Optimal quadratic element on rectangular grids for $H^1$ problems

TL;DR

This paper introduces a new quadratic finite element method on rectangular grids for $H^1$ problems, achieving optimal convergence and providing lower bounds for eigenvalues, with promising numerical results.

Contribution

It presents a reduced quadratic rectangular Morley element with proven convergence rates and eigenvalue bounds, enhancing finite element analysis for $H^1$ problems.

Findings

Convergence rate of $O(h^2)$ in energy norm on uniform grids.

Lower bounds for eigenvalues are obtained.

Numerical results demonstrate the method's potential.

Abstract

In this paper, a piecewise quadratic finite element method on rectangular grids for the $H^1$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is $O(h^2)$ in the energy norm on uniform grids. Besides, a lower bound of the $L^2$-norm error is also proved, which makes the capacity analysis of this scheme more clear. On the other hand, for the eigenvalue problem, the numerical eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented, which show the potential of the proposed finite element.

Tables

Table 1. Table 1. Eigenvalues computed by the RRM element on nonuniform grids.

$h$	${\lambda }_{1,h}$	order	${\lambda }_{2,h}$	${\lambda }_{3,h}$	order	${\lambda }_{4,h}$	order	${\lambda }_{5,h}$	${\lambda }_{6,h}$	order
$0.3375$	17.178		38.419	39.047		47.604		65.324	78.711
$0.1688$	18.982	1.690	46.125	46.126	1.696,1.599	68.932	1.564	90.210	90.236	1.966,1.181
$0.084375$	19.539	1.892	48.437	48.437	1.768, 1.768	75.941	1.662	96.105	96.106	1.638,1.633
$0.0421875$	19.688	1.969	49.112	49.112	1.928, 1.928	78.157	1.884	98.004	98.004	1.873,1.873
$0.02109375$	19.726	1.992	49.288	49.288	1.980,1.980	78.754	1.968	98.520	98.520	1.964,1.964
$0.010546875$	19.736	1.998	49.333	49.333	1.995,1.995	78.906	1.992	98.652	98.652	1.991,1.991
Trend	$\nearrow$		$\nearrow$	$\nearrow$		$\nearrow$		$\nearrow$	$\nearrow$
Exact ${\lambda }_j$	19.739	–	49.348	49.348		78.957		98.696	98.696

Table 2. Table 2. Eigenvalues computed by the RRM element on uniform grids.

$h\left(=h_x\right)$	${\lambda }_{1,h}$	order	${\lambda }_{2,h}$	${\lambda }_{3,h}$	order	${\lambda }_{4,h}$	order	${\lambda }_{5,h}$	${\lambda }_{6,h}$	order
$0.25$	18.559		44.961	45.655		63.427		90.249	95.913
$0.125$	19.428	1.896	48.127	48.163	1.796,1.558	74.233	1.644	96.050	96.427	1.596,0.613
$0.0625$	19.660	1.973	49.034	49.036	1.944,1.899	77.711	1.896	97.996	98.016	1.890,1.670
$0.03125$	19.719	1.993	49.269	49.269	1.986, 1.975	78.641	1.973	98.519	98.520	1.972,1.928
$0.015625$	19.734	1.998	49.328	49.328	1.996,1.994	78.878	1.993	98.652	98.652	1.993,1.983
$0.0078125$	19.738	2.000	49.343	49.343	2.000,1.996	78.937	1.998	98.685	98.685	1.998,1.996
Trend	$\nearrow$		$\nearrow$	$\nearrow$		$\nearrow$		$\nearrow$	$\nearrow$
Exact ${\lambda }_j$	19.739		49.348	49.348		78.957		98.696	98.696

Table 3. Table 3. Eigenvalues computed on L-shape domain with nonuniform subdivisions.

$h$	${\lambda }_{1,h}$	${\lambda }_{2,h}$	${\lambda }_{3,h}$	order	${\lambda }_{4,h}$	${\lambda }_{5,h}$	${\lambda }_{6,h}$
$0.675$	8.999	11.157	11.961		16.776	17.955	23.829
$0.3375$	9.637	13.948	17.219	1.626	25.337	27.892	36.616
$0.16875$	9.708	14.840	18.984	1.739	28.190	30.865	40.131
$0.084375$	9.691	15.104	19.539	1.916	29.160	31.713	41.140
$0.0421875$	9.667	15.174	19.688	1.977	29.429	31.897	41.412
$0.02109375$	9.652	15.191	19.726	1.994	29.498	31.923	41.469
$0.010546875$	9.645	15.196	19.736	1.999	29.516	31.921	41.477
Trend		$\nearrow$	$\nearrow$		$\nearrow$
${\lambda }_j\approx$	9.640	15.197	19.739		29.522	31.913	41.475

Table 4. Table 4. Eigenvalues computed on L-shape domain with uniform subdivisions.

$h_x=2h_y$	${\lambda }_{1,h}$	${\lambda }_{2,h}$	${\lambda }_{3,h}$	order	${\lambda }_{4,h}$	${\lambda }_{5,h}$	${\lambda }_{6,h}$
$0.5$	9.894	13.443	15.857		23.914	26.659	33.739
$0.25$	9.811	14.696	18.558	1.717	27.656	30.410	40.808
$0.125$	9.743	15.068	19.428	1.923	29.015	31.687	41.262
$0.0625$	9.691	15.165	19.660	1.980	29.392	31.921	41.456
$0.03125$	9.663	15.189	19.719	1.995	29.489	31.941	41.488
$0.015625$	9.650	15.195	19.734	1.999	29.513	31.930	41.485
Trend		$\nearrow$	$\nearrow$		$\nearrow$
${\lambda }_j\approx$	9.640	15.197	19.739		29.522	31.913	41.475

Equations

$$V_{h}^{\text{BL}}:=\Big{\{}w_{h}\in H^{1}(\Omega):w_{h}|_{K}\in Q_{1}(K),\ \forall K\in\mathcal{G}_{h}\Big{\}}.$$

$$V_{h}^{\rm{PS}}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{1}(K),\ \forall K\in\mathcal{G}_{h},\mbox{ and }\fint_{e}\llbracket{w_{h}}\rrbracket\,ds=0,\ \forall e\in\mathcal{E}_{h}^{i}\Big{\}}.$$

$$V_{h}^{\text{rQ}}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{K}^{\rm rQ},\ \forall K\in\mathcal{G}_{h},\mbox{ and }\fint_{e}\llbracket{w_{h}}\rrbracket=0\,ds,\ \forall e\in\mathcal{E}_{h}^{i}\Big{\}}.$$

$$V_{h}^{\rm{LTZ}}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{K}^{\rm{LTZ}},\ \forall K\in\mathcal{G}_{h},\mbox{ and }\fint_{e}\llbracket{w_{h}}\rrbracket\,ds=0,\ \forall e\in\mathcal{E}_{h}^{i}\Big{\}}.$$

$$V_{h}^{\rm{W}}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{K}^{\rm{W}},\ \forall K\in\mathcal{G}_{h},\mbox{ and }w_{h}\mbox{ is continuous at any }a\in\mathcal{N}_{h}^{i}\Big{\}}.$$

$$V_{h}^{\rm MC}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{2}(K),\mbox{ and }w_{h}\mbox{ is\ moment-continuous on }\mathcal{G}_{h}\Big{\}}.$$

$$V_{h0}^{\rm MC}:=\Big{\{}w_{h}\in V_{h}^{\rm MC}:w_{h}\mbox{ is\ moment-homogeneous on }\mathcal{G}_{h}\Big{\}}.$$

$${\int }_e[[w{]]}_{e}vds=0,\forall v\in P_1(e),e\in {\mathcal{E}}_h^i.$$

$$\displaystyle\mathcal{L}^{0}_{h}:=\Big{\{}q\in L^{2}(\Omega):q|_{K}\in P_{0}(K)\Big{\}},\quad\mathcal{L}^{0}_{h0}:=\mathcal{L}^{0}_{h}\cap L^{2}_{0}(\Omega),$$

$$\displaystyle\undertilde{V}{}^{\rm BL}_{h\bf n}:=\Big{\{}\undertilde{v}\in(V^{\rm BL}_{h})^{2}:\undertilde{v}\cdot\mathbf{n}|_{\partial\Omega}=0\Big{\}},\quad\undertilde{V}{}^{\rm rQ}_{h\bf n}:=\Big{\{}\undertilde{v}\in(V^{\rm rQ}_{h})^{2}:\int_{e}\undertilde{v}\cdot\mathbf{n}=0,\ \forall\,e\in\mathcal{E}_{h}^{b}\Big{\}},$$

$$\displaystyle\undertilde{V}{}^{\rm PS}_{h\bf n}:=(V^{\rm PS}_{h})^{2}\cap\undertilde{V}{}^{\rm rQ}_{h\bf n},\quad\undertilde{V}{}_{h\bf n}^{\rm LTZ}:=\Big{\{}\undertilde{v}{}\in(V_{h}^{\rm LTZ})^{2}:\int_{e}\undertilde{v}{}\cdot\mathbf{n}=0,\ e\in\mathcal{E}_{h}^{b}\Big{\}}.$$

$${\Pi }_h^{\mathrm{rQ}}{V}_h^{\mathrm{BL}}=V_h^{\mathrm{PS}}\text{mbox}and\text{undertilde}\Pi {}_h^{\mathrm{rQ}}\text{undertilde}V{}_{h\mathbf{n}}^{\mathrm{BL}}=\text{undertilde}V{}_{h\mathbf{n}}^{\mathrm{PS}}.$$

$$\left\{\begin{split}-\Delta u&=\rho f&\text{in}&\;\Omega,\\ u&=0&\text{on}&\;\partial\Omega.\end{split}\right.$$

$$a(u,v)=b(f,v),\forall v\in V,$$

$$\left\{\begin{split}-\Delta u&=\lambda\rho u&\text{in}&\;\Omega,\\ u&=0&\text{on}&\;\partial\Omega.\end{split}\right.$$

$$a(u,v)=\lambda b(u,v),\forall v\in V,$$

$$0<{\lambda }_1⩽{\lambda }_2⩽\cdots ⩽{\lambda }_k⩽\cdots ,\text{ satisfying }\underset{k\to \infty }{\lim }{\lambda }_k=\infty ,$$

$$u_1,u_2,\cdots ,u_k,\cdots ,\text{ satisfying }b(u_i,u_j)={\delta }_{ij}.$$

$$M({\lambda }_j)=􏰛\{w\in V:w\text{ is an eigenfunction of }\text{eqref}eq:varEigen\text{ corresponding to }{\lambda }_j\}.$$

$$a_h(u_h,v_h)=b(f,v_h),\forall v_h\in V_h,$$

$$a_h(u_h,v_h)={\lambda }_{h}b(u_h,v_h),\forall v_h\in V_h.$$

$$0<{\lambda }_{1,h}⩽{\lambda }_{2,h}⩽\cdots ⩽{\lambda }_{N,h},$$

$$u_{1,h},u_{2,h},\cdots ,u_{N,h},\text{ satisfying }b(u_{i,h},u_{j,h})={\delta }_{ij}.$$

$$\begin{split}V_{h}^{\rm{M}}:=\Big{\{}w_{h}\in L^{2}(\Omega):w_{h}|_{K}\in P_{K}^{\rm{M}},\ &w_{h}(a)\mbox{ is continuous at any }a\in\mathcal{N}_{h}^{i},\\ &\mbox{and}\fint_{e}\partial_{n_{e}}w_{h}\,ds\mbox{ is continuous across any }e\in\mathcal{E}_{h}^{i}\Big{\}}.\end{split}$$

$$|E_h(v,v_h)|\lesssim \sum _{K\in {\mathcal{G}}_h}{h}_K^2|v{|}_{2,K}|v_h{|}_{2,K}\lesssim h|v{|}_{2,\Omega }|v_h{|}_{1,h},\forall v\in H^2(\Omega )\cap H_0^1(\Omega ).$$

$$|E_h(v,v_h)|\lesssim h^{k-1}|v{|}_{k,\Omega }|v_h{|}_{1,h},\forall v\in H^k(\Omega )\cap H_0^1(\Omega ),k=2,3.$$

$$a_h(w-{\Pi }_h^{\mathrm{M}}w,{\Pi }_h^{\mathrm{M}}w)⩾\alpha h^2,$$

$$a_h(w-{\Pi }_h^{\mathrm{M}}w,w)⩾{\alpha }_1{h}^2.$$

$$a_{h}(u-\Pi_{h}^{\rm M}u,u)\raisebox{-4.2679pt}{$\;\stackrel{{\scriptstyle\raisebox{-11.09654pt}{=}}}{{{\sim}}}\;$}\big{|}u-\Pi_{h}^{\rm M}u\big{|}_{1,h}\big{|}u\big{|}_{1,h},\text{ when }h\text{ is small enough}.$$

$$\begin{array}[]{ccccccccc}0&\longrightarrow&V_{hs}^{\rm{M}}&\xrightarrow{{\rm curl}_{h}}&\undertilde{V}{}_{h\bf n}^{\rm LTZ}&\xrightarrow{{\rm div}_{h}}&\mathcal{L}_{h}^{1,-1}&\longrightarrow&0,\end{array}$$

$$\begin{array}{rl}a({\upsilon }_1,{\upsilon }_1)+a_h({\upsilon }_2,{\upsilon }_2)& =2a_h({\upsilon }_1,{\upsilon }_2)+a_h({\upsilon }_1-{\upsilon }_2,{\upsilon }_1-{\upsilon }_2)\\ & =2a_h({\upsilon }_3,{\upsilon }_2)+2a_h({\upsilon }_1-{\upsilon }_3,{\upsilon }_3)\\ & +2a_h({\upsilon }_1-{\upsilon }_3,{\upsilon }_2-{\upsilon }_3)+a_h({\upsilon }_1-{\upsilon }_2,{\upsilon }_1-{\upsilon }_2),\end{array}$$

