A Morley Type Triangular Finite Element with High Convergence for the Biharmonic Problem

Abstract

In this work, we construct a theoretical framework to develop non $C^0$ Morley type nonconforming high-convergence elements for biharmonic problems. For each element domain, $P_3$ should be included in the space of shape functions. Besides the degrees of freedom of Morley elements, we add the integrals and first-order moments of the normal derivatives on edges. The choice of degrees of freedom and shape function space guarantees the possibility of improving the convergence order. As an application, we specifically construct a Morley type element on triangular meshes. Lastly, numerical experiments are carried out to verify the feasibility of the element.

1. Introduction

In this work, we introduce a new non $C^0$ element method to handle biharmonic problems, which provides optimal convergence in $H^2$ norm. Biharmonic equations are closely related to practical physics problems. Nevertheless, the biharmonic problem still is challenging in mathematics. For a review of the biharmonic problem, see the literature [1,2].

When the conforming finite element method is employed for the elliptic problem, the error analysis only needs to consider the finite element space. Since biharmonic problems belong to fourth-order equations, typical conforming finite element methods require globally $C^1$ continuity. The practical application of conforming finite element methods is computationally difficult: either the local space dimension is quite large or the structure of the finite space is complicated. More precisely, conforming elements such as the Argyris element [3] and Bell’s triangle [4], the 16-dof Bogner–Fox–Schmit (BFS) element [5] is computationally expensive due to the high-degree polynomials involved. For fourth-order problems, computationally simple but formally complex conforming elements contain Hsieh–Clough–Tocher (HCT) elements [6], Zienkiewicz triangular elements [7], FVS quadrilateral elements [8] and so on.

To deal with the difficult problems caused by $C^1$, the nonconforming finite element method is usually used. Following Lascaux and Lesaint [9], two nonconforming finite elements that do not belong to $C^0$ were proposed. The first one is the famous Morley element [10,11]. For biharmonic problems, the Morley element has the least number of degrees of freedom on the triangular mesh. Simultaneously, this element is proved to have the optimal convergence order in the discrete $H_2$ and $H_1$-seminorms. The second element, known as Fraeijs de Veubeke triangle [12], selects averages as some degrees of freedom. The 12-dof Adini element [13] is a commonly used network on a rectangular grid, which contains $P_3$ on the rectangle domain. On a rectangular grid, the incomplete biquadratic element [14] is an analogy of the Morley element on a rectangular grid. Hu et al. [15] developed Powell–Sabin type macro elements on two-dimensional and three-dimensional rectangles. For quadrilateral meshes, Veubeke [8] proposed a novel biharmonic solution by splitting a convex quadrilateral into four non-overlapping triangles. Park and Sheen [16] offered a quadrilateral Morley element to handle biharmonic problems. However, the appearance of discontinuities will lead to consistency errors, so to ensure the well-posedness and convergence of finite elements, some basic continuity of the element space is still necessary. In addition, the accuracy of nonconforming element methods is usually low. And achieving higher accuracy using nonconforming finite element methods remains difficult.

Subsequently, many papers [17] conducted in-depth research on the superconvergence estimation for conforming finite elements. Nevertheless, the superconvergence of nonconforming finite element schemes is still at a relatively primitive stage. For biharmonic equations, Luo and Lin [18] presented the asymptotic error expansion for the Adini element. Mao and Shi [19] proved the superconvergence for the Morley element and incomplete biquadratic nonconforming element.

Using superconvergence methods, or by building an element where the shape function space includes $P_3$ or the internal penalty method, a second-order convergent numerical solution to the biharmonic equation can be obtained. Research on interior penalty methods can be found in [20]. Sun, Song, and Liu [21] created a high nonconforming method, where the penalty parameter is estimated accurately. On simplicial triangulation, Gao et al. [22] proposed a nonconforming element that converges at a rate of $O\left(h^2\right)$. They imposed $C^1$ interelement continuity on $C^0$ elements. Chen et al. [23,24] built a theoretical framework of $C^0$ nonconforming elements for the fourth-order elliptic problem and constructed a series of three-dimensional elements. They successfully constructed second-order convergent nonconforming elements for biharmonic equations. The virtual element method [25,26,27] also proposed high-convergence elements for biharmonic problems, but they cannot express the finite element spaces explicitly.

This paper proposes a method to construct non $C^0$ nonconforming elements for biharmonic equations. We force the shape function space to include $P_3$ in order to get the approximation error. The Morley element method on triangular meshes we have mentioned above is first-order-convergent. We enrich the degrees of freedom of the Morley element. More precisely, the degrees of freedom are chosen as $\left\{u\left(P\right),{\int }_{E}u\mathrm{d}s,{\int }_E{\displaystyle \frac{\partial u}{\partial \mathit{n}}}p\mathrm{d}s,\forall p\in P_1\left(E\right)\right\}$, where $\mathit{n}$ and P denote the outward unit normal vector to edge E and the vertex of E, respectively. The $C^0$ continuity requirement is relaxed, and we replace it with other constraints. The moments $\left\{{\int }_E{\displaystyle \frac{\partial u}{\partial \mathit{n}}}p\mathrm{d}s,\forall p\in P_1\left(E\right)\right\}$ are required to be continuous along the edge E in order to obtain the consistency error.

This work is arranged as follows. We introduce the setting of the biharmonic problem, as well as some symbols and preparations in Section 2. In Section 3, theoretical error estimates are constructed for this problem. Then, in Section 4, we present our triangular Morley type element and prove the unisolvency. Section 5 presents two numerical experiments to verify the validity of the theory. Lastly, we draw some conclusions.

