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

1. Problem Statement

The paper addresses the theoretical analysis of nonconforming finite element methods (and related polytopic methods) for nonlinear problems. A critical component in proving the well-posedness and convergence of these methods is the establishment of Sobolev–Poincaré inequalities for piecewise $W^{1,p}$ functions (functions that are in $W^{1,p}$ on each element but may be discontinuous across element interfaces).

Key Challenges:

• General Meshes: Existing literature often restricts analysis to simplicial or Cartesian meshes, or assumes specific polynomial spaces. There is a lack of rigorous inequalities for general polytopic meshes (elements with arbitrary numbers of facets) in arbitrary dimensions.
• Explicit Constants: Many existing proofs rely on compactness arguments or contradiction, yielding constants that are not explicit. For adaptive and $p$-/$hp$-version convergence analysis, constants must be explicitly dependent on geometric properties (e.g., shape regularity, domain diameter).
• Mixed Boundary Conditions: Standard tools often assume homogeneous Dirichlet conditions on the entire boundary or no boundary conditions. The analysis of nonlinear problems often requires handling mixed boundary conditions (Dirichlet on part of the boundary, Neumann on the rest).
• Regularity: Previous results often focused on $L^2$ or specific $L^p$ cases. This work aims for general $p \in [1, \infty)$ and maximal Lebesgue regularity.

2. Methodology

The authors develop a novel analytical framework that avoids enriching operators (used in some prior works) and continuous embeddings of bounded variation spaces. Instead, they generalize two fundamental tools from Sobolev space theory:

A. Novel Sobolev–Trace Inequalities

The authors derive trace inequalities for piecewise functions on general polytopic meshes.

• Approach: They decompose a polytopic element $K$ into a set of non-overlapping simplices. They utilize a lowest-order Raviart–Thomas function on these simplices to apply the divergence theorem.
• Result: They establish inequalities bounding the $L^q$ norm of a function on the boundary $\partial K$ in terms of its $L^s$ and $W^{1, s/(s-q+1)}$ norms in the interior.
• Key Feature: The constants are explicit and depend on the dimension $d$, the mesh regularity parameter $\gamma$, and the exponents $p, q, s$.

B. $W^{1,p}$-Stable Right-Inverses of the Divergence (Babuška–Aziz Inequalities)

To prove Poincaré inequalities, one typically needs a right-inverse of the divergence operator.

• Novelty: The authors construct a right-inverse for the divergence operator satisfying mixed boundary conditions (homogeneous Dirichlet on a subset $\Gamma_N$, free on $\Gamma_D$).
• Technique:
  • For convex domains, they extend the domain by reflection across the Neumann boundary to create a larger star-shaped domain, apply the standard right-inverse, and restrict back.
  • For non-convex domains, they construct an extended domain $\Omega_{ext}$ such that the union is star-shaped. They carefully handle the extension of the source term $f$ to ensure the resulting vector field has the correct boundary behavior.
• Result: They provide explicit bounds for the stability constant $C_{BA}$ depending on the domain's diameter, the radius of the inscribed ball, and the geometry of the extended domain.

3. Key Contributions

1. Generalization to Polytopic Meshes: The results hold for families of general polytopic meshes in any dimension $d$, covering simplicial, Cartesian, and agglomerated meshes with arbitrary numbers of facets.
2. Explicit Constants: All constants in the derived inequalities are fully explicit. They depend on:
   • Geometric properties of the domain (diameter $h_\Omega$, inscribed ball radius $\rho$).
   • Mesh regularity parameters (shape regularity $\gamma$).
   • The exponents $p$ and $d$.
3. Mixed Boundary Conditions: The derivation of the Babuška–Aziz inequality with mixed boundary conditions is a significant theoretical advancement, filling a gap in the literature regarding explicit constants for non-convex domains with partial boundary constraints.
4. Broken Sobolev Spaces: The inequalities are established for broken $W^{1,p}$ spaces (not just piecewise polynomials), making them applicable to non-polynomial methods (e.g., Trefftz, Extended FEM).
5. Two Types of Inequalities:
   • First Kind: Optimal Lebesgue exponents ($L^{p^*}$), suitable for high-regularity analysis but requiring stronger norms on the RHS.
   • Second Kind: Weaker norms involving jump terms scaled by facet sizes, specifically designed for nonconforming finite elements (like Crouzeix–Raviart) and Discontinuous Galerkin (DG) methods.

4. Main Results

The paper presents four main theorems/corollaries:

• Theorem 1.1 & Corollary 1.2 (Trace Inequalities): Establishes the trace inequality for piecewise functions on polytopic elements with explicit constants involving the Euler Gamma function and mesh regularity $\gamma$.
• Theorem 1.4 (Divergence Right-Inverse): Proves the existence of a $W^{1,p}$-stable right-inverse of the divergence with mixed boundary conditions. It provides explicit upper bounds for the stability constant, distinguishing between convex and non-convex domains.
• Theorem 1.5 (First Kind Sobolev–Poincaré):
  $$\|v\|_{L^{p^*}(\Omega)} \leq C_1 \|\nabla_h v\|_{L^p(\Omega)} + C_2 \left( \sum_{F} \|[[v]]\|_{L^{p^\sharp}(F)}^p \right)^{1/p}$$
  This inequality controls the $L^{p^*}$ norm by the broken gradient and jump terms. It is optimal for exponents but requires careful handling of inverse estimates for standard DG applications.
• Theorem 1.6 (Second Kind Sobolev–Poincaré):
  $$\|v\|_{L^{p_1^*}(\Omega)} \leq C_3 \|\nabla_h v\|_{L^p(\Omega)} + C_4 \left( \sum_{F} \tilde{h}_F^{1-p} \|[[v]]\|_{L^p(F)}^p \right)^{1/p}$$
  This version uses weaker norms and scaled jump terms, making it directly applicable to Crouzeix–Raviart and DG methods without requiring high-degree inverse inequalities.
• Corollaries 1.7 & 1.8: Provide "averaged" versions of the above inequalities where jump terms are projected onto constants ($\Pi^0_F [[v]]$). These are particularly useful for analyzing methods where jumps are controlled by their mean values.

5. Significance and Applications

• Nonlinear Problems: The explicit dependence on $p$ and geometry allows for the analysis of nonlinear PDEs (e.g., $p$-Laplacian, Navier-Stokes) using nonconforming methods on general meshes.
• Polytopic Methods: The results support the rigorous analysis of PolyDG and Virtual Element Methods (VEM) on meshes with arbitrary polygonal/polyhedral elements, which are increasingly popular for complex geometries and mesh coarsening.
• Convergence Analysis: The explicit constants are crucial for proving convergence rates in $p$- and $hp$-versions of DG methods, where constants depending on the polynomial degree can otherwise hinder the proof.
• Robustness: The inequalities are robust with respect to the number of facets per element (in the second kind inequality), allowing for highly agglomerated meshes.

In summary, this paper provides a foundational theoretical toolkit for the analysis of modern, high-order, nonconforming numerical methods on general unstructured meshes, bridging the gap between abstract functional analysis and practical computational requirements.