Full text

Zeng and C.-S. Zhang are partially supported by National Key Research and Development Program 2016YFB0201304, China. C.-S. Zhang is also supported by the Key Research Program of Frontier Sciences of CAS. S. Zhang is partially supported by National Natural Science Foundation, 11471026 and 11871465, China

1. Introduction

The design and capacity analysis of the discretization schemes for the source problem (say, the boundary value problem) and the eigenvalue problem are key issues in numerical analysis and, in general, of approximation theory. When the approximation of functions in Sobolev spaces is performed using piecewise polynomials defined on a domain partition, lower-degree polynomials are often preferred in order to achieve a simpler interior structure. A finite element scheme with polynomials of the total degree no more than $k$, denoted by $P_{k}$, is called optimal if it achieves $\mathcal{O}(h^{k+1-m})$ accuracy in the energy norm for $H^{m}$ elliptic problems. In this paper, we present an optimal quadratic element scheme for the $H^{1}$ problems, including the source problem and the eigenvalue problem, on rectangular grids, and present its error analysis.

The study of optimal finite element schemes has been attracting wide interests. For the case wherein the grid comprises simplexes, there are already some systematic results. It is known that the Lagrange finite elements of arbitrary degree on domains of arbitrary dimension are optimal conforming elements for second-order elliptic problems. At the same time, a systematic family of minimal-degree nonconforming finite elements is proposed by [27], where $m$-th degree polynomials work for $2m$-th order elliptic problems in $R^{n}$ for any $n\geqslant m$. Known as the Wang-Xu or Morley-Wang-Xu family, these elements are constructed based on the perfect matching between the dimension of $m$-th degree polynomials and the dimension of $(n-k)$-subsimplexes with $1\leqslant k\leqslant m$. The generalisation to the cases where $n<m$ is attracting increasing research interest (see, e.g., [29]). These spaces can be naturally used for both the source problem and the eigenvalue problem. On the other hand, to clarify the capacity of the schemes clearly, some kinds of extremal analysis have also been conducted, including, e.g., lower bounds of the error estimates and guaranteed bounds of the computed eigenvalues. We refer to, e.g., [20] for a general analysis of the lower bounds of the discretization error for piecewise polynomials, and [22, 12, 11, 10] for specific analysis with certain finite element schemes. We refer to, e.g.,[18, 1, 37, 34, 33, 21, 17, 9, 4, 5] for the computed guaranteed bounds of certain eigenvalue problems. The extremal analysis is naturally used on or ready to be generalized to optimal schemes.

When the grid comprises shapes other than simplexes, the design of optimal schemes becomes more complicated. We would like to recall that, $Q_{k}$ (rather than $P_{k}$) polynomials are used for $2k$-th order problems on $\mathbb{R}^{n}$ rectangular grids by [14], which form minimal conforming element spaces. For biharmonic equation, some low-degree rectangular elements have been designed, including the rectangular Morley element and incomplete $P_{3}$ element. Very recently, a space, consisting of exactly piecewise quadratic polynomials, is constructed and shown convergent for the biharmonic equation on general quadrilateral grids, which forms a convergent scheme of the minimal degree [36]. Also, there have been several rectangle elements for $H^{1}$ problems in the literature [16, 13, 15, 28]. In [16], an enriched quadratic nonconforming element on rectangles is introduced, and second-order error is shown, which is generalized to higher orders by [13]. Another second-order quadratic element is given by [15], where the spline technique is used, but the shape function space on a cell is not exactly $P_{2}$. The Wilson element [28] is the first quadratic quadrilateral nonconforming element. Despite its superior performance in practice, as shown in [25], its global asymptotic convergence rate is the same as that of the bilinear element, due to low internal continuity. Generally, this deficiency can be compensated by equipping the piecewise quadratic polynomial with second order moment-continuity across the internal edges. In this way, the moment-continuous (MC) element space is defined. However, it is proved in Appendix A, that the MC element space possesses essentially the same accuracy as that of the bilinear element space, and thus it fails to reach second-order convergence rate. To our best knowledge, it remains open whether an optimal scheme can be constructed with degrees higher than minimal even on rectangular grids and for $H^{1}$ problems.

In this paper, we study the optimal finite element construction for the $H^{1}$ problems, and present a finite element space comprising piecewise quadratic polynomials on uniform rectangular grids that can provide $\mathcal{O}(h^{2})$ convergence in energy norm for the source and eigenvalue problems. The computed eigenvalues are lower bounds of the exact ones, which can be proved theoretically and verified numerically. Only rectangular grids are taken into consideration herein, but if a quadrilateral grid is only a sufficiently small perturbation of a uniform one, then an optimal convergence rate could be expected on it. Moreover, the finite element functions cannot be described with free rein cell by cell. Similar to the elements described in [7, 23, 36] and in many spline-type methods, the number of continuity restrictions of the finite element function is greater than the dimension of the local polynomial space. We believe this difficulty is not abnormal for low-degree schemes. In general, these cells can share interfaces with more neighbour cells, and more continuity restrictions will strengthen the requirement for higher-degree polynomials, generally higher than the order of the underlying Sobolev space. Thus, constructing consistent finite elements in the formulation of Ciarlet’s triple is difficult with $m$-th degree polynomials for $H^{m}$ problems even on rectangular grids. Here we utilize some non-standard technical approaches to overcome the difficulty for both implementation and especially theoretical analysis.

The main difficulty is that the local interpolation is too difficult, if ever possible, to be established, which plays a fundamental role in the approximation error analysis for the source problem and the guaranteed bounds analysis for the eigenvalue problem. Notice that the space constructed herein can be viewed as a reduced rectangular Morley element space. Similar to the approach in [36] but with technical modifications, we can determine that the finite element functions are discrete stream functions of the discrete divergence-free functions constructed in a study [23]; using this exact relation, we can perform the approximation estimation indirectly. Also, the discretization of the eigenvalue problem can be viewed as an inner approximation of the rectangular Morley element scheme, and this helps avoid the direct dependence on an interpolation. This newly-designed routine method of theoretical analysis can be potentially used to find out other optimal schemes.

Finally, we remark that, two examples, namely the rectangular Morley (RM) element and the reduced rectangular Morley (RRM) element, are reported in this paper that when used for the eigenvalue problem, errors of the eigenvalues and eigenfunctions are of the same order. This unusual performance is due to the fact that no nontrivial conforming finite element subspace can be found contained in these two spaces.

The rest of the paper is organized as follows. In Section 2, some preliminaries are given and some related low-degree rectangle elements are reviewed. In Section 3, the rectangular Morley element is revisited. In Section 4, a reduced rectangular Morley element scheme is presented for both source problem and eigenvalue problem. In Section 5, the convergence analysis and lower bound properties are shown for the RRM element scheme. In Section 6, some concluding remarks and discussions are given. In contrast to a general implementation approach in Section 5, concise sets of basis functions of the MC element and the RRM element are presented in the appendix.

2. Preliminaries

2.1. Notations

Throughout this paper, we use $\Omega$ for a simply-connected polygonal domain in $\mathbb{R}^{2}$. We use $\nabla$, ${\rm curl}$, ${\rm div}$, and $\nabla^{2}$ for the gradient operator, curl operator, divergence operator, and Hessian operator, respectively. As usual, we use $H^{2}(\Omega)$, $H^{2}_{0}(\Omega)$, $H^{1}(\Omega)$, $H^{1}_{0}(\Omega)$, and $L^{2}(\Omega)$ for certain Sobolev spaces. Specifically, we denote \displaystyle L^{2}_{0}(\Omega):=\Big{\{}w\in L^{2}(\Omega):\int_{\Omega}wdx=0\Big{\}}, \undertilde{H}{}^{1}_{0}(\Omega):=\big{(}H^{1}_{0}(\Omega)\big{)}^{2}, and \undertilde{H}{}^{1}_{\bf n}(\Omega):=\Big{\{}v\in\big{(}H^{1}(\Omega)\big{)}^{2}:v\cdot{\bf n}\big{|}_{\partial\Omega}=0\Big{\}}. Denote, by $\undertilde{H}{}^{-1}(\Omega)$ and $H^{-1}(\Omega)$, the dual spaces of $\undertilde{H}{}^{1}_{0}(\Omega)$ and $H^{1}_{0}(\Omega)$, respectively. We use $``\undertilde{~{}}"$ for vector valued quantities in the present paper, and $\undertilde{v}{}^{1}$ and $\undertilde{v}{}^{2}$ for the two components of the function $\undertilde{v}$. We utilize the subscript $``\cdot_{h}"$ to indicate the dependence on grids. Particularly, an operator with the subscript ’$``\cdot_{h}"$ implies the operation is done cell by cell. Finally, $\lesssim$, $\gtrsim$, and $\;\stackrel{{\scriptstyle\raisebox{-11.09654pt}{=}}}{{{\sim}}}\;$ respectively denote $\leqslant$, $\geqslant$, and $=$ up to a generic positive constant, which might depend on the shape-regularity of subdivisions, but not on the mesh-size $h$ [30].

Let $\mathcal{G}_{h}$ be in a regular family of quadrilateral grids of domain $\Omega$. Let $\mathcal{N}_{h}$ be the set of all vertices, $\mathcal{N}_{h}=\mathcal{N}_{h}^{i}\cup\mathcal{N}_{h}^{b}$, with $\mathcal{N}_{h}^{i}$ and $\mathcal{N}_{h}^{b}$ comprising the interior vertices and the boundary vertices, respectively. Similarly, let $\mathcal{E}_{h}=\mathcal{E}_{h}^{i}\bigcup\mathcal{E}_{h}^{b}$ be the set of all the edges, with $\mathcal{E}_{h}^{i}$ and $\mathcal{E}_{h}^{b}$ comprising the interior edges and the boundary edges, respectively. For an edge $e$, $\mathbf{n}_{e}$ is a unit vector normal to $e$ and $\boldsymbol{\tau}_{e}$ is a unit tangential vector of $e$ such that $\mathbf{n}_{e}\times\boldsymbol{\tau}_{e}>0$. On the edge $e$, we use $\llbracket\cdot\rrbracket_{e}$ for the jump across $e$. If $e\subset\partial\Omega$, then $\llbracket\cdot\rrbracket_{e}$ is the evaluation on $e$. The subscript ${\cdot}_{e}$ can be dropped when there is no ambiguity brought in.

2.2. Some rectangular finite element spaces

Suppose that $K\subset\mathbb{R}^{2}$ represents a rectangle with sides parallel to the two axis respectively. Let $(x_{K},y_{K})$ be the barycenter of $K$. Let $h_{x,K}$, $h_{y,K}$ be the length of $K$ in the $x$ and $y$ directions, respectively. Let $a_{i}$ and $e_{i}$ denote an vertex and an edge of K, respectively. Let $h:=\max\limits_{K\in\mathcal{G}_{h}}\max\{h_{x,K},h_{y,K}\}$ be the mesh size of $\mathcal{G}_{h}$. When $\mathcal{G}_{h}$ is uniform, we denote $h_{x}:=h_{x,K}$ and $h_{y}:=h_{y,K}$. Let $P_{l}(K)$ denote the space of polynomials on $K$ of total degree no bigger than $l$. Let $Q_{l}(K)$ denote the space of polynomials of degree no bigger than $l$ in each variable. Similarly, we define spaces $P_{l}(e)$ and $Q_{l}(e)$ on an edge $e$.