2. The Model Problem and Some Preliminaries

We first introduce some notations. Let $\mathsf{\Omega }\subset {\mathbb{R}}^2$ be a bounded polygonal domain, and $\mathit{n}$ and $\mathit{\tau }$ denote the unit outer normal and tangential vectors on boundaries. The inner product on $L^2\left(\mathsf{\Omega }\right)$ will be denoted by $(·,·)$. For $m\ge 0$, $H^m\left(\mathsf{\Omega }\right)$ denotes the Sobolev space, and the corresponding norm and seminorm are presented by ${\parallel ·\parallel }_{m,\mathsf{\Omega }}$ and ${|·|}_{m,\mathsf{\Omega }}$, respectively. The space $H_0^m\left(\mathsf{\Omega }\right)$ is the closure in $H^m\left(\mathsf{\Omega }\right)$ of $C_0^{\infty }\left(\mathsf{\Omega }\right)$. Specifically, the following important space is given

$$\begin{array}{c}\hfill H_0^2\left(\mathsf{\Omega }\right)=\{u\in H^2\left(\mathsf{\Omega }\right)|u={\displaystyle \frac{\partial u}{\partial \mathit{n}}}=0\mathrm{on}\partial \mathsf{\Omega }\}.\end{array}$$

We denote by $\partial \mathsf{\Omega }$ the boundary of $\mathsf{\Omega }$. We discuss the biharmonic problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\mathsf{\Delta }}^{2}u& =f\mathrm{in}\mathsf{\Omega },\hfill \\ \hfill u& =0\mathrm{on}\partial \mathsf{\Omega },\hfill \\ \hfill {\displaystyle \frac{\partial u}{\partial \mathit{n}}}& =0\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(1)

$\mathsf{\Delta }$ is a standard Laplace operator. The weak form of (1) is to search $u\in H_0^2\left(\mathsf{\Omega }\right)$ satisfying

$$\begin{array}{c}\hfill a(u,v)=(f,v),\forall v\in H_0^2\left(\mathsf{\Omega }\right),\end{array}$$

(2)

where

$$\begin{array}{c}\hfill a(u,v)=(\mathsf{\Delta }u,\mathsf{\Delta }v)+(1-\sigma )\left[2\left({\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}},{\displaystyle \frac{{\partial }^{2}v}{\partial x\partial y}}\right)-\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)-\left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\right)\right].\end{array}$$

Here, $\sigma$ represents Poisson’s ratio with numerical range $[0,{\displaystyle \frac{1}{2}}]$.

Let S be a polygonal domain. In addition to $u\in H^3\left(S\right)\cap H_0^2\left(S\right)$ and $v\in H_0^2\left(S\right)$,

$$\begin{array}{c}\hfill 2{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}},{\displaystyle \frac{{\partial }^{2}v}{\partial x\partial y}}\right)}_S-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)}_S-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\right)}_S=0,\end{array}$$

(3)

since

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2& {\left({\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}},{\displaystyle \frac{{\partial }^{2}v}{\partial x\partial y}}\right)}_S-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)}_S-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\right)}_S\hfill \\ & ={〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \mathit{\tau }}},{\displaystyle \frac{\partial v}{\partial \mathit{\tau }}}〉}_{\partial S}-{〈{\displaystyle \frac{{\partial }^{2}u}{\partial {\mathit{\tau }}^2}},{\displaystyle \frac{\partial v}{\partial \mathit{n}}}〉}_{\partial S}.\hfill \end{array}\end{array}$$

(4)

3. Convergence Theorems

Assume that $\left\{{\mathcal{T}}_h\right\}$ is a family of regular convex polygonal subdivisions of $\mathsf{\Omega }$. ${\mathcal{E}}_h$ represents the set of edges. Let $V_{h0}$ denote the finite element space with boundary conditions of (1) in some way.

Then the nonconforming finite element solution of (2) is to search $u_h\in V_{h0}$ satisfying

$$\begin{array}{c}\hfill a_h(u_h,v_h)=(f,v_h),\forall v_h\in V_{h0},\end{array}$$

(5)

where

$$\begin{array}{c}\hfill a_h(u_h,v_h)=\sum _{K\in {\mathcal{T}}_h}{a}_K(u_h,v_h).\end{array}$$

(6)

Let $\left|\right|·{\left|\right|}_{m,h}$ and ${|·|}_{m,h}$ represent the norm and seminorm associated with partition:

$$\begin{array}{c}\hfill {\left|\left|w\right|\right|}_{m,h}={\left[\sum _{K\in {\mathcal{T}}_h}{\left|\left|w\right|\right|}_{m,K}^2\right]}^{1/2};{\left|w\right|}_{m,h}={\left[\sum _{K\in {\mathcal{T}}_h}{\left|w\right|}_{m,K}^2\right]}^{1/2}.\end{array}$$

For any integer $m\ge 0$, $P_m\left(K\right)$ represents the set of polynomials with degree less than or equal to m defined on K. Define global interpolation operator ${\mathsf{\Pi }}_h:H_0^2\left(\mathsf{\Omega }\right)\to V_{h0}$, ${\mathsf{\Pi }}_K={\mathsf{\Pi }}_h{|}_K,\forall K\in {\mathcal{T}}_h$. If ${\mathsf{\Pi }}_{K}w=w$ on K, $\forall w\in P_k\left(K\right)$, $\forall K\in {\mathcal{T}}_h$, by the Bramble–Hilbert lemma, one has

$$\begin{array}{c}\hfill |w-{\mathsf{\Pi }}_K{w|}_{j,K}\le C{h}^{k+1-j}{\left|w\right|}_{k+1,K},0\le j\le m\mathrm{for}v\in H^m\left(K\right).\end{array}$$

(7)

Here, C is independent of h.

