Rectangular Morley Elements: Analysis & Convergence

• Rectangular Morley elements are nonconforming finite element spaces defined on d-rectangles using vertex values and face-normal derivative averages.
• They deliver rigorous convergence for second- and fourth-order problems, with enhanced superconvergence achieved via corrected interpolation and postprocessing.
• They provide asymptotically sharp lower bounds for eigenvalues, offering a robust tool for spectral approximation in finite element analysis.

Rectangular Morley elements are nonconforming finite elements on rectangles, rectangular boxes, and, more generally, $d$-rectangles. Their defining pattern is the combination of vertex-value degrees of freedom with edge- or face-averaged normal-derivative degrees of freedom. They were introduced for fourth-order problems such as the biharmonic and plate equations, but have also been analyzed for second-order elliptic equations, singularly perturbed problems, and eigenvalue problems. Across these settings, the literature emphasizes weak interelement continuity, low-order local polynomial spaces enriched to accommodate normal-derivative moments, and sharp results on convergence, superconvergence, and lower eigenvalue bounds on structured meshes (Hu et al., 2014, Meng et al., 2015).

1. Geometric definition and local finite element structure

In the arbitrary-dimensional formulation, a $d$-rectangle $K \subset \mathbb{R}^d$ is written as

$$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$

where $x_c$ is the barycenter of $K$ and $2h_i$ is the side length in the $x_i$-direction. The local shape function space is

$$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$

with $Q_1(K)$ the space of multilinear polynomials. The associated degrees of freedom are the $d$0 vertex values and the $d$1 face-normal-derivative averages

$d$2

Hence the total number of local degrees of freedom is $d$3, and the triple $d$4 is unisolvent (Meng et al., 2015).

In two dimensions, the biharmonic literature often writes the same local space as

$d$5

that is, $d$6. In three dimensions, one closely related biharmonic variant uses

$d$7

together with vertex values and face-normal-derivative averages as local degrees of freedom (Hu et al., 2014, Hu et al., 2015). These formulations differ slightly in presentation, but all preserve the same Morley principle: low-order polynomial reproduction plus normal-derivative moment data on codimension-one facets.

The arbitrary-dimensional analysis constructs explicit local basis functions $d$8 for vertex degrees of freedom and $d$9 for face degrees of freedom, satisfying

$K \subset \mathbb{R}^d$0

and

$K \subset \mathbb{R}^d$1

This yields the local interpolation operator

$K \subset \mathbb{R}^d$2

and the decomposition of the local space into vertex-related and face-related parts (Meng et al., 2015).

2. Global spaces, continuity constraints, and nonconformity

On a mesh $K \subset \mathbb{R}^d$3 of $K \subset \mathbb{R}^d$4-rectangles, the global rectangular Morley space for the second-order problem is

$K \subset \mathbb{R}^d$5

The homogeneous Dirichlet subspace is

$K \subset \mathbb{R}^d$6

The associated broken bilinear form is

$K \subset \mathbb{R}^d$7

with broken norm

$K \subset \mathbb{R}^d$8

Because $K \subset \mathbb{R}^d$9, this is a nonconforming method for Poisson-type problems (Meng et al., 2015).

For fourth-order problems, the global space is adapted to clamped boundary conditions by imposing zero vertex values on $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$0 and vanishing edge- or face-averaged normal derivatives on the boundary. In two dimensions, a typical space is

$$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$1

and the discrete energy form is

$$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$2

The corresponding seminorm is $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$3. In three dimensions the same pattern is imposed on faces rather than edges (Hu et al., 2015, Hu et al., 2014).

This continuity structure is weaker than $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$4-conformity: functions are glued through vertex values and averaged normal derivatives, not through full $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$5 data. For fourth-order problems this produces an $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$6-nonconforming method; for second-order problems it produces an $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$7-nonconforming method. The resulting broken formulations are therefore essential rather than incidental.

3. Model problems and proven convergence regimes

The literature treats rectangular Morley elements in several distinct PDE settings. Three of the most developed are the Poisson problem on $$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$8-rectangular grids, fourth-order singular perturbation on rectangular meshes, and Shishkin-mesh discretizations of singularly perturbed reaction–diffusion/plate problems (Meng et al., 2015, Wang et al., 2011, Meng et al., 28 Aug 2025).