2.2.1. The $Q_{1}$ element

The $Q_{1}$ element is defined by $(K,P_{K}^{\text{BL}},D_{K}^{\text{BL}})$ with the following properties:

• (a)

$K$ is a rectangle;

• (b)

$P_{K}^{\text{BL}}=Q_{1}(K)$;

• (c)

for any $v\in H^{1}(K)$, D_{K}^{\text{BL}}=\big{\{}v(a_{i})\big{\}}_{i=1:4}.

Define the $Q_{1}$ element space as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{h0}^{\text{BL}}:=\Big{\{}w_{h}\in V_{h}^{\text{BL}}:w_{h}(a)=0,\ \forall a\in\mathcal{N}_{h}^{b}\Big{\}}.

2.2.2. The Park–Sheen (PS) element

The PS element [23] is a piecewise linear nonconforming finite element space for $H^{1}$ problems. It is defined as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{h0}^{\rm{PS}}:=\Big{\{}w_{h}\in V_{h}^{\rm{PS}}:\fint_{e}w_{h}\,ds=0,\ \forall e\in\mathcal{E}_{h}^{b}\Big{\}}.

2.2.3. The rotated $Q_{1}$ ($Q_{1}^{\rm{rot}}$) element

The $Q_{1}^{\rm{rot}}$ element is defined by $(K,P_{K}^{\rm{rQ}},D_{K}^{\rm{rQ}})$ with the following properties:

• (a)

$K$ is a rectangle;

• (b)

$P_{K}^{\rm{rQ}}=P_{1}(K)+\text{span}\{x^{2}-y^{2}\}$;

• (c)

for any $v\in H^{1}(K)$, D_{K}^{\rm{rQ}}=\Big{\{}\fint_{e_{i}}v\,ds\Big{\}}_{i=1:4}.

Define the $Q_{1}^{\rm{rot}}$ element space as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{h0}^{\text{rQ}}:=\Big{\{}w_{h}\in V_{h}^{\text{rQ}}:\fint_{e}w_{h}\,ds=0,\ \forall e\in\mathcal{E}_{h}^{b}\Big{\}}.

2.2.4. The Lin–Tobiska–Zhou (LTZ) element

The LTZ element([19, 35]) is defined by $(K,P_{K}^{\rm{LTZ}},D_{K}^{\rm{LTZ}})$ with the following properties:

• (a)

$K$ is a rectangle;

• (b)

$P_{K}^{\rm{LTZ}}=P_{1}(K)+\text{span}\{x^{2},y^{2}\}$;

• (c)

for any $v\in H^{1}(K)$, D_{K}^{\rm{LTZ}}=\Big{\{}\fint_{K}v\,dxdy,\ \fint_{e_{1}}v\,ds,\ \ldots,\ \fint_{e_{4}}v\,ds\Big{\}}.

Define the LTZ element space as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{h0}^{\rm{LTZ}}:=\Big{\{}w_{h}\in V_{h}^{\rm{LTZ}}:\fint_{e}w_{h}\,ds=0,\ \forall e\in\mathcal{E}_{h}^{b}\Big{\}}, and associated with $\undertilde{H}{}_{\mathbf{n}}^{1}(\Omega)$, we define \undertilde{V}{}_{h{\bf n}}^{\rm LTZ}:=\Big{\{}\undertilde{v}{}_{h}\in\big{(}V_{h}^{\rm LTZ}\big{)}^{2}:\int_{e}\undertilde{v}{}_{h}\cdot\mathbf{n}=0,\ \forall e\in\mathcal{E}_{h}^{b}\Big{\}}.

2.2.5. The Wilson element

The Wilson element is defined by $(K,P_{K}^{\rm{W}},D_{K}^{\rm{W}})$ with with the following properties:

• (a)

$K$ is a rectangle;

• (b)

$P_{K}^{\rm{W}}=P_{2}(K)$;

• (c)

for any $v\in H^{2}(K)$, D_{K}^{\rm{W}}=\Big{\{}v(a_{1}),\ \ldots,v(a_{4}),\ \fint_{K}\partial_{xx}v\,dxdy,\ \fint_{K}\partial_{yy}v\,dxdy\Big{\}}.

Define the Wilson element space as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{h0}^{\rm{W}}:=\Big{\{}w_{h}\in V_{h}^{\rm{W}}:w_{h}(a)=0,\ \forall a\in\mathcal{N}_{h}^{b}\Big{\}}.

2.2.6. The moment-continuous (MC) element

Associated with $\mathcal{G}_{h}$, the MC element space is defined as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define

[TABLE]

A piecewise quadratic polynomial function $w$ is moment-continuous of second-order, if

[TABLE]

Moreover, $w$ is moment-homogeneous of second-order, if $\int_{e}wv\,ds=0,\;\forall v\in P_{1}(e),\;e\in\mathcal{E}_{h}^{b}.$

A piecewise quadratic function $v_{h}$ belongs to $V_{h0}^{\rm MC}$ if and only if $v_{h}$ is continuous at the second-order Gauss points of any $e\in\mathcal{E}_{h}^{i}$ and vanishes on the Gauss points of any $e\in\mathcal{E}_{h}^{b}$.

Theorem 2.1**.**

If $\mathcal{G}_{h}$ is a $m\times n$ rectangular subdivision of $\Omega$, then ${\rm dim}(V_{h0}^{\rm MC})=2mn-m-n+1$.

Theorem 2.2**.**

If $\mathcal{G}_{h}$ be a $m\times n$ rectangular subdivision of $\Omega$, then ${\rm dim}(V_{h}^{\rm MC})=2mn+2m+2n$.

Detailed proof of Theorems 2.1 and 2.2 are put in Appendix A, and available sets of basis functions of $V_{h0}^{\rm MC}$ and $V_{h}^{\rm MC}$ are also presented there.

2.3. Some technical lemmas

In addition to these spaces above, we denote

[TABLE]

Let $\Pi^{\rm rQ}_{h}$ and $\undertilde{\Pi}{}^{\rm rQ}_{h}$ be the nodal interpolation associated with $V^{\rm rQ}_{h}$ and $(V{}^{\rm rQ}_{h})^{2}$, respectively.

Lemma 2.3**.**

([24, Lemma 1],[36, Lemma 6])* For the $Q_{1}^{\rm{rot}}$ element, we have*

• (1)

\big{|}\Pi^{\rm rQ}_{h}v\big{|}_{1,h}\lesssim|v|_{1,h}, $\forall v\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega);$

• (2)

\big{\|}\Pi^{\rm rQ}_{h}v-v\big{\|}_{0,\Omega}+h\big{|}\Pi^{\rm rQ}_{h}v-v\big{|}_{1,h}\lesssim h^{2}|v|_{2,\Omega}, $\forall v\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$.

Lemma 2.4**.**

([36, Lemma 7])* The following relationships hold.*

[TABLE]

2.4. $H^{1}$ elliptic problems and nonconforming finite element approximation

In this paper, we consider the following model problems:

$\bullet$ Source problem: with $f\in Q:=L^{2}(\Omega)$, $\rho\in L^{\infty}(\Omega)$, and $\rho\geqslant c_{0}>0$,

[TABLE]

Its weak form is given by: Find $u\in V:=H_{0}^{1}(\Omega)$ satisfying

[TABLE]

where $a(u,v)=\int_{\Omega}\nabla u\cdot\nabla v\,d\emph{xdy}$ and $b(f,v)=\int_{\Omega}\rho fv\,d\emph{xdy}.$

$\bullet$ Eigenvalue problem: with $\rho\in L^{\infty}(\Omega)$ and $\rho\geqslant c_{0}>0$,

[TABLE]

Its weak form is given by: Find $(\lambda,u)\in\mathbb{R}\times V$ with $\|u\|_{0,\rho}=1$, such that

[TABLE]

where $\|v\|_{0,\rho}:=b(v,v)^{\frac{1}{2}}$ defines a norm over $V$ equivalent to the usually $L^{2}$ norm.

From [2], the eigenvalue problem (2.3) has a sequence of eigenvalues

[TABLE]

and corresponding eigenfunctions

[TABLE]

For a certain eigenvalue $\lambda_{j}$ of (2.4), we define

[TABLE]

Given an discrete space $V_{h}$ defined on $\mathcal{G}_{h}$, the discretization schemes are

$\bullet$ for the source problem: Find $u_{h}\in V_{h}$, such that

[TABLE]

$\bullet$ for the eigenvalue problem: Find $(\lambda_{h},u_{h})\in\mathbb{R}\times V_{h}$ with $||u_{h}||_{0,\rho}=1$, such that

[TABLE]

Let $\text{dim}V_{h}=N$. The discrete eigenvalue problem (2.6) has a sequence of eigenvalues

[TABLE]

and corresponding eigenfunctions

[TABLE]

Lemma 2.5**.**

([6, Theorem 4.1.7]) * Suppose that $\Omega$ is a rectangular region and $\rho$ is smoothing enough. If $(\lambda_{j},u_{j})$ is an eigen-pair of (2.3), then $u_{j}\in C^{5,\alpha}(\Omega)$.*

3. The rectangular Morley (RM) element revisited

3.1. The RM element space

The RM element is defined by $(K,P_{K}^{\rm{M}},D_{K}^{\rm{M}})$ with the following properties:

• (1)

$K$ is a rectangle;

• (2)

$P_{K}^{\rm{M}}=P_{2}(K)+\text{span}\{x^{3},y^{3}\}$;

• (3)

for any $v\in H^{2}(K)$, D_{K}^{\rm{M}}=\big{\{}v(a_{i}),\fint_{e_{i}}\partial_{n_{e_{i}}}v\,ds\big{\}}_{i=1:4}.

Define the RM element space as

[TABLE]

Associated with $H_{0}^{1}(\Omega)$, we define V_{hs}^{\rm{M}}:=\Big{\{}w_{h}\in V_{h}^{\rm{M}}:w_{h}(a)=0,\forall a\in\mathcal{N}_{h}^{b}\Big{\}}.

Lemma 3.1**.**

([22, Lemmas 3.2 and 3.5])* Denote $E_{h}(w,v_{h}):=a_{h}(w,v_{h})+(\Delta w,v_{h})$ with $w\in V\text{ and }v_{h}\in V_{h}.$ The following estimates hold.*

• (a)

For any shape-regular rectangular grid, it holds for any $v_{h}\in V_{hs}^{\rm{M}}$ that

[TABLE]

• (b)

For any uniform rectangular grid, it holds for any $v_{h}\in V_{hs}^{\rm{M}}$ that

[TABLE]

For the RM element, there is a refined property of the interpolation operator $\Pi_{h}^{\rm{M}}:V\mapsto V_{hs}^{\rm{M}}$.

Lemma 3.2**.**

([22, Lemma 3.17])*

Assume that $\mathcal{G}_{h}$ is uniform. For any $w\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ with \big{\|}\frac{\partial^{2}w}{\partial x\partial y}\big{\|}_{0,\rho}\neq 0, if $h$ is small enough, then*

[TABLE]

where $\alpha>0$ is a constant independent of $h$.

Corollary 3.3**.**

Under the conditions in Lemma 3.2, there exists $\alpha_{1}>0$, such that

[TABLE]

Proof.

It follows from Lemma 3.2 and \big{|}w-\Pi_{h}^{\rm{M}}w\big{|}_{1,h}\lesssim h^{2} immediately. ∎

Hence we obtain an interesting and intuitive conclusion:

[TABLE]

By standard argument, we can prove the exact sequence which reads

[TABLE]

where $\mathcal{L}_{h}^{1,-1}=\{q\in L^{2}_{0}(\Omega):q|_{K}\in P_{1}(K)\}$.

3.2. The RM element scheme for the $H^{1}$ eigenvalue problem

3.2.1. Expanded representation of the difference between energy of states

Simple calculations yield

[TABLE]

[TABLE]

[TABLE]

Let $u$ be the solution of the source problem (2.2) or the eigenvalue problem (2.4) and $u_{h}$ be its approximation. Let $\upsilon_{1}=u$, $\upsilon_{2}=u_{h}$ and $\upsilon_{3}=\Pi_{h}u$, where $\Pi_{h}:V\mapsto V_{h0}$ is an interpolation operator. We use (3.6) to obtain an expansion of $b(-f,u-u_{h})$ and (3.7) to obtain an expansion of $\lambda-\lambda_{h}$.