On any edge E, ${[·]}_E$, $\mathit{\tau }$ and $\mathit{n}$ denote the jump of one function, unit tangential and normal vectors, respectively. We begin the error analysis with the second Strang lemma.

Lemma 1.

Assume that $u\in H_0^2\left(\mathsf{\Omega }\right)$ and $u_h\in V_{h0}$ are solutions of (2) and (5), respectively. Then,

$$\begin{array}{c}\hfill |u-u_h{|}_{2,h}\le C\left(\underset{w_h\in V_{h0}}{inf}{|u-w_h|}_{2,h}+\underset{w_h\in V_{h0}}{sup}{\displaystyle \frac{|E_h(u,w_h)|}{|w_h{|}_{2,h}}}\right),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill E_h(u,w_h)=a_h(u,w_h)-(f,w_h).\end{array}$$

(9)

For $w_h\in V_{h0}$, invoking the definition of $a_K(u,w_h)$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_K& (u,w_h)\hfill \\ & ={(\mathsf{\Delta }u,\mathsf{\Delta }{w}_h)}_K+(1-\sigma )\left[2{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}},{\displaystyle \frac{{\partial }^2{w}_h}{\partial x\partial y}}\right)}_K-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},{\displaystyle \frac{{\partial }^2{w}_h}{\partial y^2}}\right)}_K-{\left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},{\displaystyle \frac{{\partial }^2{w}_h}{\partial x^2}}\right)}_K\right]\hfill \\ & =-{(\nabla \mathsf{\Delta }u,\nabla w_h)}_K+{〈\mathsf{\Delta }u,{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}〉}_{\partial K}+(1-\sigma )\left[{〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \mathit{\tau }}},{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}〉}_{\partial K}-{〈{\displaystyle \frac{{\partial }^{2}u}{\partial {\mathit{\tau }}^2}},{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}〉}_{\partial K}\right].\hfill \end{array}\end{array}$$

(10)

$$\begin{array}{c}\hfill \begin{array}{cc}& E_1(u,w_h)=\sum _{K\in {\mathcal{T}}_h}{〈\mathsf{\Delta }u-(1-\sigma ){\displaystyle \frac{{\partial }^{2}u}{\partial {\mathit{\tau }}^2}},{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}〉}_{\partial K},\hfill \\ & E_2(u,w_h)=(1-\sigma )\sum _{K\in {\mathcal{T}}_h}{〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \mathit{\tau }}},{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}〉}_{\partial K},\hfill \end{array}\end{array}$$

the consistency difference between $a_h(u,w_h)$ and $(f,w_h)$ is

$$\begin{array}{c}\hfill a_h(u,w_h)-(f,w_h)=-\sum _{K\in {\mathcal{T}}_h}{\left(\nabla \mathsf{\Delta }u,\nabla w_h\right)}_K-(f,w_h)+E_1(u,w_h)+E_2(u,w_h).\end{array}$$

(11)

Based on the analysis of the above paragraphs, we are ready to give the following error estimate.

Theorem 1.

Assume that $u\in H^4\left(\mathsf{\Omega }\right)\cap H_0^2\left(\mathsf{\Omega }\right)$ and $u_h\in V_{h0}$ are solutions of (2) and (5), respectively. If $V_{h0}$ satisfies the following conditions:

(H1) 

$\forall K\in {\mathcal{T}}_h$, $\forall v\in P_3\left(K\right)$, ${\mathsf{\Pi }}_{K}v=v$,

(H2) 

$\forall v_h\in V_{h0}$, $v_h$ is continuous at the interior vertices and zero at the vertices on $\partial \mathsf{\Omega }$,

(H3) 

$\forall v_h\in V_{h0}$, ${\int }_E{\left[v_h\right]}_E\mathrm{d}s=0$,

(H4) 

$\forall v_h\in V_{h0}$, ${\int }_E{\left[{\displaystyle \frac{\partial u}{\partial \mathit{n}}}\right]}_{E}p\mathrm{d}s=0,\forall p\in P_1\left(E\right)$, $\forall E\in {\mathcal{E}}_h$,

then,

$$\begin{array}{ccc}\hfill |u-u_h{|}_{2,h}& \le & C{h}^2{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0).\hfill \end{array}$$

(12)

Proof. 

