[Paper Review] Discrete compactness for the p-version of discrete differential forms

Daniele Boffi, Martin Costabel+3 more|arXiv (Cornell University)|Sep 28, 2009

Advanced Numerical Methods in Computational Mathematics27 references4 citations

TL;DR

This paper establishes the discrete compactness property for the $p$-version of $H(\operatorname{curl})$-conforming finite element methods using generalized Nédélec edge elements on polyhedral domains in $\mathbb{R}^d$. By leveraging a smoothed Poincaré lifting operator and projection-based interpolation, it proves that widely used edge finite element families converge spectrally and without spurious modes for the Maxwell eigenvalue problem in 2D and 3D.

ABSTRACT

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.

Motivation & Objective

• To establish the discrete compactness property for the $p$-version of $H(\operatorname{curl})$-conforming finite element methods in 2D and 3D.
• To provide a general framework for analyzing the $p$-version of discrete differential forms of order $\ell$ on polyhedral domains.
• To verify that standard edge finite element families satisfy the discrete compactness property under $p$-refinement.
• To extend the convergence analysis of the Maxwell eigenvalue problem to variable material coefficients and general $p$-refinement strategies.
• To demonstrate that the $p$-version of Nédélec elements on triangles, tetrahedra, parallelograms, and parallelepipeds yields spectrally correct approximations without spurious modes.

Proposed method

• Utilizes a recently developed smoothed Poincaré lifting operator to ensure continuity and compatibility with discrete differential forms.
• Applies projection-based interpolation operators with $p$-uniform boundedness in $L^2$-norm, derived from recent error estimates for $p$-version finite elements.
• Establishes the discrete compactness property via a set of natural assumptions on finite element spaces and interpolation operators.
• Relies on trace theorems and Sobolev regularity estimates to ensure well-defined traces on lower-dimensional facets for $\ell$-forms.
• Uses abstract framework from discrete differential forms theory to generalize the analysis beyond standard finite element spaces.
• Validates the approach through application to $H(\operatorname{curl})$-conforming edge elements, including Nédélec elements of first and second kind on various elements.

Experimental results

Research questions

• RQ1 Do standard $p$-version edge finite element families satisfy the discrete compactness property for the Maxwell eigenvalue problem?
• RQ2 Can the discrete compactness property be established for $H(\operatorname{curl})$-conforming $p$-refined finite elements using a general framework for discrete differential forms?
• RQ3 What conditions on interpolation operators and lifting operators ensure convergence of the $p$-version discretization for non-elliptic eigenvalue problems?
• RQ4 To what extent can the theory be extended to general material coefficients $\epsilon$ and $\mu$ in the Maxwell eigenvalue problem?
• RQ5 Why does the current approach not extend to $hp$-refinement or $d > 3$ dimensions?

Key findings

• The $p$-version of Nédélec edge elements on triangles, tetrahedra, parallelograms, and parallelepipeds satisfies the discrete compactness property.
• The discrete compactness property ensures that the $p$-version Galerkin approximation of the Maxwell eigenvalue problem converges spectrally and without spurious modes.
• The smoothed Poincaré lifting operator enables uniform continuity and compatibility with discrete differential forms, forming a key technical tool.
• The analysis applies to general material coefficients $\epsilon$ and $\mu$ that are uniformly positive definite, extending beyond the normalized case.
• The theory does not cover $hp$-refinement due to technical challenges in maintaining $L^2$-boundedness of projection operators under variable polynomial degrees.
• Extension to $d > 3$ dimensions is obstructed by trace regularity limitations and the lack of $p$-uniform boundedness results for higher-dimensional polyhedral elements.