$\bullet$ For the source problem:

Let $u$ and $u_{h}$ be the solutions of (2.2) and (2.5), respectively. It holds that

[TABLE]

From $\left[2a_{h}(\Pi_{h}u,u_{h})-2a(u,u)\right]=2b(f,\Pi_{h}u-u),$ the formula (3.6) becomes

[TABLE]

Analyze the items on the right-hand-side of (3.8). Suppose that $u\in H^{3}(\Omega)\cap H_{0}^{1}(\Omega)$. With the second term $2a_{h}(u-\Pi_{h}u,\Pi_{h}u)$ not considered, the rest items in (3.8) are of high order than $|u-u_{h}|_{1,h}$. Therefore, $2a_{h}(u-\Pi_{h}u,\Pi_{h}u)$ becomes the dominant factor to determine whether $b(-f,u-u_{h})$ is of higher order than $|u-u_{h}|_{1,h}$.

$\bullet$ For the eigenvalue problem:

Let $u$ and $u_{h}$ be the solutions of (2.4) and (2.6), which satisfy $b(u,u)=b(u_{h},u_{h})=1$. It holds that

[TABLE]

Notice that $2b(u_{h},u-u_{h})=2b(u_{h},u)-b(u_{h},u_{h})-b(u,u)=-b(u-u_{h},u-u_{h})$. Thus we can obtain

[TABLE]

Based on these above, (3.7) becomes

[TABLE]

Analyze the items on the right-hand-side of (3.9). Similarly, $2a_{h}(u-\Pi_{h}u,\Pi_{h}u)$ is also the dominant factor to determine whether $\lambda-\lambda_{h}$ is of higher order than $|u-u_{h}|_{1,h}$.

3.2.2. Analysis of the RM element for the eigenvalue problem.

Based on the error estimates of the rectangular Morley element scheme for the source problem (see [22]), the following estimates for the eigenvalue problem follows by standard argument.

Theorem 3.4**.**

Let $\lambda_{j}$ be the $j$-th eigenvalue of (2.4), and $(\lambda_{j,h}^{\rm{M}},u_{j,h}^{\rm{M}})\in\mathbb{R}\times V_{hs}^{\rm{M}}$ be the $j$-th eigen-pair of (2.6) with $\|u_{j,h}^{\rm{M}}\|_{0,\rho}=1$. It holds that

• (a)

if $M(\lambda_{j})\subset H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$, then there exists $u_{j}\in M(\lambda_{j})$ with $\|u_{j}\|_{0,\rho}=1$, such that

[TABLE]

• (b)

if the mesh is uniform and $M(\lambda_{j})\subset H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$, then there exists $u_{j}\in M(\lambda_{j})$ with $\|u_{j}\|_{0,\rho}=1$, such that \big{|}u_{j}-u_{j,h}^{\rm{M}}\big{|}_{1,h}\lesssim h^{2}.

Moreover, we obtain the lower-bound property of eigenvalue approximations by the RM element.

Theorem 3.5**.**

Let $\lambda_{j}$ and $\lambda_{j,h}^{\rm{M}}$ be an exact eigenvalue and its approximation by the RM element. Suppose that $u_{j}\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ and the mesh is uniform. When $h$ is small enough, we have

[TABLE]

where $C_{\rm M}$ is a positive constant independent of $h$.

Proof.

We have the basic expansion by [34, 33], which generalizes the identity introduced by [1],

[TABLE]

From Theorem 3.4, the first two terms can be bounded as

[TABLE]

From a standard interpolation theory in [26], the assumption \big{\|}u_{j,h}^{\rm{M}}\big{\|}_{0,\rho}=1, and Theorem 3.4 (b), the third and last terms have the estimates below

[TABLE]

When $h$ is small enough, it follows from Lemma 3.2 that

[TABLE]

Thus, $a_{h}(u_{j}-\Pi_{h}^{\rm{M}}u_{j},\Pi_{h}^{\rm{M}}u_{j})$ becomes the dominant term on the right-hand-side of (3.11). Hence the result. ∎

4. Reduced rectangular Morley element space for $H^{1}$ problems

4.1. Reduced rectangular Morley element space

We introduce an reduced rectangular Morley (RRM) element space by

[TABLE]

and, associated with $H_{0}^{1}(\Omega)$, define

[TABLE]

Theorem 4.1**.**

If $\mathcal{G}_{h}$ is a $m\times n$ rectangular subdivision of $\Omega$, then ${\rm dim}(V_{hs}^{\rm{R}})=mn+1$.

Detailed proof of Theorems 4.1 and an available set of basis functions of $V_{hs}^{\rm{R}}$ are put in Appendix B.

For any function $v_{h}$ in the RRM element space, the number of continuity restrictions across internal edges is greater than {\rm dim}\big{(}P_{2}(K)\big{)}, which makes it a nontrivial task to find out a set of basis functions of $V_{hs}^{\rm{R}}$, and it is not even easy to tell if the space contains non-zero functions. Actually, the proof of Theorem 4.1 in Appendix B ensures that the RRM element space is non-zero. From the analysis therein, the supports of the basis functions in $V_{hs}^{\rm{R}}$ are not completely local, making it complicated to construct an interpolation operator from $V$ to $V_{hs}^{\rm{R}}$, which, however, plays a fundamental role in the approximation error analysis.

Remark 4.2**.**

Since a non-convex domain which can be covered by a rectangular subdivision can be considered as a combination of several rectangular regions, a nontrivial RRM element space can still be expected on it.

4.2. Approximation property of the RRM element space

The main result of this subsection is the theorem below.

Theorem 4.3**.**

Given $u\in H^{1}_{0}(\Omega)\cap H^{3}(\Omega)$, we have

[TABLE]

We postpone the proof of Theorem 4.3 after some technical lemmas. Let \undertilde{f}\in\big{(}\undertilde{H}{}^{1}_{\bf n}(\Omega)\big{)}^{\prime}. We firstly consider the regularity of the Stokes problem: Find $(\undertilde{u},p)\in\undertilde{H}{}^{1}_{\bf n}(\Omega)\times L^{2}_{0}(\Omega)$, such that

[TABLE]

Lemma 4.4**.**

Let $\Omega$ be a rectangle. If $\undertilde{f}\in\undertilde{L}^{2}(\Omega)$, then $(\undertilde{u},p)\in\undertilde{H}^{2}(\Omega)\times H^{1}(\Omega)$.

Proof.

As ${\rm div}\undertilde{u}=0$, there exists a unique $\varphi\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$, such that ${\rm curl}\varphi=\undertilde{u}$. Moreover, $\varphi$ solves the biharmonic equation:

[TABLE]

By the regularity theory of the biharmonic equation (see [3, Theorem 2 ]), we have $\varphi\in H^{3}(\Omega)$, and $\|\varphi\|_{3,\Omega}\lesssim\sup\limits_{\psi\in H^{1}_{0}(\Omega)\setminus\{0\}}{{\displaystyle\rule[0.0pt]{0.0pt}{11.0pt}(\undertilde{f},{\rm curl}\psi)\over\displaystyle\rule[-3.0pt]{0.0pt}{11.0pt}\|\psi\|_{1,\Omega}}}\lesssim\|\undertilde{f}\|_{0,\Omega}$. Furthermore, $\nabla p=\undertilde{f}+\Delta\undertilde{u}$, and $\|p\|_{1,\Omega}\lesssim\|\undertilde{f}+\Delta\undertilde{u}\|_{0,\Omega}\lesssim\|\undertilde{f}\|_{0,\Omega}$. The proof is completed. ∎

A related finite element problem is to find $(\undertilde{u}{}_{h},q_{h})\in\undertilde{V}{}^{\rm BL}_{h\bf n}\times\mathcal{L}^{0}_{h0}$, such that

[TABLE]

To ensure the convergence of the finite element scheme in Theorem 4.5, we need the following hypothesis:

Hypothesis RT

A rectangular grid $\mathcal{G}_{h}$ is called to satisfy the hypothesis Hypothesis RT if and only if it is generated by refining a grid $\mathcal{G}_{4h}$ twice.

Theorem 4.5**.**

Let $\mathcal{G}_{h}$ be a grid that satisfies Hypothesis RT. Let $(\undertilde{u},p)$ and $(\undertilde{u}{}_{h},p_{h})$ be the solutions of (4.3) and (4.4), respectively. If $(\undertilde{u},p)\in\undertilde{H}^{2}(\Omega)\times H^{1}(\Omega)$, then

[TABLE]

and further

[TABLE]

Based on Lemma 4.4, the proof of Theorem 4.5 is just a duplication of the proofs of [8, Theorems 3.4–3.5 and Corollary 3.2], and we omit the details here.

Theorem 4.6**.**

Let $\mathcal{G}_{h}$ be a grid that satisfies Hypothesis RT. Given $\undertilde{w}\in\undertilde{H}{}^{1}_{\bf n}(\Omega)\cap\undertilde{H}^{2}(\Omega)$ satisfying ${\rm div}\undertilde{w}=0$. It holds that

[TABLE]

Proof.

Let $\undertilde{u}$ be the exact velocity of (4.3). Denote $p_{0}\equiv 0$. It can be verified directly that the pair $(\undertilde{u},p_{0}\equiv 0)\in\undertilde{H}{}^{1}_{\bf n}(\Omega)\times L^{2}_{0}(\Omega)$ solves the equation

[TABLE]

Let $(\undertilde{y}{}_{h},p_{0h})\in\undertilde{V}{}^{\rm BL}_{h\bf n}\times\mathcal{L}^{0}_{h0}$ solve

[TABLE]

then $h|\undertilde{w}-\undertilde{y}{}_{h}|_{1,\Omega}+\|\undertilde{w}-\undertilde{y}{}_{h}\|_{0,\Omega}\leqslant Ch^{2}|\undertilde{u}|_{2,\Omega}.$ Set $\undertilde{w}{}_{h}:=\undertilde{\Pi}{}^{\rm rQ}_{h}\undertilde{y}{}_{h}$, then, from Lemma 2.4, $\undertilde{w}{}_{h}\in\undertilde{V}{}^{\rm PS}_{h\bf n}$. Moreover, it is easy to verify that

[TABLE]

namely ${\rm div}_{h}\undertilde{w}{}_{h}=0$. Furthermore,

[TABLE]

and

[TABLE]

Hence the result. ∎

Lemma 4.7**.**

For the ${\rm curl}$ of space $V_{hs}^{\rm{R}}$, it can be depicted as a special subspace of the vector Park–Sheen element space, i.e.,

[TABLE]

Proof.

Firstly, by standard argument, we can prove the exact sequence which reads

[TABLE]

where $\mathcal{L}_{h}^{1,-1}=\{q\in L^{2}_{0}(\Omega):q|_{K}\in P_{1}(K)\}$. This way, given $\undertilde{v}{}_{h}\in\undertilde{V}{}^{\rm PS}_{h\mathbf{n}}\subset\undertilde{V}{}_{h\bf n}^{\rm LTZ}$ with ${\rm div}_{h}\undertilde{v}{}_{h}=0$, there exists $w_{h}\in V_{hs}^{\rm{M}}$, such that ${\rm curl}_{h}w_{h}=\undertilde{v}{}_{h}$. Since $\undertilde{v}{}_{h}$ is piecewise linear polynomial, $w_{h}$ is piecewise quadratic, namely $w_{h}\in V_{hs}^{\rm{R}}$. On the other hand, it is evident that {\rm curl}_{h}V_{hs}^{\rm{R}}\subset\big{\{}\undertilde{z}{}_{h}\in\undertilde{V}{}^{\rm PS}_{h\bf n}:{\rm div}_{h}\undertilde{z}{}_{h}=0\big{\}}. Hence the result. ∎

Proof of Theorem 4.3

By Theorem 4.6 and Lemma 4.7,

[TABLE]

The proof is completed. ∎

5. Convergence analysis of the RRM element schemes

5.1. Optimal convergence for the source problem

For the RM element space $V_{hs}^{\rm{M}}$ and the RRM element space $V_{hs}^{\rm{R}}$, the discrete source problems are given as:

Find $u_{h}^{\rm{M}}\in V_{hs}^{\rm{M}}$, such that

[TABLE]

Find $u_{h}^{\rm{R}}\in V_{hs}^{\rm{R}}$, such that

[TABLE]

It is obvious that $V_{hs}^{\rm{R}}=V_{hs}^{\rm{M}}\cap V_{h0}^{\rm{W}}$, and we infer that the RRM element is a quadratic nonconforming element on rectangles with a second-order convergence rate in the energy norm. We will verify this assertion strictly in this section.

