Upper bound of high-order derivatives for Wachspress coordinates on polytopes

Technical Summary: Upper Bound of High-Order Derivatives for Wachspress Coordinates on Polytopes

Problem Statement
Generalized barycentric coordinates (GBCs) are fundamental to constructing finite element methods (FEM) on polytopal meshes. While the gradient bounds (first-order derivatives) of GBCs are well-established and sufficient for $H^1$ norm error estimates in second-order elliptic equations, higher-order FEM applications (e.g., fourth-order elliptic equations like plate bending) require upper bounds for high-order derivatives ($H^k$ norms, $k > 1$). Prior to this work, such bounds were unavailable in the literature, creating a theoretical gap in the error analysis of GBC-based non-conforming FEMs, such as the Morley-type element proposed in [34].

Methodology
The authors focus on Wachspress coordinates defined on simple, non-degenerate, convex $d$-dimensional polytopes. The derivation of upper bounds for arbitrary-order derivatives ($D^\alpha \phi_v$) involves the following technical steps:

1. Algebraic Decomposition: The Wachspress coordinate $\phi_v$ is expressed as a ratio $w_v / W$, where $w_v$ is a polynomial weight function and $W$ is the sum of all weights. The authors utilize the General Leibniz formula and the Multivariate Faà di Bruno formula to expand the derivatives of this quotient. This reduces the problem to estimating the derivatives of the numerator ($w_v$) and the denominator ($W$).
2. Geometric Parameters: The bounds are derived in terms of two key geometric quantities:
   • $h_K$: The diameter of the polytope $K$.
   • $h^*$: The minimum distance between non-incident vertices and facets of $K$.
3. Derivative Estimation:
   • Numerator ($w_v$): Since $w_v$ is a product of linear distance functions, its derivatives are bounded using combinatorial arguments involving the number of facets and the diameter $h_K$.
   • Denominator ($W$): A critical step involves establishing a strictly positive lower bound for $W$. The authors prove that $W(x) \geq C \frac{(h^*)^{|F|}}{h_K^d}$ by analyzing the distribution of facets relative to a point $x$, utilizing the linear precision property of GBCs and the geometry of simple polytopes.
4. Synthesis: Combining the upper bounds of the derivatives of $w_v$ and $W$ with the lower bound of $W$, the authors derive an explicit upper bound for $|D^\alpha \phi_v|$.

Key Contributions

1. Derivation of High-Order Bounds: The paper provides the first explicit upper bounds for arbitrary-order derivatives of Wachspress coordinates on simple convex polytopes. The main result (Theorem 3.7) states that when $h^* = O(h_K)$ (a condition implying the polytope is not "thin" or degenerate in a specific way), the derivatives satisfy:
   $$|D^\alpha \phi_v| \leq C h_K^{-|\alpha|}$$
   where $C$ depends on the dimension $d$, the number of vertices $|V|$, the number of facets $|F|$, and the multi-index $\alpha$.
2. Geometric Regularity Analysis: The paper systematically compares various shape-regularity conditions for convex polytopes (e.g., minimum edge length, maximum vertex count, chunkiness parameter). It establishes logical implications between these conditions, showing that the assumption $h^* = O(h_K)$ (Assumption H1) is practically reasonable and implies other standard regularity conditions used in FEM.
3. Clarification of Sharpness: The authors clarify that while the derived bounds are sharp in asymptotic order when $h^* = O(h_K)$, they may not be sharp when $h^* \ll h_K$. Numerical experiments suggest that in the latter case, the behavior of derivatives depends on the direction of the "short" edges relative to the derivative direction.

Numerical Results
The authors validate their theoretical findings through 2D numerical experiments:

• Mesh Quality: They demonstrate that Centroidal Voronoi Tessellations (CVT), after eliminating "short" edges, naturally satisfy the condition $h^* = O(h_K)$.
• Verification of Bounds: For random polygons satisfying $h^* = O(h_K)$, the computed values of $\Lambda_\alpha = \max \sum |D^\alpha \phi_i|$ align with the theoretical prediction $O(h_K^{-|\alpha|})$.
• Degenerate Cases: Experiments on polygons where $h^* \ll h_K$ reveal that the theoretical upper bound is conservative. The actual growth of derivatives depends on the orientation of the short edges, with some directional derivatives exhibiting slower growth rates than the general bound suggests.
• Application: The paper includes results from a Morley-type GBC-based FEM applied to a clamped plate bending problem (fourth-order equation). The numerical convergence rates ($O(h)$ in $H^2$-seminorm and $O(h^2)$ in $H^1$-seminorm) confirm that the derived derivative bounds are sufficient to prove optimal convergence for such problems.

Significance and Claims
The paper claims to fill a critical theoretical gap in the analysis of polytopal finite element methods. By providing the necessary high-order derivative bounds, it enables the rigorous proof of optimal convergence for Wachspress-based FEMs applied to fourth-order elliptic equations. The authors emphasize that their approach avoids the highly technical, gradient-specific techniques of previous works (like [14]) by using a different algebraic strategy suitable for arbitrary orders. Furthermore, the work contributes to the understanding of polytope geometry by clarifying the relationships between various shape-regularity assumptions, asserting that the condition $h^* = O(h_K)$ is a robust and practical assumption for generating high-quality meshes in applications.