.0624e−04

8.2998e−04

1.5739e−04

Level 2

5.1961e−04

0.58

5.1174e−05

1.05

3.2384e−04

1.36

4.8506e−05

1.70

Level 3

2.6407e−04

0.98

2.0600e−05

1.31

1.3200e−04

1.29

1.3084e−05

1.89

Level 4

1.5410e−04

0.78

7.6905e−06

1.42

5.7317e−05

1.20

3.4725e−06

1.91

Level 5

6.9429e−05

1.15

2.3493e−06

1.71

2.6375e−05

1.12

9.0590e−07

1.94

Level 6

2.9944e−05

1.21

7.0584e−07

1.73

1.2943e−05

1.03

2.3296e−07

1.96

$$t=10^{-6}$$

Level 1

7.6534e−04

1.0686e−04

8.4013e−04

1.3300e−04

Level 2

6.1236e−04

0.32

6.0484e−05

0.82

2.9641e−04

1.50

4.7647e−05

1.48

Level 3

2.9091e−04

1.07

2.0736e−05

1.54

1.3129e−04

1.17

1.2717e−05

1.91

Level 4

1.4092e−04

1.05

7.1745e−06

1.54

5.6538e−05

1.22

3.4872e−06

1.87

Level 5

7.1986e−05

0.97

2.4755e−06

1.54

2.6627e−05

1.09

9.2409e−07

1.92

Level 6

3.0244e−05

1.25

7.0684e−07

1.81

1.3243e−05

1.01

2.4396e−07

1.92

6.4 General Polygonal Partitions

The numerical tests on general polygonal partitions was conducted by using the model Reissner–Mindlin problem of Sect. 6.1 on the unit square domain $$\Omega =(0,1)\times (0,1)$$. First of all, the WG algorithm 2.1 with the lowest order of element was used for computing numerical approximations of $$\varvec{\theta }$$ and w on regular polygonal partitions obtained from the usual uniform triangulations by adding the middle point of each edge as a new node; see Fig. 5a, b. Denote this type of partitions by $$\mathcal {T}_h^6$$. It should be noted that each element in $$\mathcal {T}_h^6$$ has 6 edges, though it appears to have 3 edges in geometry. The relative errors and convergence rates are shown in Table 9, and the results are in great consistency with theory for the numerical approximation $$\varvec{\theta }_h$$. For $$w_h$$, a superconvergence is clearly indicated in the discrete $$H^1$$-norm for the thickness $$t=10^{-3}$$ and $$t=10^{-6}$$.
Table 9

Numerical results of the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on the polygonal partitions $$\mathcal {T}_h^6$$ shown as in Fig. 5