Theorem 5.1**.**

Let $\mathcal{G}_{h}$ be a grid that satisfies Hypothesis RT. Let $u$ and $u_{h}^{\rm{R}}$ be the solutions of (2.2) and (5.2), respectively. It holds that

• (a)

if $u\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$, then \big{|}u-u_{h}^{\rm{R}}\big{|}_{1,h}\lesssim h|u|_{2,\Omega}\;\text{and}\;\big{\|}u-u_{h}^{\rm{R}}\big{\|}_{0,\rho}\lesssim h^{2}|u|_{2,\Omega};

• (b)

if $u\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ and the mesh is uniform, then \big{|}u-u_{h}^{\rm{R}}\big{|}_{1,h}\lesssim h^{2}|u|_{3,\Omega}.

Proof.

(a) By the Strang lemma, we have

[TABLE]

For the first term in the right hand side of (5.3), we have from Theorem 4.3 that

[TABLE]

For the second term, we have from Lemma 3.1 and $V_{hs}^{\rm{R}}\subset V_{hs}^{\rm{M}}$ that

[TABLE]

Submit (5.4) and (5.5) into (5.3), and we obtain \big{|}u-u_{h}^{\rm R}\big{|}_{1,h}\lesssim h|u|_{2,\Omega},\mbox{ where }u\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega).

Given $g\in Q$, let $\phi_{g}\in V$ and $\phi_{gh}\in V_{hs}^{\rm{R}}$ be the solutions of the two problems below, respectively,

[TABLE]

By the Nitsche-Lascaux-Lesaint lemma (see e.g., [26, Theorem 5.3.1]), it holds that

[TABLE]

For the first term in the right side of (5.6), we have

[TABLE]

where we utilize the regularity of solution on a convex domain, namely, $|\phi_{g}|_{2,\Omega}\lesssim\|g\|_{0,\rho}$.

For the second term, we notice that

[TABLE]

Let $\Pi_{h0}^{\rm p}:L^{2}(\Omega)\mapsto V_{hs}^{\rm{M}}$ be an average projection operator defined in [26]. From [26, Theorem 3.5.4], we have \big{|}u-\Pi_{h0}^{\rm p}u\big{|}_{1,K}\lesssim h_{K}|u|_{2,K} and \big{|}\Pi_{h0}^{\rm p}u\big{|}_{2,K}\lesssim|u|_{2,K}. Thus we obtain

[TABLE]

The last inequality holds due to the fact that \big{|}u-u_{h}^{\rm R}\big{|}_{1,h}\lesssim h|u|_{2,\Omega}. Submitting (5.9) into (5.8), it yields \big{|}E_{h}(\phi_{g},u_{h}^{\rm R})\big{|}\lesssim h^{2}|\phi_{g}|_{2,\Omega}|u|_{2,\Omega}\lesssim h^{2}||g||_{0,\rho}|u|_{2,\Omega}. Similarly, it holds that \big{|}E_{h}(u,\phi_{gh})\big{|}\lesssim h^{2}|u|_{2,\Omega}||g||_{0,\rho}. Thus we have

[TABLE]

Submit (5.7) and (5.10) into (5.6), and we obtain \big{\|}u-u_{h}^{\rm R}\big{\|}_{0,\rho}\lesssim h^{2}|u|_{2,\Omega},\mbox{ where }u\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega).

(b) If $u\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ and the mesh is uniform, then $\inf\limits_{v_{h}\in V_{hs}^{\rm{R}}}|u-v_{h}|_{1,h}\lesssim h^{2}|u|_{3,\Omega}.$ From Lemma   3.1, $|E_{h}(u,v_{h})|\lesssim h^{2}|u|_{3,\Omega}|v_{h}|_{1,h},\forall v_{h}\in V_{h}^{\rm{R}}.$ Hence we have $|u-u_{h}|_{1,h}\lesssim h^{2}|u|_{3,\Omega}.$ ∎

It shows that the error estimate in the energy norm can be $\mathcal{O}(h^{2})$ when the mesh is uniform. However, the convergence rate in $L^{2}$-norm can not be improved on uniform grids. Actually, there exists a lower bound stated in the following Theorem 5.2.

Theorem 5.2**.**

Let $\mathcal{G}_{h}$ be a uniform grid satisfying Hypothesis RT. Let $u$ and $u_{h}^{\rm{R}}$ be the solutions of (2.2) and (5.2), respectively. If $\|f\|_{0,\rho}\neq 0\text{ and }u\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$, then $||u-u_{h}^{\rm R}||_{0,\rho}\gtrsim h^{2}$, provided that $h$ is small enough.

Proof.

Let $u_{h}^{\rm{M}}$ be the solution of (5.1). Then

[TABLE]

given that $h$ is small enough, where $\delta>0$ is a constant independent of $h$; see [22, Lemma 3.18]. Noticing that $V_{hs}^{\rm{R}}\subset V_{hs}^{\rm{M}}$, we obtain

[TABLE]

A simple division yields

[TABLE]

Owing to the orthogonality and \big{|}u_{h}^{\rm{M}}-u_{h}^{\rm{R}}\big{|}_{1,h}\leqslant\big{|}u_{h}^{\rm{M}}-u\big{|}_{1,h}+\big{|}u-u_{h}^{\rm{R}}\big{|}_{1,h}\lesssim h^{2}, it holds that

[TABLE]

A combination of (5.11), (5.13) and (5.14) leads to the following lower bound, provided that $h$ is small enough,

[TABLE]

where $\gamma>0$ is a constant independent of $h$. Therefore we have

[TABLE]

Hence the result. ∎

Remark 5.3**.**

[22, Remark 3.16]* For rectangle domain, the condition $\|f\|_{0,\rho}\neq 0$ implies that \big{\|}\frac{\partial^{2}u}{\partial x\partial y}\big{\|}_{0,\rho}\neq 0. In fact, if \big{\|}\frac{\partial^{2}u}{\partial x\partial y}\big{\|}_{0,\rho}=0, then $u$ is of the form $u=h(x)+g(y)$, for some function $h(x)$ with respect to $x$ and $g(y)$ with respect to $y$. Then, the boundary condition, i.e., $u=0$ on $\partial\Omega$, indicates $u\equiv 0$, which contradicts $\|f\|_{0,\rho}\neq 0$.*

5.2. Analysis of the scheme for the eigenvalue problem

With the associated spaces $V_{hs}^{\rm{M}}$ and $V_{hs}^{\rm{R}}$, the discrete eigenvalue problems are given as:

Find $(\lambda_{h}^{\rm{M}},u_{h}^{\rm{M}})\in\mathbb{R}\times V_{hs}^{\rm{M}}$ with \big{\|}u_{h}^{\rm{M}}\big{\|}_{0,\rho}=1, such that

[TABLE]

Find $(\lambda_{h}^{\rm{R}},u_{h}^{\rm{R}})\in\mathbb{R}\times V_{hs}^{\rm{R}}$ with \big{\|}u_{h}^{\rm{R}}\big{\|}_{0,\rho}=1, such that

[TABLE]

From Theorem 5.4, the convergence results of the eigenvalue problem is obtained by standard argument (see, e.g., [31, 9, 34, 32]).

Theorem 5.4**.**

Let $\mathcal{G}_{h}$ be a grid satisfying Hypothesis RT. Let $\lambda_{j}$ be the $j$-th eigenvalue of (2.4), and $(\lambda_{j,h}^{\rm{R}},u_{j,h}^{\rm{R}})\in\mathbb{R}\times V_{hs}^{\rm{R}}$ be the $j$-th eigen-pair of (5.16) with \big{\|}u_{j,h}^{\rm{R}}\big{\|}_{0,\rho}=1. It holds that

• (a)

if $M(\lambda_{j})\subset H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$, then there exists $u_{j}\in M(\lambda_{j})$ with $\|u_{j}\|_{0,\rho}=1$, such that

[TABLE]

• (b)

if the mesh is uniform and $M(\lambda_{j})\subset H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$, then there exists $u_{j}\in M(\lambda_{j})$ with $\|u_{j}\|_{0,\rho}=1$, such that \big{|}u_{j}-u_{j,h}^{\rm{R}}\big{|}_{1,h}\lesssim h^{2}.

Similar to the basic relation between $V$ and its conforming approximation $V_{h}^{C}$ for eigenvalue problems in [2], the following relation holds.

Lemma 5.5**.**

Let $(\lambda_{h}^{\rm{M}},u_{h}^{\rm{M}})$ be an eigen-pair of (5.15) with \big{\|}u_{h}^{\rm{M}}\big{\|}_{0,\rho}=1. Denote, the Rayleigh quotient, by $R(v_{h}):={\displaystyle\rule[0.0pt]{0.0pt}{11.0pt}a_{h}(v_{h},v_{h})\over\displaystyle\rule[-3.0pt]{0.0pt}{11.0pt}b(v_{h},v_{h})}$. For any $v_{h}\in V_{hs}^{\rm{M}}$ with $\|v_{h}\|_{0,\rho}=1$, it holds that

[TABLE]

Proof.

From \|v_{h}\|_{0,\rho}=\big{\|}u_{h}^{\rm{M}}\big{\|}_{0,\rho}=1 and $\lambda_{h}^{\rm{M}}=a_{h}(u_{h}^{\rm{M}},u_{h}^{\rm{M}})$, it is equivalent to prove that

[TABLE]

where we utilize again the assumption: $b(v_{h},v_{h})=b(u_{h}^{\rm{M}},u_{h}^{\rm{M}})$.∎

Theorem 5.6**.**

Let $(\lambda_{j},u_{j})$, $(\lambda_{j,h}^{\rm{R}},u_{j,h}^{\rm{R}})$ and $(\lambda_{j,h}^{\rm{M}},u_{j,h}^{\rm{M}})$ be the $j$-th exact eigen-pair and its discrete approximations with \|u_{j}\|_{0,\rho}=\big{\|}u_{j,h}^{\rm{R}}\big{\|}_{0,\rho}=\big{\|}u_{j,h}^{\rm{M}}\big{\|}_{0,\rho}=1. Assume that $u_{j}\in H_{0}^{1}(\Omega)\cap H^{3}(\Omega)$ and the mesh is uniform. Provided that $h$ is small enough, we have

[TABLE]

where $C_{\rm R}$ is a positive constant independent of $h$.

Proof.

Since $V_{hs}^{\rm{R}}\subset V_{hs}^{\rm{M}}$, the second inequality, or $\lambda_{j,h}^{\rm{R}}\geqslant\lambda_{j,h}^{\rm{M}}$, holds from the minimum-maximum principle [2]. Let $v_{h}=u_{j,h}^{\rm{R}}$ in Lemma 5.5. We obtain

[TABLE]

From Theorem 3.4, Theorem 5.4, and the triangle inequality, it holds that

[TABLE]

A combination of (3.10), (5.19), and (5.20) yields that

[TABLE]

Hence the result. ∎

Remark 5.7**.**

To the best of our knowledge, the RM element and the RRM element are the only two elements, by which eigenvalue approximations have the same convergence rates as that of eigenfunction approximations in the energy norm.**

5.3. Implementation and numerical results

5.3.1. Implementation

Since constructing clearly a basis functions of $V_{hs}^{\rm{R}}$ on an arbitrary grid is sophisticated, we now present an available approach how (5.2) and (5.16) can be implemented. We start with the fact that V_{hs}^{\rm{R}}=\Big{\{}w_{h}\in V_{h0}^{\rm{W}}:\int_{e}\llbracket{\partial_{n_{e}}w_{h}}\rrbracket=0,\forall e\in\mathcal{E}_{h}^{i}\Big{\}}. Define $P_{0}(\mathcal{E}_{h}^{i})$ the space of piecewise constant functions defined on $\mathcal{E}_{h}^{i}$.

An equivalent formulation of (5.2) is to find $(u_{h},\delta_{h})\in V_{h0}^{\rm{W}}\times P_{0}(\mathcal{E}_{h}^{i})$, such that

[TABLE]

An equivalent formulation of (5.16) is to find $(\lambda_{h},u_{h},\delta_{h})\in\mathbb{R}\times V_{h0}^{\rm{W}}\times P_{0}(\mathcal{E}_{h}^{i})$ with $\|u_{h}\|_{0,\rho}=1$, such that

[TABLE]

Problem (5.21) admits a solution $(u_{h},\delta_{h})$, where $u_{h}$ solves problem (5.2), and problem (5.22) admits a solution $(\lambda_{h},u_{h},\delta_{h})$, where $(\lambda_{h},u_{h})$ solves problem (5.16).