Denote by $P_l^E$ the $L^2$-projection from $L^2\left(E\right)$ to $P_l\left(E\right)$ and $P_l^K$ is the $L^2$-projection from $L^2\left(K\right)$ to $P_l\left(K\right)$. Denote by $w_h^I$ the lowest-order $C^0$-conforming interpolation of $w_h$. We first consider the consistency difference $E_h(u,w_h)$. The first two terms in (11) are bounded as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|-\sum _{K\in {\mathcal{T}}_h}{\left(\nabla \mathsf{\Delta }u,\nabla w_h\right)}_K-(f,w_h)\right|& =\left|-\sum _{K\in {\mathcal{T}}_h}{\int }_{\partial K}{\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}{w}_h\mathrm{d}s\right|=\left|-\sum _{E\in {\mathcal{E}}_h}{\int }_E{\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}\left[w_h\right]\mathrm{d}s\right|\hfill \\ & =\left|-\sum _{E\in {\mathcal{E}}_h}{\int }_E\left({\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}-P_0^E\left({\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}\right)\right)\left[w_h\right]\mathrm{d}s\right|\hfill \\ & =\left|-\sum _{E\in {\mathcal{E}}_h}{\int }_E\left({\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}-P_0^E\left({\displaystyle \frac{\partial \mathsf{\Delta }u}{\partial \mathit{n}}}\right)\right)[w_h-w_h^I]\mathrm{d}s\right|\hfill \\ & \le C{h}^2{\left|u\right|}_4{\left|w_h\right|}_{2,h}.\hfill \end{array}\end{array}$$

(13)

Abbreviating ${\eta }_E=\left(\mathsf{\Delta }u-(1-\sigma ){\displaystyle \frac{{\partial }^{2}u}{\partial {\mathit{\tau }}^2}}\right){|}_E$ for each edge $E\in {\mathcal{E}}_h$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |E_1(u,w_h)|& =\left|\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈\mathsf{\Delta }u-(1-\sigma ){\displaystyle \frac{{\partial }^{2}u}{\partial {\mathit{\tau }}^2}},{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}〉}_E\right|\hfill \\ & =\left|\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈{\eta }_E-P_l^E\left({\eta }_E\right),{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}〉}_E\right|\hfill \\ & =\left|\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈{\eta }_E-P_l^E\left({\eta }_E\right),{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}-P_0^E\left({\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}\right)〉}_E\right|\hfill \\ & \le \sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{\left|{\eta }_E-P_l^E\left({\eta }_E\right)\right|}_{0,E}{\left|{\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}-P_0^E\left({\displaystyle \frac{\partial w_h}{\partial \mathit{n}}}\right)\right|}_{0,E}\hfill \\ & \le \sum _{K\in {\mathcal{T}}_h}C{h}^{l+{\displaystyle \frac{1}{2}}}{\left|u\right|}_{l+3,K}{h}^{{\displaystyle \frac{1}{2}}}{\left|w_h\right|}_{2,K}\hfill \\ & \le \left\{\begin{array}{c}\hfill {Ch\left|u\right|}_3{\left|w_h\right|}_{2,h},l=0,\\ \hfill C{h}^2{\left|u\right|}_4{\left|w_h\right|}_{2,h},l=1.\end{array}\right.\hfill \end{array}\end{array}$$

(14)

Since ${\int }_E\left[{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}\right]p\mathrm{d}s=-{\int }_E{\displaystyle \frac{\partial p}{\partial \mathit{\tau }}}\left[w_h\right]\mathrm{d}s=0$, $\forall p\in P_1\left(E\right)$, similarly, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |E_2(u,w_h)|& =\left|(1-\sigma )\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }},{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}〉}_E\right|\hfill \\ & =\left|(1-\sigma )\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}-P_l^E\left({\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}\right),{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}〉}_E\right|\hfill \\ & =\left|(1-\sigma )\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{〈{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}-P_l^E\left({\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}\right),{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}-P_0^E\left({\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}\right)〉}_E\right|\hfill \\ & \le (1-\sigma )\sum _{K\in {\mathcal{T}}_h}\sum _{E\in \partial K\cap {\mathcal{E}}_h}{\left|{\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}-P_l^E\left({\displaystyle \frac{{\partial }^{2}u}{\partial \mathit{n}\partial \tau }}\right)\right|}_{0,E}{\left|{\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}-P_0^E\left({\displaystyle \frac{\partial w_h}{\partial \mathit{\tau }}}\right)\right|}_{0,E}\hfill \\ & \le C{h}^{l+1}{\left|u\right|}_{l+3}{\left|w_h\right|}_{2,h},l=0,1.\hfill \end{array}\end{array}$$

(15)

Collecting the above estimates, here $l=1$, we have

$$\begin{array}{c}\hfill |E_h(u,w_h)|\le C{h}^2\left({\left|u\right|}_4+{\left|\right|f\left|\right|}_0\right){\left|w_h\right|}_{2,h}.\end{array}$$

(16)

Since ${\mathsf{\Pi }}_K$ preserves functions in $P_3\left(K\right)$, (7) implies the following result

$$\begin{array}{c}\hfill \underset{w_h\in V_{h0}}{inf}|u-w_h{|}_{2,h}\le |u-{\mathsf{\Pi }}_h{u|}_{2,h}\le C{h}^2{\left|u\right|}_4.\end{array}$$

(17)

A combination of (16) and (17) completes the proof. □

Next, we derive the error estimates in the $H^1$- and $L^2$-norm. We shall consider the duality problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\mathsf{\Delta }}^2\varphi & =h\mathrm{in}\mathsf{\Omega },\hfill \\ \hfill \varphi & =0\mathrm{on}\partial \mathsf{\Omega },\hfill \\ \hfill {\displaystyle \frac{\partial \varphi }{\partial \mathit{n}}}& =0\mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.\end{array}$$

(18)

Since $\mathsf{\Omega }$ is a convex polygon, it holds (see [28])

$$\begin{array}{c}\hfill {\left|\right|\varphi \left|\right|}_3\le {C\left|\right|h\left|\right|}_{-1},\forall h\in H^{-1}\left(\mathsf{\Omega }\right),\end{array}$$

(19)

and we assume that

$$\begin{array}{c}\hfill {\left|\right|\varphi \left|\right|}_4\le C\left|\right|h\left|\right|,\forall h\in L^2\left(\mathsf{\Omega }\right).\end{array}$$

(20)

Remark 1.

