High accuracy analysis of two nonconforming plate elements

Abstract

We prove the superconvergence of Morley element and the incomplete biquadratic nonconforming element for the plate bending problem. Under uniform rectangular meshes, we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method.

Superconvergence of both the Crouzeix–Raviart and Morley elements

“…ln h| 1/2 |σ| 1,∞,Ω ) w h 1,Ω . (3.16) A substitution of (3.9) and (3.16) into (3.7) concludes |I 11 | (h 2 |σ| 5 2 ,Ω + κh 2 | ln h| 1/2 |σ| 1,∞,Ω ) σ RT − Π RT σ 0,Ω , which completes the proof.Similar arguments for the sums I 12 , I21 and I 22 prove a full one order superconvergence property for the RT element as follows. Suppose that (σ, u) is the solution to (3.2) with σ ∈ H 5 2 (Ω, R 2 ), and (σ RT , u RT ) is the solution to (3.4) on a uniform triangulation T h .…”

“…So far, most of superconvergence analysis results for nonconforming elements are focused on rectangular or nearly parallelogram triangulations, see [15,19,23]. There are a few superconvergence results for nonconforming elements on triangular meshes [14,18,21]. In [14], a half order superconvergence was analyzed for the Crouzeix-Raviart (CR for short hereinafter) element and the Morley element.…”

Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations

“…From the tables and figures, we can see the superconvergent behaviors of the numerical solutions. Besides, in our examples, the exact solution u i (x, y), i = 1, 2, or u i (x, y, z), i = 3, 4, don't satisfy the boundary condition ∂ 3 ui ∂n 3 = 0, i = 1, · · · , 4, which are need for superconvergence of second order in twodimensional case [6,17]. However, our results still have the superconvergent property, which are coincide with our theoretical analysis.…”

“…In [3], Chen first established the supercloseness of the corrected interpolation of the incomplete biquadratic element [29,21] on uniform rectangular meshes. By using similar corrected interpolations as in [3], Mao et al [17] first proved one and a half-order superconvergence for the Morley element [19] and the incomplete biquadratic nonconforming element on uniform rectangular meshes. In a recent paper [6], Hu and Ma proposed a new method by using equivalence between the Morley element and the first order Hellan-Herrmann-Johnson element and obtained one and a half-order superconvergence for the Morley element on uniform mesh.…”

Forty Years of the Crouzeix‐Raviart element

“…There are many topics, such as estimates , superconvergence , higher order nonconforming finite elements , rectangular nonconforming finite elements , and applications to eigenvalue problems , that have not been discussed in this article. Perhaps a survey of these results would appear before the fiftieth anniversary of .…”