Remark 5.8**.**

It’s worth mentioning that formulations (5.21) and (5.22) can be applied to an arbitrary quadrilateral grid [36]. For the case wherein the grid comprises rectangles, a detailed construction process of basis functions of $V_{hs}^{\rm R}$ is given in Appendix B, based on which (5.2) and (5.16) can be implemented directly in elliptic formulation.**

5.3.2. Numerical experiments

Let $\Omega=(0,1)^{2}$. We consider nonuniform meshes (see Figure 1) with \frac{h_{x,K}}{h_{y,K}}\in\Big{\{}\frac{0.35}{0.65},\frac{0.65}{0.35},\frac{1}{1}\Big{\}}, and uniform meshes with $\frac{h_{x,K}}{h_{y,K}}=\frac{1}{2}$. Numerical examples of the source problem and the eigenvalue problem are given below.

Example 1 for the source problem.

Consider (2.1) with $f=2\pi^{2}sin(\pi x)sin(\pi y)$. The exact solution $u$ is computed as $u=sin(\pi x)sin(\pi y)$. Apply (5.21) to get the discrete solutions on uniform and nonuniform meshes. From Figure 2, the convergence rate is $\mathcal{O}(h)$ in the energy norm, and $\mathcal{O}(h^{2})$ in $L^{2}$-norm, on a nonuniform mesh. Both rates reach $\mathcal{O}(h^{2})$ order on uniform grids.

Example 2 for the eigenvalue problem.

Consider (2.3) with $\rho=1$. Then we have the exact eigenfunctions $u_{k,l}=2sin(k\pi x)sin(l\pi y)$ and eigenvalues $\lambda_{k,l}=(k^{2}+l^{2})\pi^{2}\ (k,l\in\mathbb{N}^{+}).$ Arrange them by increasing order. Apply (5.22) to get the the smallest six discrete eigenvalues. From Figure 3, the convergence rates of eigenvalues almost reach $\mathcal{O}(h^{2})$ order in both nonuniform and uniform cases. Moreover, from tables 1 and 2, the eigenvalue approximations by the RRM element converge monotonically from below to the exact ones.

6. Conclusions and discussions

6.1. Concluding remarks

In this paper, we present a reduced rectangular Morley element scheme for $H^{1}$ problems. Technically, the exactness relation between the RRM element and the PS element is figured out, and the approximation error estimate is established by an auxiliary Stokes problem. For the source problem, the convergence rate of this scheme is $\mathcal{O}(h)$ in the energy norm and $\mathcal{O}(h^{2})$ in $L^{2}$-norm, on general meshes. The error estimate in the energy norm reaches $\mathcal{O}(h^{2})$ order on uniform grids. Besides, a lower bound of the $L^{2}$-norm error is proved, and the best $L^{2}$-norm error estimate is at most $\mathcal{O}(h^{2})$. For the eigenvalue problem, the discrete eigenvalues by the RM element and the RRM element are both proved to be lower bounds of the exact ones. In fact, the inequality (3.1), reads $a_{h}(w-\Pi_{h}^{\rm M}w,\Pi_{h}^{\rm M}w)\geqslant\alpha h^{2}$, or (3.2), reads $a_{h}(w-\Pi_{h}^{\rm M}w,w)\geqslant\alpha_{1}h^{2}$, is the dominant factor for the RRM element to yield these lower bounds, where $\Pi_{h}^{\rm M}$ is the interpolation operator for the RM element.

Roughly speaking, for schemes which provide the lower bounds of the eigenvalues, a smaller space provides a better approximation. This can be viewed as a motivation for the optimal space.

6.2. Further discussions

In this paper, we mainly focus on the convex domain (rectangle domain) case. Also, for the eigenvalue problem, we pay special attention to the computation of eigenvalues. Some more numerical experiments illustrate that the schemes can perform even better than the theoretical description in this paper. These can stimulate further research, and we list part of them below.

(1) Consider the eigenvalue problem (2.3) with $\rho=1$ on $\Omega=(0,1)^{2}$. From Figure 4, right, the convergence rate of the first eigenfunction in $L^{2}$-norm reaches $\mathcal{O}(h^{3})$ order on uniform grids, while in Theorem 5.4 we can only derive \big{\|}u_{j}-u_{j,h}^{\rm R}\big{\|}_{0,\rho}\leqslant\big{|}u_{j}-u_{j,h}^{\rm R}\big{|}_{1,h}\lesssim h^{2}. Although, there exists a lower bound of the $L^{2}$-norm error for the source problem, it may not holds for the eigenvalue problem, and it is possible that the convergence rate of the eigenfunctions in $L^{2}$-norm may be higher than that in the energy norm.

(2) Consider the source problem (2.1) on L-shape domain: $\Omega=(0,2)^{2}\backslash[1,2]^{2}$. From Figure 5, the convergence rates are consistent with the results derived on $\Omega=(0,1)^{2}$. Although Theorem 5.1 for the source problem is based on the assumption of convex domain, this example implies that the RRM element may also be applicable to non-convex regions.

(3) Consider the eigenvalue problem (2.3) with $\rho=1$ on L-shape domain. The eigenfunctions and eigenvalues are unknown, and eigenfunctions may have singularities around the reentrant corner. From [2], the third eigenfunction is analytic: $u_{3}=sin(\pi x)sin(\pi x)$. We present the errors of $u_{3}$ in Figure 6, and observe that the convergence rates for error on L-shape domain are the same as that on a rectangle region. Moreover, the third eigenvalue computed by the RRM scheme satisfies $\lambda_{3,h}\leqslant\lambda_{3}=2\pi^{2}$ and the convergence rate is $\mathcal{O}(h^{2})$. The perfermance of the smallest six eigenvalues with nonuniform and uniform subdivisions are listed in Tables 3 and 4.

These examples suggest that the RRM element may have better numerical applications, and these will be studied in our future work.

Appendix A Construction of basis functions for the moment-continuous (MC) element space

Let $\Omega\subset\mathbb{R}^{2}$ be a polygonal region subdivided into a rectangular grid $\mathcal{G}_{h}$. Define the moment-continuous (MC) element spaces as

[TABLE]

[TABLE]

They have the following equivalent definitions:

[TABLE]

[TABLE]

In this section, we will present available sets of basis functions of $V_{h}^{\rm MC}$ and $V_{h0}^{\rm MC}$.

A.1. Compatibility conditions

Let $K$ be a rectangle with $a_{i}$ the vertices and $g_{ij}$ the Gauss–Legendre points on the boundary (see Figure 7), where $i=1,2,3,4$ and $j=1,2$. Let $\theta_{1}=\frac{1}{2}(1-\sqrt{\frac{1}{3}})$ and $\theta_{2}=\frac{1}{2}(1+\sqrt{\frac{1}{3}})$ be the coordinates of second-order Gauss-Legendre points on $[0,1]$. By a pure linear algebra argument, we have the following description of $P_{2}(K)$.

Lemma A.1**.**

Given $\mathbb{g}_{ij}\in\mathbb{R}$, $i=1,2,3,4$, $j=1,2$. There exists $p\in P_{2}(K)$, such that $p(g_{ij})=\mathbb{g}_{ij}$, if and only if the following compatibility conditions are satisfied on $K$,

[TABLE]

Proof.

To prove (A.5), we connect the vertices $a_{1}$ and $a_{3}$. Let $g_{131}$ and $g_{132}$ be the two Gauss points on $a_{1}a_{3}$. Then we obtain that (see [7, (2)])

[TABLE]

and

[TABLE]

Thus (A.5) follows. Since the two directional derivatives of $p$ belongs to $P_{1}(K)$, (A.6) and (A.7) hold. ∎

A set of basis functions of $P_{2}(K)$ are listed below. Note that with $\varphi_{a_{i}}$ being the bilinear functions and $\varphi_{xx}$ and $\varphi_{yy}$ being the bubbles in two directions, these six functions form a set of basis functions with respect to the Wilson element.

Local basis functions of $P_{2}(K)$:

[TABLE]

Lemma A.2**.**

Let $p\in P_{2}(K)$. The following results hold.

1. (1)

If $p(g_{41})=p(g_{42})=0$, then $p\in{\rm span}\{\varphi_{a_{2}},\varphi_{a_{3}},\varphi_{xx},\varphi_{0,K}\};$ 2. (2)

If $p(g_{11})=p(g_{12})=p(g_{41})=p(g_{42})=0$, then $p\in{\rm span}\{\varphi_{a_{3}},\varphi_{0,K}\};$ 3. (3)

If $p(g_{11})=p(g_{12})=p(g_{21})=p(g_{22})=p(g_{41})=p(g_{42})=0$, then $p\in{\rm span}\{\varphi_{0,K}\}$.

Remark A.3**.**

A polynomial $p\in P_{2}(K)$ is uniquely determined by its evaluations on $g_{ij}$ only up to $\varphi_{0,K}(x,y):=\frac{1}{\xi^{2}}\varphi_{xx,K}+\frac{1}{\eta^{2}}\varphi_{yy,K}-2\theta_{1}\theta_{2}$ multiplicated by a constant.**

A.2. Patterns of supports of basis functions in $V_{h}^{\rm MC}$ and $V_{h0}^{\rm MC}$

Suppose that the domain is divided into $m\times n$ rectangles; see Figure 8. In $x$ direction, it is decomposed to $m$ rows, each being $\mathcal{G}_{i}$ ($1\leqslant i\leqslant m$), and in $y$ direction, it is decomposed to $n$ columns, each being $\mathcal{G}^{j}$ ($1\leqslant j\leqslant n$). The vertices are labeled by $X_{i}^{j}$, and the cells by $K^{j}_{i}$. That is, $K^{j}_{i}=\mathcal{G}_{i}\cap\mathcal{G}^{j}$, and it has four vertices as $X^{j-1}_{i-1}$, $X^{j-1}_{i}$, $X^{j}_{i-1}$, and $X^{j}_{i}$.

We call the support set of a basis function a pattern. Below we present four kinds of patterns sequentially, namely, cell patterns in Lemma A.4, vertex patterns in Lemma A.5, column patterns and row patterns in Lemma A.6.

Lemma A.4**.**

Let $\varphi_{0,K_{i}^{j}}$ be defined in Remark A.3 on $K_{i}^{j}$, $1\leqslant i\leqslant m$, $1\leqslant j\leqslant n$. Then, $\varphi_{0,K_{i}^{j}}\in V_{h0}^{MC}$.

Lemma A.5**.**

Let $\varphi_{X_{i}^{j}}$ denote a function defined on the support patch $\omega$ associated with $X_{i}^{j}$, which is bilinear on every element in $\omega$. Then, $\varphi_{X_{i}^{j}}\in V_{h}^{MC}$, $\forall X_{i}^{j}\in\mathcal{N}_{h}$, and $\varphi_{X_{i}^{j}}\in V_{h0}^{MC}$, $\forall X_{i}^{j}\in\mathcal{N}_{h}^{i}$.

Lemma A.6**.**

Let $\omega$ be a patch with boundaries $\Gamma_{l}$ $(l=1:4)$, anticlockwise; see Figure 9. Let $V_{h}^{\rm MC}(\omega)$ denote a moment-continuous space restricted on $\omega$.

(a) Let $\omega=\mathcal{G}_{i}$ be the union of elements in the $i$-th column; see Figure 9 (Left). We define

[TABLE]

Let $\varphi_{i}^{x}$ be a function defined on $\omega$, which is equal to $\varphi_{xx,K_{i}^{j}}$ on every $K_{i}^{j}\in\omega$. Then $\varphi_{i}^{x}\in V_{{\rm col},i}^{\rm MC}$, and furthermore, V_{{\rm col},i}^{\rm MC}={\rm{span}}\big{\{}\varphi_{i}^{x}\big{\}}\oplus{\rm{span}}\big{\{}\varphi_{0,K_{i}^{j}}\big{\}}_{j=1}^{n}.

(b) Let $\omega:=\mathcal{G}^{j}$ be the union of elements in the $j$-th row; see Figure 9 (Right). We define

[TABLE]

Let $\varphi_{j}^{y}$ be a function defined on $\omega$, which is equal to $\varphi_{yy,K_{i}^{j}}$ on every $K_{i}^{j}\in\omega$. Then $\varphi_{j}^{y}\in V_{{\rm row},j}^{\rm MC}$, and furthermore, V_{{\rm row},j}^{\rm MC}={\rm{span}}\big{\{}\varphi_{j}^{y}\big{\}}\oplus{\rm{span}}\big{\{}\varphi_{0,K_{i}^{j}}\big{\}}_{i=1}^{m}.

Proof.