Problem setting	Mesh assumption	Proven estimate
$$K = \Big\{ x = (x_1,\dots,x_d)^T \;\Big|\; x_i = x_{i,c} + \xi_i h_i,\; -1 \le \xi_i \le 1,\; i=1,\dots,d \Big\},$$9 with RM in arbitrary dimension	general shape-regular $x_c$0-rectangular grids	$x_c$1
$x_c$2 with RM in arbitrary dimension	uniform grids	$x_c$3
$x_c$4 with RM in arbitrary dimension	divisionally uniform grids	$x_c$5
$x_c$6 with RM in arbitrary dimension	convex domain	$x_c$7
$x_c$8	convex domain, rectangular mesh	$x_c$9
$K$0	Shishkin mesh	$K$1

For the second-order Poisson problem, the arbitrary-dimensional analysis proves $K$2 convergence in energy norm and $K$3 convergence in $K$4 norm on general $K$5-rectangular grids, while uniform grids improve the energy-norm rate to $K$6. It also proves that the $K$7-norm rate cannot be improved: if $K$8 with $K$9, $2h_i$0, and $2h_i$1, then for sufficiently small $2h_i$2,

$2h_i$3

with $2h_i$4 independent of $2h_i$5 (Meng et al., 2015).

For the singularly perturbed fourth-order model

$2h_i$6

the non-$2h_i$7 rectangular Morley element and a $2h_i$8 extended high order rectangular Morley element are shown to be uniformly convergent in the energy norm with respect to $2h_i$9. Under convexity,

$x_i$0

with a constant independent of $x_i$1 (Wang et al., 2011).

For the Shishkin-mesh method applied to

$x_i$2

on $x_i$3, the rectangular Morley discretization yields

$x_i$4

In the critical regime $x_i$5, this becomes $x_i$6. The same paper contrasts this with an Adini-element bound that gives $x_i$7 in that regime on the same mesh (Meng et al., 28 Aug 2025).

4. Superconvergence, corrected interpolation, and postprocessing

A distinctive feature of rectangular Morley elements is the availability of full second-order superconvergence for biharmonic problems after suitable correction and postprocessing. For both the two- and three-dimensional first-order rectangular Morley elements, the superconvergence analysis proceeds through three ingredients: second-order consistency-error estimates, a corrected canonical interpolation operator, and a macro-element postprocessing operator (Hu et al., 2015).

For the biharmonic equation with clamped boundary conditions, the paper proves in two dimensions that

$x_i$8

for all $x_i$9, and proves the analogous bound in three dimensions. The canonical interpolation $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$0 is then corrected elementwise by subtracting a specially designed term $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$1, producing $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$2 in two dimensions and $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$3 in three dimensions. The corrected interpolants satisfy the supercloseness estimates

$$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$4

in two dimensions and

$$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$5

in three dimensions (Hu et al., 2015).

The final postprocessing uses a macro-element operator $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$6, defined on $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$7 rectangular blocks in two dimensions and $$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$8 blocks in three dimensions. It satisfies

$$P_M(K) := Q_1(K) + \operatorname{span}\{x_i^2, x_i^3 \mid 1 \le i \le d\},$$9

and, combined with supercloseness, yields

$Q_1(K)$0

The paper states that, for first order nonconforming finite element methods of both two and three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions, and also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems (Hu et al., 2015).

A plausible interpretation is that this rectangular theory parallels the mixed-method viewpoint developed for the triangular Morley element, where equivalence with the lowest-order Hellan–Herrmann–Johnson method and stress postprocessing lead first to half-order and then to full one-order superconvergence on structured meshes (Hu et al., 2014, Hu et al., 2018). The rectangular results are direct rather than merely transferred, but the common reliance on patch symmetry and postprocessed Hessians is evident.