For $h\in H^{-1}\left(\mathsf{\Omega }\right)$ the regularity estimate (19) is always correct. For $h\in L^2\left(\mathsf{\Omega }\right)$, there exists q $(0\le q\le 1)$ that satisfies

$$\begin{array}{c}\hfill {\left|\right|\varphi \left|\right|}_{3+q}\le C\left|\right|h\left|\right|,\forall h\in L^2\left(\mathsf{\Omega }\right).\end{array}$$

The maximum angle in Ω determines the value of q. Moreover, there exists a ${\theta }_0<\pi$ satisfying $q=1$, and thus $\varphi \in H^4\left(\mathsf{\Omega }\right)$ holds.

Theorem 2.

As Theorem 1, suppose $u\in H^4\left(\mathsf{\Omega }\right)\cap H_0^2\left(\mathsf{\Omega }\right)$ satisfy (2) and (5), respectively. $V_{h0}$ satisfies the same conditions as Theorem 1. Then,

$$\begin{array}{c}\hfill |u-u_h{|}_{1,h}\le C{h}^3{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0).\end{array}$$

(21)

Proof. 

Let $e_h=u-u_h$ and denote by $e_h^I\in H_0^1\left(\mathsf{\Omega }\right)$ the lowest-order $C^0$-conforming interpolation of $e_h$. Setting $g=-\mathsf{\Delta }{e}_h^I\in H^{-1}\left(\mathsf{\Omega }\right)$ in (18), $\varphi \in H_0^2\left(\mathsf{\Omega }\right)$ satisfies

$${\mathsf{\Delta }}^2\varphi =-\mathsf{\Delta }{e}_h^I,$$

then, (19) implies that

$$\begin{array}{c}\hfill {\left|\right|\varphi \left|\right|}_3\le \left|\right|\mathsf{\Delta }{e}_h^I{\left|\right|}_{-1}\le C{\left|e_h^I\right|}_1.\end{array}$$

(22)

Then, it yields that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |e_h^I{|}_1^2& =(g,e_h^I)=({\mathsf{\Delta }}^2\varphi ,e_h^I)=-(\nabla \mathsf{\Delta }\varphi ,\nabla e_h^I)\hfill \\ & =\sum _{K\in {\mathcal{T}}_h}{(\nabla \mathsf{\Delta }\varphi ,\nabla (e_h-e_h^I))}_K-\sum _{K\in {\mathcal{T}}_h}{(\nabla \mathsf{\Delta }\varphi ,\nabla e_h)}_K=I_1+I_2.\hfill \end{array}\end{array}$$

(23)

First, we use the estimate (7) to obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |I_1|\le \sum _{K\in {\mathcal{T}}_h}{\left|\varphi \right|}_{3,K}|e_h-e_h^I{|}_{1,K}\le {Ch\left|\varphi \right|}_3{\left|e_h\right|}_{2,h}.\end{array}\end{array}$$

(24)

From the above Theorem 1, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |e_h{|}_{2,h}=|u-u_h{|}_{2,h}\le C{h}^2{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0).\end{array}\end{array}$$

(25)

Together with (24), one has

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |I_1|\le C{h}^3{\left|\varphi \right|}_3{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0).\end{array}\end{array}$$

(26)

Observing (11), we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill I_2=-\sum _{K\in {\mathcal{T}}_h}(\nabla \mathsf{\Delta }\varphi ,\nabla e_h)=a_h(\varphi ,e_h)-E_1(\varphi ,e_h)-E_2(\varphi ,e_h).\end{array}\end{array}$$

(27)

Recalling the proof of Theorem 1, we can bound the last two terms in the above equation

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |E_i(\varphi ,e_h){|\le Ch|\varphi |}_3{\left|e_h\right|}_{2,h}i=1,2.\end{array}\end{array}$$

(28)

For $a_h(\varphi ,e_h)$, the interpolation ${\mathsf{\Pi }}_h$ serves to get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_h(\varphi ,e_h)=& a_h({\mathsf{\Pi }}_h\varphi ,e_h)+a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h)\hfill \\ \hfill =& a_h(u,{\mathsf{\Pi }}_h\varphi )-(f,{\mathsf{\Pi }}_h\varphi )+a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h)\hfill \\ \hfill =& a_h(u,{\mathsf{\Pi }}_h\varphi -\varphi )-(f,{\mathsf{\Pi }}_h\varphi -\varphi )+a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h).\hfill \end{array}\end{array}$$

(29)

The first two terms are estimated as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(u,{\mathsf{\Pi }}_h\varphi -\varphi )-(f,{\mathsf{\Pi }}_h\varphi -\varphi )|=|E_h(u,{\mathsf{\Pi }}_h\varphi -\varphi )|& \le C{h}^2{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0\left)\right|{\mathsf{\Pi }}_h{\varphi -\varphi |}_{2,h}\hfill \\ & \le C{h}^3{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0{\left)\right|\varphi |}_3.\hfill \end{array}\end{array}$$

(30)

The estimate of the last term in (29) is:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h)|& \le C|\varphi -{\mathsf{\Pi }}_h{\varphi |}_{2,h}|e_h{|}_{2,h}\le {Ch\left|\varphi \right|}_3{\left|e_h\right|}_{2,h}.\hfill \end{array}\end{array}$$