We present the proof of (a), and omit the proof of (b), which can be obtained similarly. Suppose that $V_{{\rm col},i}^{\rm MC}\neq\varnothing$. Let $v_{h}$ be a function in $V_{{\rm col},i}^{\rm MC}$. Denote by $\delta_{i}$ the value of $v_{h}$ on a Gauss point in Figure 9. According to Lemma A.1, it holds on the element $K_{i}^{1}$ that

[TABLE]

It follows from (A.9) - (A.11) that $\delta_{1}=\delta_{2}=\delta_{3}=\delta_{4}.$ Apply Lemma A.1 to $v_{h}$ on $K_{i}^{2}$, and we obtain $\delta_{3}=\delta_{4}=\delta_{5}=\delta_{6}.$ Repeat the process for the whole row, and we have $\delta_{1}=\delta_{2}=\cdots=\delta_{2n+1}=\delta_{2n+2}.$ By definition, it is obvious that $\varphi_{i}^{x}\in V_{{\rm col},i}^{\rm MC}$. From Remark A.3, we derive that V_{{\rm col},i}^{\rm MC}=\text{span}\big{\{}\varphi_{i}^{x}\big{\}}\oplus\text{span}\big{\{}\varphi_{0,K_{i}^{j}}\big{\}}_{j=1}^{n}. ∎

Here and throughout this paper, we do not distinct $\varphi_{0,K_{i}^{j}}$, $\varphi_{X_{i}^{j}}$, $\varphi_{i}^{x}$, $\varphi_{j}^{y}$ and their respective extension onto the whole domain by zero. Thus we also obtain $\varphi_{i}^{x}\in V_{h}^{\rm MC}$ and $\varphi_{j}^{y}\in V_{h}^{\rm MC}$.

A.3. Structure of the MC element space

Here we will present the construction of basis functions in spaces $V_{h0}^{\rm MC}$ and $V_{h}^{\rm MC}$.

Theorem A.7**.**

Let $\mathcal{G}_{h}$ be a $m\times n$ rectangular subdivision of $\Omega$. Then, $V_{h0}^{\rm MC}=V^{\rm BL}_{h0}\oplus{\rm{span}}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}.$

Proof.

It is obvious that $V^{\rm BL}_{h0}\cap\text{span}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}=0$. We only have to show that $V_{h0}^{\rm MC}\subset V^{\rm BL}_{h0}\oplus\text{span}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}$, i.e., any function in the former is the combination of functions in the latter.

Here we use a sweeping procedure. Let $v_{h}\in V_{h0}^{\rm MC}$. First, by Lemma A.2, we have that $v_{h}|_{K_{1}^{1}}=\alpha_{1}^{1}\cdot\varphi_{X_{1}^{1}}|_{K_{1}^{1}}+\gamma_{1}^{1}\cdot\varphi_{0,K_{1}^{1}}$ with some constants $\alpha_{1}^{1}$ and $\gamma_{1}^{1}$. Therefore, $v_{h}=v_{h}^{1,1}+\alpha_{1}^{1}\cdot\varphi_{X_{1}^{1}}|_{K_{1}^{1}}+\gamma_{1}^{1}\cdot\varphi_{0,K_{1}^{1}}$ with $v_{h}^{1,1}\in V_{h0}^{\rm MC}$ and $v_{h}^{1,1}$ vanishing on $K_{1}^{1}$. Second, $v_{h}^{1,1}=v_{h}^{1,2}+\alpha_{1}^{2}\cdot\varphi_{X_{1}^{2}}+\gamma_{1}^{2}\varphi_{0,K_{1}^{2}}$ with $v_{h}^{1,2}\in V_{h0}^{\rm MC}$ and $v_{h}^{1,2}$ vanishing on $K_{1}^{1}$ and $K_{1}^{2}$. Furthermore, repeat this process on all the cells of the first column, and we obtain that

[TABLE]

where $v_{h}^{1}\in V_{h0}^{\rm MC}$ and $v_{h}^{1}$ vanishes on the whole column $\mathcal{G}_{1}$. Finally, we repeat the process from $\mathcal{G}_{1}$ to $\mathcal{G}_{n}$, and obtain that

[TABLE]

Hence the result. ∎

Theorem A.8**.**

Let $\mathcal{G}_{h}$ be a $m\times n$ rectangular subdivision of $\Omega$. Then, we have

[TABLE]

Proof.

It is obvious that $V_{h}^{\rm MC}\supset V^{\rm BL}_{h}+(\text{span}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}\oplus\text{span}\{\varphi_{i}^{x}\}_{i=1}^{m}\oplus\text{span}\{\varphi_{j}^{y}\}_{j=1}^{n})$. Here we have noted that, if $\varphi_{1}+\varphi_{2}+\varphi_{3}=0$ with $\varphi_{1}\in{\rm{span}}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}$, $\varphi_{2}\in\text{span}\{\varphi_{i}^{x}\}_{i=1}^{m}$, and $\varphi_{3}\in\text{span}\{\varphi_{j}^{y}\}_{j=1}^{n}$, then $\varphi_{1}=\varphi_{2}=\varphi_{3}=0$. We only have to show the other direction. Let $v_{h}\in V_{h}^{\rm MC}$.

First, by (A.8), there exists unique constants $\alpha_{0}^{0},\alpha_{0}^{1},\alpha_{1}^{0},\alpha_{1}^{1},\kappa_{1}$, and $\sigma_{1}$, such that

[TABLE]

Thus, we have $v_{h}=v_{h}^{1,1}+\alpha_{0}^{0}\varphi_{X_{0}^{0}}+\alpha_{0}^{1}\varphi_{X_{0}^{1}}+\alpha_{1}^{0}\varphi_{X_{1}^{0}}+\alpha_{1}^{1}\varphi_{X_{1}^{1}}+\kappa_{1}\varphi_{1}^{x}+\sigma_{1}\varphi_{1}^{y}$ with $v_{h}^{1,1}\in V_{h}^{\rm MC}$ and $v_{h}^{1,1}|_{K_{1}^{1}}=0$. Second, by Lemma A.2, we have $v_{h}^{1,1}|_{K_{1}^{2}}=\alpha_{0}^{2}\varphi_{X_{0}^{2}}|_{K_{1}^{2}}+\alpha_{1}^{2}\varphi_{X_{1}^{2}}|_{K_{1}^{2}}+\gamma_{1}^{2}\varphi_{0,K_{1}^{2}}+\sigma_{2}\varphi_{2}^{y}|_{K_{1}^{2}}.$ Therefore, we obtain $v_{h}^{1,1}=v_{h}^{1,2}+\alpha_{0}^{2}\varphi_{X_{0}^{2}}+\alpha_{1}^{2}\varphi_{X_{1}^{2}}+\gamma_{1}^{2}\varphi_{0,K_{1}^{2}}+\sigma_{2}\varphi_{2}^{y}$ with $v_{h}^{1,2}\in V_{h}^{\rm MC}$ and $v_{h}^{1,2}|_{K_{1}^{1}\cup K_{1}^{2}}=0$.

Furthermore, repeat this process on the column $\mathcal{G}_{1}$, and we obtain

[TABLE]

where $v_{h}^{1}\in V_{h}^{\rm MC}$ and $v_{h}^{1}|_{\mathcal{G}_{1}}=0$.

Similarly, repeat this process on the row $\mathcal{G}^{1}$, and we have

[TABLE]

with $\tilde{v}_{h}^{1,1}\in V_{h}^{\rm MC}$, and $v_{h}^{1,1}|_{\mathcal{G}_{1}\cup\mathcal{G}^{1}}=0$. Finally, by the same technique as used in the proof of Theorem A.7, we can prove that

[TABLE]

A combination of (A.12), (A.13), and (A.14) leads to

[TABLE]

Hence the result. ∎

Remark A.9**.**

From the above two theorems, it holds that ${\rm dim}(V_{h0}^{\rm MC})=2mn-m-n+1$ and ${\rm dim}(V_{h}^{\rm MC})=2mn+2m+2n$.**

Proposition A.10**.**

Define V^{(2)}_{h}:=\Big{\{}v_{h}\in H^{1}(\Omega):v_{h}|_{K}\in P_{2}(K),\forall K\in\mathcal{G}_{h}\Big{\}}, and $V^{(2)}_{h0}:=V^{(2)}_{h}\cap H^{1}_{0}(\Omega)$. That is, $V^{(2)}_{h}$ and $V^{(2)}_{h0}$ are conforming $P_{2}$ element spaces. Then,

1. (1)

$V^{(2)}_{h}=V^{\rm BL}_{h}+{\rm span}\{\varphi_{i}^{x},\varphi^{y}_{j}\}$, and $\ V^{(2)}_{h0}=V^{\rm BL}_{h0};$ 2. (2)

$V_{h}^{\rm MC}=V^{(2)}_{h}+{\rm span}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}$, and $V_{h0}^{\rm MC}=V^{(2)}_{h0}+{\rm span}\{\varphi_{0,K}\}_{K\in\mathcal{G}_{h}}.$

Proposition A.11**.**

Let $V_{h}^{\rm W}$ be the Wilson element space and $V_{h0}^{\rm W}$ be its homogeneous subspace. Then

1. (1)

$\nabla_{h}V_{h}^{\rm MC}\subset\nabla_{h}V_{h}^{\rm W}$, and $\nabla_{h}V_{h0}^{\rm MC}\subset\nabla_{h}V_{h0}^{\rm W};$ 2. (2)

$V_{h}^{\rm MC}\setminus V_{h}^{\rm W}\neq\varnothing$, $V_{h}^{\rm W}\setminus V_{h}^{\rm MC}\neq\varnothing$, $V_{h0}^{\rm MC}\setminus V_{h0}^{\rm W}\neq\varnothing$, and $V_{h0}^{\rm W}\setminus V_{h0}^{\rm MC}\neq\varnothing$.

Appendix B Construction of basis functions for the reduced rectangular Morley (RRM) element space

Let $\Omega\in\mathbb{R}^{2}$ be rectangle domain divided by $m\times n$ rectangles. Define the reduced rectangular Morley element spaces as

[TABLE]

[TABLE]

In this section, we will present an available set of basis functions of $V_{hs}^{\rm{R}}$.

B.1. Compatibility conditions

By a pure linear algebra argument, the following description holds for $P_{2}(K)$.

Lemma B.1**.**

([36, Lemma 15])* Let $K$ be a rectangle with vertices $a_{i}$ and edges $\Gamma_{l}$ ($i,l=1:4$). Denote its length and width by $L$ and $H$, respectively; see Figure 13 (Left). Then, given $\alpha_{i},\beta_{i}\in\mathbb{R}$, there exists a uniquely $p\in P_{2}(K)$ satisfying*

[TABLE]

if and only if the following compatibility conditions are satisfied on $K$,

[TABLE]

Recall the definitions of $\mathcal{G}_{h}$, $\mathcal{G}_{i}$, and $\mathcal{G}^{j}$ in Appendix A. Also, the vertices are labeled by $X_{i}^{j}$, the midpoints on any edge by $Y^{j}_{i}$ and $Z^{j}_{i}$, and the cells by $K^{j}_{i}$; see Figure 13 (Right). Next we present some local or global functions in $V_{hs}^{\rm{R}}$ by giving their value on $X_{i}^{j}\in\mathcal{E}_{h}^{i}$ and derivative on the midpoint of $e\in\mathcal{E}_{h}$.

B.2. Patterns of supports of basis functions in $V_{hs}^{\rm{R}}$

Associated with $\mathcal{G}_{h}$, we present some patterns, i.e., the support sets of basis functions in the RRM element space. To begin with, we introduce some notations. An interior edge, denoted by $e_{{\rm bot},i}$ $(2\leqslant i\leqslant m-1)$, is called a bottom interior edge if its two endpoints are interior points, and its lower opposite edge is on the bottom of $\mathcal{G}_{h}$. A top interior edge $e_{{\rm top},i}$, a left interior edge $e_{{\rm lef},j}$, and a right interior edge $e_{{\rm rig},j}$ are defined in a similar way $(2\leqslant j\leqslant n-1)$.

In the following lemmas, we always denote, by $\omega$, a generic patch with boundaries $\Gamma_{l}$ ($l=1:4$), anticlockwise.

Lemma B.2**.**

Let $e_{{\rm bot},i}$ ($2\leqslant i\leqslant m-1$) be a bottom interior edge with endpoints $X_{i-1}^{1}$ and $X_{i}^{1}$ . Let $\omega$ be a $3\times 2$ cells patch as shown in Figure 14, left. Define

[TABLE]

Likewise, we can define $V_{{\rm top},i}^{\rm R}$ (see Figure 14, right), $V_{{\rm lef},j}^{\rm R}$, and $V_{{\rm rig},j}^{\rm R}$ (see Figure 15, left and right), where $2\leqslant i\leqslant m-1$ and $2\leqslant j\leqslant n-1$. Then we have ${\rm dim}(V_{{\rm bot},i}^{\rm R})={\rm dim}(V_{{\rm top},i}^{\rm R})={\rm dim}(V_{{\rm lef},j}^{\rm R})={\rm dim}(V_{{\rm rig},j}^{\rm R})=1$.