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

1/4

5.5218e−04

6.9246e−05

5.2104e−04

5.2220e−05

1/8

1.9159e−04

1.53

2.2333e−05

1.63

1.8815e−04

1.47

1.4047e−05

1.89

1/16

7.5829e−05

1.34

5.9936e−06

1.90

8.2169e−05

1.20

3.6087e−06

1.96

1/32

3.4710e−05

1.13

1.5264e−06

1.97

3.9413e−05

1.06

9.0874e−07

1.99

1/64

1.6919e−05

1.04

3.8341e−07

1.99

1.9488e−05

1.02

2.2760e−07

2.00

$$t=10^{-3}$$

1/4

4.4280e−04

5.6744e−05

5.3045e−04

7.5682e−05

1/8

1.5273e−04

1.54

1.6518e−05

1.78

1.4874e−04

1.83

1.8953e−05

1.99

1/16

6.9247e−05

1.14

4.2877e−06

1.95

3.8649e−05

1.94

4.7206e−06

2.01

1/32

3.3809e−05

1.03

1.0826e−06

1.99

9.7971e−06

1.98

1.1785e−06

2.00

1/64

1.6803e−05

1.01

2.7135e−07

2.00

2.4652e−06

1.99

2.9452e−07

2.00

$$t=10^{-6}$$

1/4

4.4278e−04

5.6742e−05

5.3045e−04

7.5682e−05

1/8

1.5274e−04

1.54

1.6518e−05

1.78

1.4874e−04

1.83

1.8953e−05

2.00

1/16

6.9257e−05

1.14

4.2889e−06

1.95

3.8646e−05

1.94

4.7206e−06

2.01

1/32

3.3815e−05

1.03

1.0819e−06

1.99

9.7979e−06

1.98

1.1786e−06

2.00

1/64

1.6816e−05

1.01

2.6693e−07

2.02

2.4692e−06

1.99

2.9475e−07

2.00

In our final numerical experiment, we shall deform the uniform polygonal partitions in Fig. 5 by perturbing the mid-point of each edge to produce a set of irregular polygonal partitions $$\mathcal {T}_h^8$$ shown as in Fig. 6. It should be pointed out that $$\mathcal {T}_h^8$$ consists of non-convex polygons. The WG algorithm 2.1 with the lowest order of element was employed to produce numerical approximations for $$\varvec{\theta }$$ and w on $$\mathcal {T}_h^8$$. The corresponding numerical results are presented in Table 10. Readers are invited to draw their own conclusions on the performance of the WG finite element method proposed and analyzed in this paper.
Table 10

Numerical results for the WG element $$P_1(T)P_1(e)P_1(T)P_0(e)$$ on general polygonal partitions $$\mathcal {T}_h^8$$ shown as in Fig. 6

h

$$\frac{{|||}\mathbf{e}_h{|||}_{\theta }}{{|||}\mathbf{Q}_h{\varvec{\theta }}{|||}_{\theta }}$$

Rate

$$\frac{\Vert \mathbf{e}_0\Vert }{\Vert \mathbf{Q}_0{\varvec{\theta }}\Vert }$$

Rate

$$\frac{{|||} \xi _h{|||}_w}{{|||} Q_hw{|||}_w}$$

Rate

$$\frac{\Vert \xi _0\Vert }{\Vert Q_0w\Vert }$$

Rate

$$t=1$$

1/4

5.9684e−04

7.3550e−05

6.4424e−04

6.3826e−05

1/8

2.1602e−04

1.47

2.4601e−05

1.58

3.1924e−04

1.01

1.8470e−05

1.79

1/16

8.6088e−05

1.33

6.9087e−06

1.83

1.5994e−04

1.00

4.9065e−06

1.91

1/32

3.9091e−05

1.14

1.7862e−06

1.95

8.7866e−05

0.86

1.2964e−06

1.92

1/64

1.9016e−05

1.04

4.5066e−07

1.99

4.5458e−05

0.95

3.3237e−07

1.96

$$t=10^{-3}$$

1/4

7.3749e−04

7.5066e−05

6.0148e−04

8.5699e−05

1/8

3.5604e−04

1.05

2.8930e−05

1.38

2.0381e−04

1.56

2.2932e−05

1.90

1/16

2.0418e−04

0.80

1.0683e−05

1.44

1.0780e−04

0.92

6.1746e−06

1.89

1/32

9.6140e−05

1.09

3.2826e−06

1.70

5.4583e−05

0.98

1.5665e−06

1.98

1/64

4.5551e−05

1.08

1.0157e−06

1.70

2.8984e−05

0.91

4.0035e−07

1.97

$$t=10^{-6}$$

1/4

7.1054e−04

7.6957e−05

5.9248e−04

9.2136e−05

1/8

4.0828e−04

0.80

3.4285e−05

1.17

2.5963e−04

1.19

2.4299e−05

1.92

1/16

2.0805e−04

0.97

1.0878e−05

1.66

1.0078e−04

1.37

6.0760e−06

2.00

1/32

1.0308e−04

1.01

3.5763e−06

1.60

5.6399e−05

0.84

1.5946e−06

1.93

1/64

4.5607e−05

1.18

1.0437e−06

1.78

2.9250e−05

0.95

4.0213e−07

1.99

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