(31)

Substituting (30) and (31) into (29) leads to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(\varphi ,e_h)|& \le C{h}^3{\left|\varphi \right|}_3{\left|\right(\left|u\right|}_4+{\left|\right|f\left|\right|}_0),\hfill \end{array}\end{array}$$

(32)

which, together with the estimate (28), implies that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |I_2|\le C{h}^3{\left|\varphi \right|}_3{\left|\right(\left|u\right|}_4+{\left|\right|f\left|\right|}_0).\end{array}\end{array}$$

(33)

Plugging (26) and (33) into (23), we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |e_h^I{|}_1^2\le C{h}^3{\left|\varphi \right|}_3{\left|\right(\left|u\right|}_4+{\left|\right|f\left|\right|}_0)\le C{h}^3|e_h^I{|}_1{\left|\right(\left|u\right|}_4+{\left|\right|f\left|\right|}_0),\end{array}\end{array}$$

(34)

Finally, applying the triangle inequality leads to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |e_h{|}_{1,h}& \le |e_h-e_h^I{|}_{1,h}+{\left|e_h^I\right|}_1\hfill \\ & \le Ch|e_h{|}_{2,h}+{\left|e_h^I\right|}_1\hfill \\ & \le C{h}^3{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0),\hfill \end{array}\end{array}$$

(35)

which completes the proof. □

Theorem 3.

Suppose $u\in H^4\left(\mathsf{\Omega }\right)\cap H_0^2\left(\mathsf{\Omega }\right)$ satisfy (2) and (5), respectively. $V_{h0}$ satisfies the same conditions as Theorem 1. Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|\right|u-u_h{\left|\right|}_0& \le C{h}^4{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0).\hfill \end{array}\end{array}$$

(36)

Proof. 

For convenience, the same notations as in the proof of Theorem 2 are adopted. Setting $g=e_h\in L^2\left(\mathsf{\Omega }\right)$ in (18), $\varphi \in H_0^2\left(\mathsf{\Omega }\right)$ satisfies

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathsf{\Delta }}^2\varphi =e_h,\end{array}\end{array}$$

(37)

and then (20) implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left|\right|\varphi \left|\right|}_4& \le C\left|\right|e_h\left|\right|.\hfill \end{array}\end{array}$$

(38)

Observing (11) again, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left|\right|e_h{\left|\right|}^2=({\mathsf{\Delta }}^2\varphi ,e_h)=a_h(\varphi ,e_h)-E_1(\varphi ,e_h)-E_2(\varphi ,e_h).\end{array}\end{array}$$

(39)

For the last two terms, arguments similar to the proof of Theorem 1 yield

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |E_i(\varphi ,e_h)|& \le C{h}^2{\left|\right|\varphi \left|\right|}_4|e_h{|}_{2,h}\le C{h}^4{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0{\left)\right|\left|\varphi \right||}_4,i=1,2.\hfill \end{array}\end{array}$$

(40)

For $a_h(\varphi ,e_h)$, we proceed as in the proof of Theorem 2 and obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(\varphi ,e_h)|& =|a_h(u,{\mathsf{\Pi }}_h\varphi -\varphi )-(f,{\mathsf{\Pi }}_h\varphi -\varphi )+a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h)|\hfill \\ & \le C{h}^4{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0{\left)\right|\varphi |}_4,\hfill \end{array}\end{array}$$

(41)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(u,{\mathsf{\Pi }}_h\varphi -\varphi )-(f,{\mathsf{\Pi }}_h\varphi -\varphi )|& \le C{h}^2{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0\left)\right|{\mathsf{\Pi }}_h{\varphi -\varphi |}_{2,h}\hfill \\ & \le C{h}^4{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0{\left)\right|\varphi |}_4,\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_h(\varphi -{\mathsf{\Pi }}_h\varphi ,e_h)|& \le C|\varphi -{\mathsf{\Pi }}_h{\varphi |}_{2,h}|e_h{|}_{2,h}\le C{h}^4{\left(\right|u|}_4+{\left|\right|f\left|\right|}_0{\left)\right|\varphi |}_4.\hfill \end{array}\end{array}$$

Combining (38)–(41) completes the proof. □

4. A New Morley Type Element with High Convergence

In this section, we introduce our new nonconforming element and prove the unisolvency. Let K be a triangle where $V_1,V_2,V_3$ denote the vertices, $E_j$ designates the edge between $V_j$ and $V_{j+1}$ modulo 3, $j=1,2,3$, and ${\mathit{n}}_j$ denotes the unit outward normal to $E_j$, $j=1,2,3$.

Definition 1.

The triangular finite element $(K,P_K,{\mathsf{\Phi }}_K)$ is defined as follows:

• K is a triangle.
• $P_K=P_3\left(K\right)+\mathrm{Span}\{x^{3}y,x{y}^3\}$.
• The degrees of freedom are given by ${\mathsf{\Phi }}_K=\left\{u\left(V_j\right),{\int }_{E_j}u\mathrm{d}s,{\int }_{E_j}{\displaystyle \frac{\partial u}{\partial {\mathit{n}}_j}}{q}_j\mathrm{d}s:j=1,2,3\right\}$,

where $q_i\in P_1\left(E_i\right)$ is taken such that $q_i\left(V_i\right)=-1$ and $q_i\left(V_{i+1}\right)=1$.