Proof.

First, we consider $V_{{\rm bot},i}^{\rm R}$. Let the geometric features of $\omega$ be represented as in Figure 14, left. Given $\varphi\in V_{{\rm bot},i}^{\rm R}$, denote by $x_{i}^{j}:=\varphi(X_{i}^{j})$, $y_{i}^{j}:=\partial_{y}\varphi(Y_{i}^{j})$, and $z_{i}^{j}:=\partial_{x}\varphi(Z_{i}^{j})$. Apply conditions (B.3) and (B.4) on every element, we have, row by row,

[TABLE]

Rewrite the system after adjusting the order,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

It is straightforward to verify that \eqref{eq:3}-\eqref{eq:1}-\eqref{eq:5}-\big{[}\eqref{eq:9}-\eqref{eq:7}-\eqref{eq:11}\big{]}=0. The system admits a one-dimension solution space. Thus we obtain ${\rm dim}(V_{{\rm bot},i}^{\rm R})=1$. Likewise, we have ${\rm dim}(V_{{\rm top},i}^{\rm R})={\rm dim}(V_{{\rm lef},j}^{\rm R})={\rm dim}(V_{{\rm rig},j}^{\rm R})=1\ (\forall\ 2\leqslant i\leqslant m-1,2\leqslant j\leqslant n-1$). ∎

With similar procedures in Lemma B.3, the following two lemmas can be obtained.

Lemma B.3**.**

Let $X_{m-1}^{n-1}$ be an interior node in the northeast corner. Let $\omega$ be a $2\times 2$ patch as shown in Figure 16, left. Define V_{X}^{\rm R}:=\Big{\{}v_{h}\in V_{hs}^{\rm{R}}(\omega):\fint_{e}\partial_{n_{e}}v_{h}\,ds=0,\ \forall e\subset\Gamma_{l}\ (l=1,2)\Big{\}}. Then ${\rm dim}(V_{X}^{\rm R})=1$.

Lemma B.4**.**

([36, Lemma 15])* Let $\omega$ be a $3\times 3$ patch as shown in Figure 16, right. Define V_{K_{i}^{j}}^{\rm R}:=\Big{\{}v_{h}\in V_{hs}^{\rm{R}}(\omega):\fint_{e}\partial_{n_{e}}v_{h}\,ds=0,\ \forall e\subset\Gamma_{i}\ (1\leqslant l\leqslant 4)\Big{\}}, where $2\leqslant i\leqslant m-1$ and $\ 2\leqslant j\leqslant n-1$. Then ${\rm dim}(V_{K_{i}^{j}}^{\rm R})=1$.*

Lemma B.5**.**

Let $\omega$ be a $3\times 1$ or $1\times 3$ patch in Figure 17, left and right, respectively. Define

[TABLE]

Then, ${\rm dim}(V_{\omega})=1$ and $v_{h}(X_{1})=0$ implies $v_{h}\equiv 0$ on $\omega.$

Lemma B.6**.**

If we denote $\omega=\mathcal{G}_{i}$ the union of elements in the $i$-th column with boundaries $\bigcup_{l=1}^{4}\Gamma_{l}$ (see Figure 18, left), and define

[TABLE]

then ${\rm dim}(V_{{\rm col},i}^{\rm R})=1$. If $\omega=\mathcal{G}^{j}$ (see Figure 18, right), and define

[TABLE]

then ${\rm dim}(V_{{\rm row},j}^{\rm R})=1$.

Proof.

Obviously, $V_{{\rm col},i}^{\rm R}\neq\varnothing$. Assume that $v_{h}\in V_{{\rm col},i}^{\rm R}$. We denote $y_{i}^{j}:=v_{h}(Y_{i}^{j})$, where $1\leqslant j\leqslant n+1$; see Figure 18 (Left). Apply (B.3) and (B.4) to $v_{h}$ on each element in $\mathcal{G}_{i}$, and we obtain $y_{i}^{j}=-y_{i}^{j+1}\ (1\leqslant j\leqslant n)$. It proves that ${\rm dim}(V_{{\rm col},i}^{\rm R})=1$. Similarly, assume that $w_{h}\in V_{{\rm col},i}^{\rm R}$ and denote $z_{i}^{j}:=w_{h}(Z_{i}^{j})$ $(1\leqslant i\leqslant m+1)$. We have $z_{i}^{j}=-z_{i+1}^{j}\ (1\leqslant i\leqslant m)$; see Figure 18 (Right). Thus we obtain ${\rm dim}(V_{{\rm row},j}^{\rm R})=1$. ∎

Remark B.7**.**

Here and throughout this paper, we do not distinct between $V_{{\rm bot},i}^{\rm R},$ $V_{{\rm top},i}^{\rm R},$ $V_{{\rm lef},j}^{\rm R},$ $V_{{\rm rig},j}^{\rm R},$ V_{X}^{\rm R},$$V_{K_{i}^{j}}^{\rm R}, $V_{{\rm col},i}^{\rm R},$ $V_{{\rm row},j}^{\rm R}$ and their respective extension onto the whole domain by zero. Each of them is a subspace in $V_{hs}^{\rm{R}}$.**

B.3. Structure of the RRM element space

Theorem B.8**.**

Let $\mathcal{G}_{h}$ be a $m\times n$ rectangular subdivision of $\Omega$. Define V_{btlr}^{\rm R}:=\oplus_{i,j}\Big{(}V_{{\rm bot},i}^{\rm R}\oplus V_{{\rm top},i}^{\rm R}\oplus V_{{\rm lef},j}^{\rm R}\oplus V_{{\rm rig},j}^{\rm R}\Big{)}, $V_{K}^{\rm R}:=\oplus_{i,j}V_{K_{i}^{j}}^{\rm R}$, and $V_{{\rm glob}}^{\rm R}:=V_{{\rm col},m-1}^{\rm R}\oplus V_{{\rm col},m}^{\rm R}\oplus V_{{\rm row},n-1}^{\rm R}\oplus V_{{\rm row},n}^{\rm R}$, where $2\leqslant i\leqslant m-1$, and $2\leqslant j\leqslant n-1$. Then we have $V_{hs}^{\rm{R}}=V_{btlr}^{\rm R}\oplus V_{{\rm glob}}^{\rm R}\oplus V_{K}^{\rm R}\oplus V_{X}^{\rm R}$.

Proof.

Here we utilize the sweeping procedure again, and divide the proof process into four steps.

Step 1. Given $v\in V_{hs}^{\rm{R}}$. We begin with the boundaries of $\mathcal{G}_{h}$; see Figure 19. Recall the definition of $e_{{\rm bot},i}$, $e_{{\rm top},i}$, $e_{{\rm lef},j}$, and $e_{{\rm rig},j}$. Associated with $e_{{\rm bot},2}$, $V_{{\rm bot},2}^{\rm R}$ is a subspace of $V$ defined in Lemma B.2. There exists a unique function $\varphi_{b,1}\in V_{{\rm bot},2}^{\rm R}$, such that $\varphi_{b,1}(Y_{1}^{1})=v(Y_{1}^{1})$. Then, $v_{1}=v-\varphi_{b,1}$ with $v_{1}\equiv 0$ on the bottom edge of $K_{1}^{1}$. Repeat the procedure for $Y_{2}^{1}$, …, $Y_{m-2}^{1}$, and we obtain $v_{m-2}=v-\sum_{i=2}^{m-1}\varphi_{b,i}$ with $v_{m-2}$ vanishing on the first $m-2$ edges on the bottom boundary of $\mathcal{G}_{h}$. Similarly, we obtain functions $\varphi_{t,i}$, $\varphi_{l,j}$, and $\varphi_{r,j}$. Therefore, $w_{1}=v-\sum_{i=2}^{m-1}\varphi_{b,i}-\sum_{i=2}^{m-1}\varphi_{t,i}-\sum_{j=2}^{n-1}\varphi_{l,j}-\sum_{j=2}^{n-1}\varphi_{r,j}$ with $w_{1}$ vanishing on the dotted edges; see Figure 19 (Middle).

Step 2. There exists uniquely $\varphi_{{\rm col},i}$ in $V_{{\rm col},i}^{\rm R}$ and $\varphi_{{\rm row},j}$ in $V_{{\rm row},j}^{\rm R}$, which satisfy $\varphi_{{\rm col},i}(Y_{i}^{1})=w_{1}(Y_{i}^{1})$, and $\varphi_{{\rm row},j}(Z_{1}^{j})=w_{1}(Z_{1}^{j})$, respectively. Then $w_{2}=w_{1}-\sum_{i=m-1}^{m}\varphi_{{\rm col},i}-\sum_{j=n-1}^{n}\varphi_{{\rm row},j}$ with $w_{2}$ vanishing on the dotted edges; see Figure 19 (Right).

Step 3. Consider elements in the first column $\mathcal{G}_{1}$; see Figure 17. Note that $K_{2}^{2}$ is the only interior element whose $3\times 3$ patch contains $K_{1}^{1}$. There exists uniquely $\varphi_{1}^{1}\in V_{K_{2}^{2}}^{\rm R}$, such that $\varphi_{1}^{1}|_{K_{1}^{1}}=w_{2}|_{K_{1}^{1}}$. Therefore, $w_{2}-\varphi_{1}^{1}$ vanishes on $K_{1}^{1}$. Next, there exists a unique $\varphi_{1}^{2}\in V_{K_{2}^{3}}^{\rm R}$, such that $w_{2}-\varphi_{1}^{1}-\varphi_{1}^{2}$ vanishes on on $K_{1}^{1}\cup K_{1}^{2}$. Repeating the procedure for $\mathcal{G}_{1}$, we obtain $w_{2}-\sum_{j=1}^{n-2}\varphi^{j}_{1}$, which vanishes on $\cup_{j=1}^{n-2}K_{1}^{j}$. Notice that $\cup_{j=n-2}^{n}K_{1}^{j}$ is a $1\times 3$ patch $\omega$ stated in Lemma B.5; see Figure 20 (Left). Since $w_{2}-\sum_{j=1}^{n-2}\varphi^{j}_{1}$ vanishes on $X_{1}^{n-2}$, it vanishes on $\omega$ and further $\mathcal{G}_{1}$.

By repeating the procedure along the column $\mathcal{G}_{i}\ (i=2,\ \ldots,\ m-2)$, we derive $w_{3}:=w_{2}-\sum_{i=1}^{m-2}\sum_{j=1}^{n-2}\varphi^{j}_{i}$, which vanishes on $\bigcup_{i=1}^{m-2}\mathcal{G}_{i}$. Especially, $w_{3}(X_{m-2}^{j})=0$ ($1\leqslant j\leqslant n-2$). Since $\cup_{i=m-2}^{m}K_{i}^{1}$ forms a $1\times 3$ patch $\omega$ (see Figure 20, middle) and $w_{3}(X_{m-2}^{1})=0$, thus it vanishes on $\omega$. Find other $1\times 3$ or $3\times 1$ patch $\omega$, which satisfies the conditions in Lemma B.5, in columns $\mathcal{G}_{m-2}$, $\mathcal{G}_{m-1}$, and $\mathcal{G}_{m}$. Therefore, we derive that $w_{3}$ vanishes on $\mathcal{G}_{h}$ except for a $2\times 2$ patch in the northeast corner; see Figure 20 (Right).

Step 4. From Lemma B.3, there exists a unique function $\varphi_{m-1}^{n-1}\in V_{X}^{\rm R}$ and $w_{3}-\varphi_{m-1}^{n-1}=0$. This way, we have verified that $v=(\sum_{i=2}^{m-1}\varphi_{b,i}+\sum_{i=2}^{m-1}\varphi_{t,i}+\sum_{j=2}^{n-1}\varphi_{l,j}+\sum_{j=2}^{n-1}\varphi_{r,j})+(\sum_{i=m-1}^{m}\varphi_{{\rm col},i}+\sum_{j=n-1}^{n}\varphi_{{\rm row},j})+\sum_{i=1}^{m-2}\sum_{j=1}^{n-2}\varphi^{j}_{i}+\varphi_{m-1}^{n-1}$, and this representation by these $mn+1$ functions is unique. ∎

Remark B.9**.**

From Theorem B.8, it holds that ${\rm dim}V_{hs}^{\rm{R}}=mn+1$.**

Proposition B.10**.**

It holds for $V_{h}^{\rm{W}}$, $V_{h0}^{\rm{W}}$, $V_{h}^{\rm{M}}$, $V_{hs}^{\rm{M}}$, $V_{h}^{\rm{R}}$, and $V_{hs}^{\rm{R}}$ that

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