Interpolation error analysis using a new geometric parameter

TL;DR

The article advances interpolation error analysis by introducing a new geometric parameter $H_T := \frac{\prod_{i=1}^d h_i}{|T|_d} h_T$ to capture anisotropic element quality. Building on a two-step affine mapping between a reference element, an intermediate shape, and the actual element, it derives a semi-regular mesh condition $\frac{H_T}{h_T} \le \gamma_0$ that generalizes the familiar maximum-angle condition and relates to the classical shape-regularity through equivalences in 2D and 3D. A central result is the equivalence between this new condition and the maximum-angle criteria (Synge-type in 2D and a 3D analogue), establishing rigorous interpolation error bounds under significantly relaxed geometric assumptions, including detailed scaling analyses (Part 1). The work also characterizes good versus bad elements in isotropic and anisotropic settings, provides numerical examples, and discusses FE generation within the Bramble–Hilbert framework, thereby enabling reliable, efficient mesh design for anisotropic FE computations. Overall, the paper delivers a theoretically grounded pathway to optimal interpolation rates on anisotropic meshes, with practical implications for mesh generation and error control in finite element simulations.

Abstract

This article presents novel proof methods for estimating interpolation errors, predicated on the understanding that one has already studied foundational error analysis using the finite element method.

Figures (23)

• Figure 1: New parameters $\widetilde{\mathscr{H}}_i$, $i=1,2,3$
• Figure 2: Affine mapping $\Phi_{T}$ and vectors $r_i$, $i=1,2$
• Figure 3: (Type i) Vectors $r_i$, $i=1,2,3$
• Figure 4: (Type ii) Vectors $r_i$, $i=1,2,3$
• Figure 5: Tetrahedra

Theorems & Definitions (221)

• Theorem 1.1
• Proof
• Theorem 1.2
• Proof
• Theorem 1.3
• Proof
• Theorem 1.4
• Proof
• Theorem 1.5
• Proof