Theorem 4.

The degrees of freedom $P_K$ are unisolvent for ${\mathsf{\Phi }}_K$.

Proof. 

For $j=1,\dots ,3$, we have

$$\begin{array}{ccc}\hfill {\int }_{E_j}\nabla \varphi \mathrm{d}s& =& (0,0),\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill {\int }_{E_j}\nabla \varphi p\mathrm{d}s& =& (0,0),\hfill \end{array}$$

(43)

since

$$\begin{array}{ccc}\hfill {\int }_{E_j}{\displaystyle \frac{\partial \varphi }{\partial \mathit{\tau }}}\mathrm{d}s& =& \varphi \left(V_{j+1}\right)-\varphi \left(V_j\right)=0.\hfill \end{array}$$

We prove the unisolvency with the help of a reference triangle $\widehat{\mathsf{\Delta }}$ whose vertices are denoted by ${\widehat{V}}_1=[0,0],{\widehat{V}}_2=[1,0],{\widehat{V}}_3=[0,1]$. Correspondingly, we denote the edges of the reference triangle by $\widehat{E_j}$ and $\widehat{{\mathit{n}}_{\mathit{j}}}$ is the unit normal to the edge $\widehat{E_j},j=1,2,3$. We only need to consider the unisolvency on the reference triangle since there exists a unique map F from the reference triangle to an arbitrary triangle and ${\nabla }_{\widehat{x},\widehat{y}}\widehat{\varphi }=D{\left(F\right)}^T{\nabla }_{x,y}\varphi$, where $D\left(F\right)$ represents the Jacobian matrix of F. Define $\widehat{\mathsf{\Phi }}(\widehat{x},\widehat{y})=\mathrm{Span}\{1,\widehat{x},\widehat{y},\widehat{x}\widehat{y},{\widehat{x}}^2,{\widehat{y}}^2,{\widehat{x}}^2\widehat{y},\widehat{x}{\widehat{y}}^2,{\widehat{x}}^3,{\widehat{y}}^3,{\widehat{x}}^3\widehat{y},\widehat{x}{\widehat{y}}^3\}$. Define a matrix

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A=[& \widehat{\mathsf{\Phi }}\left({\widehat{V}}_1\right)\widehat{\mathsf{\Phi }}\left({\widehat{V}}_2\right)\widehat{\mathsf{\Phi }}\left({\widehat{V}}_3\right){\int }_{{\widehat{E}}_1}\widehat{\mathsf{\Phi }}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_2}\widehat{\mathsf{\Phi }}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_3}\widehat{\mathsf{\Phi }}\mathrm{d}\widehat{s}\hfill \\ & {\int }_{{\widehat{E}}_1}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_1}}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_2}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_2}}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_3}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_3}}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_1}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_1}}\widehat{x}\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_2}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_2}}(\widehat{x}-\widehat{y})\mathrm{d}\widehat{s}{\int }_{{\widehat{E}}_3}{\displaystyle \frac{\partial \widehat{\mathsf{\Phi }}}{\partial {\widehat{\mathit{n}}}_3}}\widehat{y}\mathrm{d}\widehat{s}{]}^T.\hfill \end{array}\end{array}$$

Precisely,

$$\begin{array}{c}\hfill A=\left[\begin{array}{cccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 1& 0& 0& 0& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 1& 0& 0& 0& 1& 0& 0\\ 1& 1/2& 0& 0& 1/3& 0& 0& 0& 1/4& 0& 0& 0\\ \sqrt{2}& \sqrt{2}/2& \sqrt{2}/2& \sqrt{2}/6& \sqrt{2}/3& \sqrt{2}/3& \sqrt{2}/12& \sqrt{2}/12& \sqrt{2}/4& \sqrt{2}/4& \sqrt{2}/20& \sqrt{2}/20\\ 1& 0& 1/2& 0& 0& 1/3& 0& 0& 0& 1/4& 0& 0\\ 0& 0& 1& 1/2& 0& 0& 1/3& 0& 0& 0& 1/4& 0\\ 0& 1& 1& 1& 1& 1& 2/3& 2/3& 1& 1& 1/2& 1/2\\ 0& 1& 0& 1/2& 0& 0& 0& 1/3& 0& 0& 0& 1/4\\ 0& 0& 1/2& 1/3& 0& 0& 1/4& 0& 0& 0& 1/5& 0\\ 0& 0& 0& 0& 1/3& -1/3& 1/6& -1/6& 1/2& -1/2& 1/5& -1/5\\ 0& 1/2& 0& 1/3& 0& 0& 0& 1/4& 0& 0& 0& 1/5\end{array}\right],\end{array}$$

(44)

and here, for convenience, we ignore the positive and negative sign for calculus since it does not affect the final determinant. By a simple computation, we obtain

$$det\left(A\right)={\displaystyle \frac{\sqrt{2}}{24883200}}\ne 0.$$

This completes the proof. □

5. Numerical Experiments

Now, we will use the proposed element method to calculate two examples. Here, we take $\sigma =0$. Set $\mathsf{\Omega }={[0,1]}^2\subset {\mathbb{R}}^2$. We first divide the unit into $n\times n$ uniform squares, and then connect the secondary diagonals of each square. Thus, two types of triangulation are used: uniform triangular meshes (see Figure 1 as an example) and randomly perturbed triangular meshes (see Figure 2 as an example).

Example 1.