5. Eigenvalue approximation and lower-bound phenomena

Rectangular Morley methods have a pronounced spectral feature: they often produce lower bounds for eigenvalues. For biharmonic eigenvalue problems,

$Q_1(K)$1

the rectangular Morley element is analyzed in both two and three dimensions on uniform rectangular or cuboidal meshes. The discrete eigenproblem uses the elementwise Hessian bilinear form

$Q_1(K)$2

and the corresponding discrete eigenvalues are shown to be smaller than the exact ones for sufficiently small $Q_1(K)$3. The analysis combines an eigenvalue error identity, refined properties of the canonical interpolation operator, and a saturation condition. In two dimensions the key interpolation term is of higher order; in three dimensions it is negative and of second order, and the proof therefore introduces a novel decomposition of the leading terms. The final conclusion is that rectangular Morley elements yield asymptotically sharp lower bounds for biharmonic eigenvalues in both two and three dimensions, while eigenfunction errors remain first order in the energy norm and eigenvalue errors are second order (Hu et al., 2014).

A related lower-bound phenomenon appears in the reduced rectangular Morley element for $Q_1(K)$4 eigenvalue problems. On rectangular grids, the reduced space

$Q_1(K)$5

is a purely quadratic Morley-type subspace. For an $Q_1(K)$6 rectangular partition, $Q_1(K)$7. On uniform grids, the resulting eigenvalues satisfy

$Q_1(K)$8

hence

$Q_1(K)$9

The paper therefore identifies the reduced rectangular Morley method as a lower-bound eigenvalue scheme for second-order elliptic operators on rectangular meshes (Zeng et al., 2019).

These lower-bound properties are not incidental. In both the biharmonic and the $d$00 settings, the analyses emphasize the absence of a nontrivial conforming subspace inside the rectangular Morley family, which changes the usual relation between energy and eigenvalue errors and makes lower-bound behavior structurally plausible.

6. Variants, extensions, and computational viewpoints

The rectangular Morley family includes several important variants. One is the extended high order rectangular Morley element, introduced for singular perturbation problems. On each rectangle,

$d$01

with

$d$02

and 12 degrees of freedom given by four vertex values, four edge-midpoint values, and four edge-normal-derivative integrals. The resulting global space is $d$03, remains nonconforming for $d$04, and is uniformly convergent for the singularly perturbed model problem (Wang et al., 2011).

Another variant is the reduced rectangular Morley element, which removes the cubic enrichment and uses a globally constrained piecewise $d$05 space on rectangles. For source problems on uniform rectangular grids it achieves $d$06 convergence in the energy norm, and the paper proves a lower bound $d$07, showing that the $d$08-rate cannot be improved (Zeng et al., 2019).

Rectangles also appear naturally inside more general Morley-type frameworks. A weak Galerkin extension of the Morley element to general polytopal partitions treats rectangles as a special polygonal case. After Schur complement elimination of interior unknowns, a rectangle has exactly $d$09 boundary degrees of freedom, namely four edge-value and four edge-normal data, and the method delivers $d$10 energy convergence and $d$11 $d$12-convergence on shape-regular polytopal meshes (Li et al., 2022). Likewise, a nonconforming virtual element method with Morley degrees of freedom extends the lowest-order Morley element to polygons, with local degrees of freedom given by vertex values and edge-normal integrals, and supplies a conforming companion operator $d$13 for a priori and a posteriori analysis (Carstensen et al., 2022).

Finally, reference-element implementation is subtler for Morley-type elements than for affine-equivalent Lagrange families. A general transformation framework shows that Morley basis functions can be mapped by enriching the nodal set with tangential derivative data, constructing a compatible nodal completion, and using a basis transformation of the form

$d$14

That analysis is carried out for the triangular Morley element rather than for rectangles, but it suggests a closely analogous strategy for rectangular Morley elements based on vertex values, edge-midpoint normal derivatives, and auxiliary tangential derivative nodes (Kirby, 2017).

In this broader perspective, rectangular Morley elements occupy a distinctive position within finite element theory. They are low-order and nonconforming, yet they support arbitrary-dimensional second-order analysis, uniformly stable singular-perturbation discretizations, second-order superconvergence after postprocessing, and lower-bound spectral approximation. The structured geometry of rectangular grids is not merely a convenience: it is the mechanism behind many of their strongest analytical properties.