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On discrete Sobolev inequalities for nonconforming finite elements under a semi-regular mesh condition

Hiroki Ishizaka

TL;DR

The paper addresses discrete Sobolev embeddings for nonconforming CR and discontinuous CR finite element spaces on anisotropic, semi-regular meshes in 2D and 3D. It develops a mesh-robust framework using a two-step affine mapping, Bogovskii operator, and Raviart–Thomas interpolation, combined with anisotropic trace and face-weighted integration-by-parts, to prove $L^q-L^p$ and related inequalities with constants independent of mesh geometry. The results extend classical shape-regular theory to highly anisotropic partitions and provide a robust foundation for stability and error analysis of nonconforming and DG methods on such meshes, including adaptive scenarios. The work removes convexity constraints present in prior duality-based arguments by leveraging a RT/Bogovskii-based approach, and it identifies open directions for $q>p$ regimes and higher-order CR elements.

Abstract

We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh condition, this discrete Sobolev inequality is applicable to all pairs $(q,p)$ that align with the local Sobolev embedding, including scenarios where $q \leq p$. Importantly, the constant is influenced solely by the domain and the semi-regular parameter, ensuring robustness against variations in aspect ratios and interior angles of the mesh. The proof employs an anisotropy-sensitive trace inequality that leverages the element height, a two-step affine/Piola mapping approach, the stability of the Raviart--Thomas interpolation, and a discrete integration-by-parts identity augmented with weighted jump/trace terms on faces. This Sobolev inequality serves as a mesh-robust foundation for the stability and error analysis of nonconforming and discontinuous Galerkin methods on highly anisotropic meshes.

On discrete Sobolev inequalities for nonconforming finite elements under a semi-regular mesh condition

TL;DR

The paper addresses discrete Sobolev embeddings for nonconforming CR and discontinuous CR finite element spaces on anisotropic, semi-regular meshes in 2D and 3D. It develops a mesh-robust framework using a two-step affine mapping, Bogovskii operator, and Raviart–Thomas interpolation, combined with anisotropic trace and face-weighted integration-by-parts, to prove and related inequalities with constants independent of mesh geometry. The results extend classical shape-regular theory to highly anisotropic partitions and provide a robust foundation for stability and error analysis of nonconforming and DG methods on such meshes, including adaptive scenarios. The work removes convexity constraints present in prior duality-based arguments by leveraging a RT/Bogovskii-based approach, and it identifies open directions for regimes and higher-order CR elements.

Abstract

We derive a discrete Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh condition, this discrete Sobolev inequality is applicable to all pairs that align with the local Sobolev embedding, including scenarios where . Importantly, the constant is influenced solely by the domain and the semi-regular parameter, ensuring robustness against variations in aspect ratios and interior angles of the mesh. The proof employs an anisotropy-sensitive trace inequality that leverages the element height, a two-step affine/Piola mapping approach, the stability of the Raviart--Thomas interpolation, and a discrete integration-by-parts identity augmented with weighted jump/trace terms on faces. This Sobolev inequality serves as a mesh-robust foundation for the stability and error analysis of nonconforming and discontinuous Galerkin methods on highly anisotropic meshes.

Paper Structure

This paper contains 21 sections, 15 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mesh, mesh faces, jumps and averages
  4. Trace inequality
  5. Finite element spaces
  6. Norms
  7. Reference elements
  8. Two-dimensional case
  9. Three-dimensional case
  10. Two-step affine mapping
  11. Construct mapping $\Phi_{\widetilde{T}}: \widehat{T} \to \widetilde{T}$
  12. Construct mapping $\Phi_{T}: \widetilde{T} \to T$
  13. Two-step Piola transforms
  14. Additional notations
  15. $L^2$-orthogonal projection
  16. ...and 6 more sections

Key Result

Lemma 2.2

Let $p \in [1,\infty]$. Let $T \subset \mathbb{R}^d$ be a simplex. There exists a positive constant ${C^{Tr}(d,p)}$ such that for any $\bm v = (v^{(1)}, \ldots,v^{(d)})^{\top} \in W^{1,p}(T)$, $F \in \mathcal{F}_{T}$, and $h$, where $\ell_{T,F} := \frac{d |T|_d}{|F|_{d-1}}$ denotes the distance of the vertex of $T$ opposite to $F$ to the face. Here, the constant $C^{Tr}(d,p)$ depends only on the

Figures (3)

  • Figure 1: Two-step affine mapping and vectors $r_i$, $i=1,2$
  • Figure 2: (Type i) Vectors $r_i$, $i=1,2,3$
  • Figure 3: (Type ii) Vectors $r_i$, $i=1,2,3$

Theorems & Definitions (37)

  • Remark 2.1: Conforming meshes
  • Lemma 2.2: Trace inequality
  • Proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.8
  • Proof
  • Definition 2.9: Two-step Piola transforms
  • Lemma 2.10
  • ...and 27 more