Consider problem (1) with $f={\mathsf{\Delta }}^{2}u$, where $u={(sin\left(\pi x\right)sin\left(\pi y\right))}^2$.

The errors measured by the seminorms ${|·|}_{i,h},i=1,2$ and $L^2$ norm are shown in

Table 1. Error table for uniform triangular meshes applied to Example 1.

n	$\left|\right|\mathit{u}-{\mathit{u}}_{\mathit{h}}{\left|\right|}_0$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{1,\mathit{h}}$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{2,\mathit{h}}$
	Value	Order	Value	Order	Value	Order
4	0.0030	-	0.0504	-	1.4953	-
8	$2.3201\times {10}^{-4}$	3.6927	0.0072	2.8074	0.3935	1.9260
16	$1.7000\times {10}^{-5}$	3.7706	0.0010	2.8480	0.1016	1.9535
32	$1.1538\times {10}^{-6}$	3.8811	$1.3207\times {10}^{-4}$	2.9206	0.0258	1.9775
64	$7.4849\times {10}^{-8}$	3.9463	$1.6901\times {10}^{-5}$	2.9661	0.0065	1.9889
128	$4.4861\times {10}^{-9}$	4.0611	$2.1350\times {10}^{-6}$	2.9848	0.0016	2.0224

and

Table 2. Error table for randomly perturbed triangular meshes applied to Example 1.

n	$\left|\right|\mathit{u}-{\mathit{u}}_{\mathit{h}}{\left|\right|}_0$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{1,\mathit{h}}$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{2,\mathit{h}}$
	Value	Order	Value	Order	Value	Order
4	0.0033	-	0.0547	-	1.5881	-
8	$2.5517\times {10}^{-4}$	3.6929	0.0077	2.8286	0.4098	1.9543
16	$1.7753\times {10}^{-5}$	3.8453	0.0010	2.9449	0.1049	1.9659
32	$1.2181\times {10}^{-6}$	3.8654	$1.3816\times {10}^{-4}$	2.8556	0.0268	1.9687
64	$7.8719\times {10}^{-8}$	3.9518	$1.7656\times {10}^{-5}$	2.9681	0.0068	1.9786
128	$5.0060\times {10}^{-9}$	3.9750	$2.2300\times {10}^{-6}$	2.9850	0.0017	2.0000

, respectively. The optimal order of convergence in discrete $H^2$, $H^1$ seminorms and $L^2$ norm is indicated in Theorems 1–3. Obviously, the convergence orders in $L^2$ norm and $H^1$ and $H^2$ seminorms mathematically approximate 4, 3, and 2, respectively. In short, from the following two tables, we can find that theoretical analysis and numerical results are consistent.

Example 2.

Assume that $u=y(y-1){(x-1)}^{2}sin\left(3\pi x^2\right)sin\left(3\pi y^2\right)$ satisfies (1).

Similarly, the errors measured by the seminorms ${|·|}_{i,h},i=1,2$ and $L^2$ norm are shown in

Table 3. Error table for uniform triangular meshes applied to Example 2.

n	$\left|\right|\mathit{u}-{\mathit{u}}_{\mathit{h}}{\left|\right|}_0$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{1,\mathit{h}}$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{2,\mathit{h}}$
	Value	Order	Value	Order	Value	Order
4	0.0057	-	0.0978	-	2.8488	-
8	$4.0445\times {10}^{-4}$	3.8169	0.0121	3.0148	0.6773	2.0725
16	$4.5274\times {10}^{-5}$	3.1592	0.0024	2.3339	0.2463	1.4594
32	$3.4742\times {10}^{-6}$	3.7039	$3.5767\times {10}^{-4}$	2.7463	0.0668	1.8825
64	$2.4140\times {10}^{-7}$	3.8472	$4.8606\times {10}^{-5}$	2.8794	0.0173	1.9491
128	$1.5714\times {10}^{-8}$	3.9413	$6.3244\times {10}^{-6}$	2.9421	0.0044	1.9752

and

Table 4. Error table for randomly perturbed triangular meshes applied to Example 2.

n	$\left|\right|\mathit{u}-{\mathit{u}}_{\mathit{h}}{\left|\right|}_0$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{1,\mathit{h}}$		$|\mathit{u}-{\mathit{u}}_{\mathit{h}}{|}_{2,\mathit{h}}$
	Value	Order	Value	Order	Value	Order
4	0.0050	-	0.0902	-	2.7398	-
8	$4.3183\times {10}^{-4}$	3.5334	0.0127	2.8283	0.6970	1.9748
16	$4.5836\times {10}^{-5}$	3.2359	0.0025	2.3448	0.2482	1.4897
32	$3.6049\times {10}^{-6}$	3.6685	$3.6911\times {10}^{-4}$	2.7598	0.0685	1.8579
64	$2.4998\times {10}^{-7}$	3.8501	$5.0041\times {10}^{-5}$	2.8829	0.0177	1.9524
128	$1.6323\times {10}^{-8}$	3.9368	$6.5130\times {10}^{-6}$	2.9417	0.0045	1.9758

, respectively. And the numerical experimental results are consistent with the theoretical analysis.

6. Conclusions

In this paper, an error analysis for the biharmonic problem is given. And the error analysis provides a theoretical basis for constructing nonconforming high-order Morley type elements for biharmonic problems. Subsequently, a Morley type triangular element is presented. Numerical experiments are carried out on two types of triangular meshes. Furthermore, based on this error analysis, we can also study the Morley type element on quadrilateral meshes in the future.