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

Michele Botti, Lorenzo Mascotto

TL;DR

This paper develops new trace inequalities for piecewise $W^{1,p}$ functions on general polytopic meshes, addressing the needs of nonconforming discretizations. The authors prove three main theorems that handle different Lebesgue index regimes without relying on finite-dimensional arguments, and they provide explicit, mesh-dependent constants. The results extend trace control to highly general meshes with arbitrarily many and small facets, enabling stability and convergence analyses for $p$-version and $hp$-version nonconforming methods. By incorporating BV theory and broken Sobolev spaces, the work offers robust tools for handling minimal-regularity data in DG and CR-type schemes on polytopic grids.

Abstract

Trace inequalities are crucial tools to derive the stability of partial differential equations with inhomogeneous, natural boundary conditions. In the analysis of corresponding Galerkin methods, they are also essential to show convergence of sequences of discrete solutions to the exact one for data with minimal regularity under mesh refinements and/or degree of accuracy increase. In nonconforming discretizations, such as Crouzeix-Raviart and discontinuous Galerkin, the trial and test spaces consists of functions that are only piecewise continuous: standard trace inequalities cannot be used in this case. In this work, we prove several trace inequalities for piecewise $W^{1,p}$ functions. Compared to analogous results already available in the literature, our inequalities are established: (i) on fairly general polytopic meshes (with arbitrary number of facets and arbitrarily small facets); (ii) without the need of finite dimensional arguments (e.g., inverse estimates, approximation properties of averaging operators); (iii) for different ranges of maximal and nonmaximal Lebesgue indices.

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

TL;DR

This paper develops new trace inequalities for piecewise functions on general polytopic meshes, addressing the needs of nonconforming discretizations. The authors prove three main theorems that handle different Lebesgue index regimes without relying on finite-dimensional arguments, and they provide explicit, mesh-dependent constants. The results extend trace control to highly general meshes with arbitrarily many and small facets, enabling stability and convergence analyses for -version and -version nonconforming methods. By incorporating BV theory and broken Sobolev spaces, the work offers robust tools for handling minimal-regularity data in DG and CR-type schemes on polytopic grids.

Abstract

Trace inequalities are crucial tools to derive the stability of partial differential equations with inhomogeneous, natural boundary conditions. In the analysis of corresponding Galerkin methods, they are also essential to show convergence of sequences of discrete solutions to the exact one for data with minimal regularity under mesh refinements and/or degree of accuracy increase. In nonconforming discretizations, such as Crouzeix-Raviart and discontinuous Galerkin, the trial and test spaces consists of functions that are only piecewise continuous: standard trace inequalities cannot be used in this case. In this work, we prove several trace inequalities for piecewise functions. Compared to analogous results already available in the literature, our inequalities are established: (i) on fairly general polytopic meshes (with arbitrary number of facets and arbitrarily small facets); (ii) without the need of finite dimensional arguments (e.g., inverse estimates, approximation properties of averaging operators); (iii) for different ranges of maximal and nonmaximal Lebesgue indices.

Paper Structure

This paper contains 10 sections, 10 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An example.
  3. Polytopic meshes, broken spaces, and related inequalities
  4. Lebesgue and Sobolev spaces, and special Lebesgue indices
  5. Meshes and broken Sobolev spaces
  6. Broken seminorms and norms, and technical tools
  7. Functions of bounded variation
  8. Trace inequalities in broken Sobolev spaces
  9. Preliminary results
  10. Trace theorems

Key Result

Lemma 2.1

Let $\{\mathcal{T}_h\}$ be a family of meshes as in Section subsection:meshes-broken and $p$ be in $[1,\infty)$. There exist positive constant $C_{\rm SP}$ and $C_{\rm SP}^\sharp$ may possibly depend on $p$, $\Gamma_D$, $\Omega$, $d$, and $\gamma$ such that and, if $p$ further belongs to $[1,d)$,

Figures (1)

  • Figure 1: Behaviour of the indices in \ref{['indices']} for $p$ varying in $[1,6]$ with step $1/10$.

Theorems & Definitions (20)

  • Remark 1: Meaning of the Lebesgue indices
  • Remark 2: Comparison of broken seminorms
  • Lemma 2.1: Sobolev-Poincaré inequalities for piecewise $\mathop{\mathrm{W}}\nolimits^{1,p}$ functions
  • Lemma 2.2: Local Sobolev--trace inequality
  • Lemma 2.3: Auxiliary result
  • proof
  • Lemma 2.4: Trace inequality in $\mathop{\mathrm{BV}}\nolimits(\Omega)$
  • Lemma 2.5: Bound of the $\mathop{\mathrm{BV}}\nolimits$-norm by broken Sobolev norms
  • proof
  • Lemma 3.1: Algebraic bound
  • ...and 10 more