An Interface Green's Function Framework for Complete Discrete $W^{1,\infty}$ Analysis of Discontinuous Galerkin Methods

Technical Summary: An Interface Green's Function Framework for Complete Discrete $W^{1,\infty}$ Analysis of Discontinuous Galerkin Methods

Problem Statement
Pointwise error analysis of Discontinuous Galerkin (DG) methods has been a significant area of research, with existing literature establishing optimal error estimates in the $L^\infty$ norm and the broken $W^{1,\infty}$ seminorm. However, a critical theoretical gap remains: the broken $W^{1,\infty}$ seminorm, defined as $\|\nabla(u - u_h)\|_{L^\infty_h(\Omega)}$, fails to control discontinuities across mesh interfaces. Consequently, existing analyses do not provide direct control over interface jumps, which are intrinsic to discontinuous approximations and vital for stability, conservation, and post-processing. Furthermore, on convex polyhedral domains, where weak discrete maximum principles are currently unavailable for DG methods, deriving optimal estimates without suboptimal logarithmic factors (e.g., $|\ln h|^\beta$) has been a persistent challenge, particularly for polynomial degrees greater than one.

Methodology
The authors propose a novel interface Green's function framework to achieve a complete discrete $W^{1,\infty}$ analysis. The core methodological innovation departs from classical Green's function analyses, which typically utilize a single discrete delta function to estimate the solution or its gradient. Instead, this framework introduces interface Green's functions generated by the difference between adjacent discrete delta functions.

1. Definition of Interface Green's Functions:

   • For interior mesh interfaces, the framework defines a Green's function $\Gamma_{if}$ satisfying $-\Delta \Gamma_{if} = \delta_{h, K_+^*} - \delta_{h, K_-^*}$, where $\delta_{h, K}$ represents a smooth discrete delta function supported on element $K$. This source term naturally represents the jump of discontinuous approximations across the interface.
   • For boundary faces, a similar Green's function $\Gamma_{bf}$ is defined using a single discrete delta function.

2. Representation Formulas:
   The jumps of the numerical solutions (and their gradients) are represented using these interface Green's functions and their DG approximations. This reduces the estimation of interface quantities to the approximation error of the interface Green's functions themselves.

3. Key Analytical Ingredient: Local Energy Estimates:
   The proof relies on a new local energy estimate for these interface Green's functions, established via dyadic decomposition of the domain.

   • The authors employ a Taylor expansion argument with respect to the source variable.
   • Crucially, the difference between neighboring discrete delta functions ($\delta_{h, K_+^*} - \delta_{h, K_-^*}$) creates a cancellation effect at the level of the singular source.
   • This cancellation removes the dominant singular contribution, yielding sharper local estimates (specifically, an additional factor of $h$) compared to classical estimates.
   • This improvement allows the derivation of interface estimates without introducing logarithmic factors that typically arise from direct applications of discrete trace inequalities combined with existing maximum-norm estimates.

Key Contributions and Results
The paper establishes a "complete" discrete $W^{1,\infty}$ theory, defined by the norm:
$$|||v|||_{W^{1,\infty}_h(\Omega)} = \max_{K \in \mathcal{T}_h} \|\nabla v\|_{L^\infty(K)} + \max_{F \in \mathcal{E}_h} h_F^{-1} \|[v]\|_{L^\infty(F)}$$
This norm simultaneously measures the broken gradient and the scaled interface jumps.

The framework is applied to the following methods on convex polyhedral domains for the Poisson equation:

• Symmetric Interior Penalty DG (IPDG) method.
• Hybrid Mixed DG method.
• Two representative Hybridizable DG (HDG) methods (HDG-1 and HDG-2).

Main Theoretical Results:

• Optimal Jump Estimates: The authors prove optimal maximum-norm estimates for the interface jumps $[u - u_h]$ and numerical fluxes, removing the logarithmic factor $\beta$ that previously appeared in estimates on convex polyhedral domains.
• Complete Error Bounds: They derive optimal error estimates for the broken gradients and interface jumps simultaneously, completing the pointwise analysis for these methods.
• Extension to Fourth-Order Problems: The framework is extended to the $C^0$ interior penalty method for the biharmonic equation. The authors derive new maximum-norm estimates for the jumps of the normal derivative, a quantity previously untreated in pointwise analysis for this method.

Significance and Claims
The paper claims that its primary contribution is methodological. By introducing interface Green's functions based on the difference of discrete delta functions, the authors overcome the fundamental barrier of the lack of weak discrete maximum principles for DG methods on convex polyhedral domains.

• Removal of Logarithmic Factors: The framework successfully eliminates the suboptimal logarithmic factors inherited from classical trace-inequality approaches, providing optimal estimates for polynomial degrees greater than one.
• Generality: The construction depends on the local discontinuous structure of the approximation rather than the specific numerical formulation. The authors assert that the methodology is sufficiently general to be applicable to a broad class of discontinuous and nonconforming finite element methods.
• Completeness: The work provides the first complete discrete $W^{1,\infty}$ analysis that simultaneously controls broken gradients and interface jumps for the specified DG methods, filling a critical gap in the theoretical understanding of pointwise approximations for discontinuous methods.