[Paper Review] Optimal Superconvergence Analysis for the Crouzeix-Raviart and the Morley elements

Jun Hu, Limin Ma+1 more|arXiv (Cornell University)|Aug 28, 2018

Advanced Numerical Methods in Computational Mathematics19 references3 citations

TL;DR

This paper presents an optimal superconvergence analysis for the Crouzeix-Raviart and Morley finite elements by introducing a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for their associated mixed finite elements (Raviart–Thomas and Hellan–Herrmann–Johnson). The method enables a full one-order superconvergence result on uniform and mildly-structured meshes, resolving a gap between prior half-order estimates and observed numerical performance.

ABSTRACT

In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first order mixed Raviart--Thomas element and the mixed Hellan--Herrmann--Johnson element, respectively. This, in particular, allows for proving a full one order superconvergence result for these two mixed finite elements. Finally, a full one order superconvergence result of both the Crouzeix-Raviart element and the Morley element follows from their special relations with the first order mixed Raviart--Thomas element and the mixed Hellan--Herrmann--Johnson element respectively. Those superconvergence results are also extended to mildly-structured meshes.

Motivation & Objective

• To close the gap between observed full one-order superconvergence in numerical tests and prior half-order theoretical estimates for nonconforming finite elements.
• To develop a new superconvergence analysis technique that overcomes limitations in existing Sobolev space-based estimates for boundary terms.
• To extend optimal superconvergence results to mildly-structured (α,σ)-meshes, relaxing the quasi-uniformity assumption.
• To establish a rigorous theoretical foundation for the observed superconvergence behavior of the Crouzeix-Raviart and Morley elements.
• To demonstrate the effectiveness of the method through numerical experiments on various mesh types, including uniform, piecewise uniform, and Delaunay meshes.

Proposed method

• Employ a discrete Helmholtz decomposition of the error between the canonical interpolation and the finite element solution for the first-order Raviart–Thomas and Hellan–Herrmann–Johnson mixed finite elements.
• Use the decomposition to enable cancellation of boundary terms sharing a common vertex in key error estimates.
• Apply the improved analysis to the mixed elements, which then implies optimal superconvergence for the nonconforming Crouzeix-Raviart and Morley elements via their known equivalence relations.
• Generalize the results to mildly-structured (α,σ)-meshes by assuming |ln h_K| ≈ |ln h| for all elements K, a weaker condition than quasi-uniformity.
• Avoid variational error expansions used in prior work, enabling broader applicability and simpler extension to non-uniform meshes.
• Validate the theoretical findings with numerical tests on uniform, piecewise uniform, and Delaunay meshes for the Morley element in plate bending problems.

Experimental results

Research questions

• RQ1 Can a full one-order superconvergence result be rigorously proven for the Crouzeix-Raviart and Morley elements, resolving the discrepancy with prior half-order estimates?
• RQ2 Can the analysis of boundary terms in mixed finite element methods be improved beyond existing Sobolev space-based estimates that are known to be non-improvable?
• RQ3 Does the discrete Helmholtz decomposition technique enable sufficient cancellation in boundary terms to achieve optimal superconvergence?
• RQ4 Can the superconvergence result be extended to mildly-structured (α,σ)-meshes without requiring full quasi-uniformity?
• RQ5 How do the theoretical superconvergence rates compare with numerical results on different mesh types, including Delaunay and piecewise uniform meshes?

Key findings

• A full one-order superconvergence result is proven for both the Crouzeix-Raviart and Morley elements on uniform meshes, resolving the prior half-order estimate gap.
• The improved analysis achieves optimal convergence rates of order h^{1+ρ} for the Raviart–Thomas and Hellan–Herrmann–Johnson mixed finite elements, improving upon the prior half-order result.
• The discrete Helmholtz decomposition enables critical cancellation of boundary terms, which is key to achieving optimal superconvergence.
• The superconvergence result extends to mildly-structured (α,σ)-meshes under the condition |ln h_K| ≈ |ln h|, which is weaker than quasi-uniformity.
• Numerical tests confirm the theoretical rates: the post-processing error for the Morley element converges at order 2.04 on uniform meshes, approaching optimal order 2.
• On piecewise uniform and Delaunay meshes, the interpolation error remains optimal (order ~1.9), but post-processing error is suboptimal (~1.6), indicating sensitivity to mesh structure in post-processing.