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Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes

Michele Botti, Lorenzo Mascotto

TL;DR

This work addresses establishing Sobolev--Poincaré inequalities for piecewise $W^{1,p}$ functions on general polytopic meshes in any dimension, with fully explicit constants depending on domain geometry and mesh properties. The authors develop Sobolev--trace inequalities and BA inequalities with mixed boundary conditions to prove two main broken-space inequalities, along with averaged variants suitable for nonconforming discretizations. The results generalize classical broken Poincaré-type inequalities to polytopic meshes and provide tools for rigorous analysis of nonconforming and polyDG methods for nonlinear problems, including explicit dependence on mesh geometry. The findings facilitate well-posedness and convergence analyses of polytopic methods under mixed boundary conditions, with immediate relevance to Crouzeix–Raviart-type discretizations and other nonpolynomial Galerkin approaches.

Abstract

We establish Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincaré inequalities for piecewise $W^{1,p}$ functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and Babuška-Aziz inequalities with mixed boundary conditions. We provide estimates with constants having an explicit dependence on the geometric properties of the domain and the underlying family of polytopic meshes.

Sobolev-Poincaré inequalities for piecewise $W^{1,p}$ functions over general polytopic meshes

TL;DR

This work addresses establishing Sobolev--Poincaré inequalities for piecewise functions on general polytopic meshes in any dimension, with fully explicit constants depending on domain geometry and mesh properties. The authors develop Sobolev--trace inequalities and BA inequalities with mixed boundary conditions to prove two main broken-space inequalities, along with averaged variants suitable for nonconforming discretizations. The results generalize classical broken Poincaré-type inequalities to polytopic meshes and provide tools for rigorous analysis of nonconforming and polyDG methods for nonlinear problems, including explicit dependence on mesh geometry. The findings facilitate well-posedness and convergence analyses of polytopic methods under mixed boundary conditions, with immediate relevance to Crouzeix–Raviart-type discretizations and other nonpolynomial Galerkin approaches.

Abstract

We establish Sobolev-Poincaré inequalities for piecewise functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincaré inequalities for piecewise functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and Babuška-Aziz inequalities with mixed boundary conditions. We provide estimates with constants having an explicit dependence on the geometric properties of the domain and the underlying family of polytopic meshes.

Paper Structure

This paper contains 16 sections, 9 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Domains of interest and Sobolev spaces
  3. Meshes and broken Sobolev spaces
  4. Main technical tools
  5. Main results
  6. Proof of Sobolev--trace inequalities
  7. Babuška--Aziz inequalities with mixed boundary conditions
  8. Proof of the main results
  9. Proof of the first main result
  10. Proof of the second main result
  11. Proof of the third main result
  12. Proof of the fourth main result
  13. Inequalities involving the right-inverse of the divergence
  14. Results in the Banach setting
  15. Results in the Hilbertian setting
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{\mathcal{T}_n\}$ be a family of meshes as in Section subsection:meshes-broken, $q$ be in $[1,\infty)$, $s$ be in $[q,\infty)$, and $\gamma$ be as in regularity-mesh. For all $K$ in any $\mathcal{T}_n$ and all $v$ in $W^{1,\frac{s}{s-q+1}}(K) \cap L^s(K)$, we have where, given $\Gamma(\cdot)$ the Euler's Gamma function,

Figures (2)

  • Figure 1: Extended domain $\widetilde{\Omega}$ (with black and dashed green boundary) for a convex domain $\Omega$. The opaque dashed lines are used to determine the radius of the ball $B_{\widetilde{\rho}}$.
  • Figure 2: Extended domain $\widetilde{\Omega}$ (with black and dashed green boundary) for a nonconvex $\Omega$.

Theorems & Definitions (19)

  • Remark 1: On the regularity of the family of meshes
  • Theorem 1.1: Sobolev--trace inequalities
  • Corollary 1.2: Special Sobolev--trace inequalities
  • Lemma 1.3: Babuška--Aziz inequalities: homogeneous boundary conditions
  • Theorem 1.4: Babuška--Aziz inequalities: mixed boundary conditions
  • Remark 2: Babuška-Aziz inequalities for more general domains
  • Theorem 1.5: 1st kind Sobolev--Poincaré inequalities
  • Theorem 1.6: 2nd kind Sobolev--Poincaré inequalities
  • Remark 3: Norms comparison and finite dimensional spaces
  • Corollary 1.7: 1st kind Sobolev--Poincaré inequalities: averaged version
  • ...and 9 more