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Piecewise polynomial approximation on non-Lipschitz domains

D P Hewett

TL;DR

This work develops best-approximation error estimates for discontinuous piecewise polynomial spaces on non-Lipschitz meshes of non-Lipschitz domains with fractal boundaries. The authors introduce an intrinsic/extrinsic Sobolev-space framework and prove local hp-approximation results (Lemma \ref{lem:hp}) that lead to global broken-$H^m$ estimates (Theorem \ref{thm:Approx}) valid for arbitrary meshes and domains. They also derive corollaries for $L^2$-estimates in fractional spaces, dual-norm estimates, and intrinsic-space estimates, under varying regularity assumptions on the domain (e.g., Koch snowflake) and interpolation-scale conditions. The results extend hp-type approximation theory to fractal geometries, with potential applications to dG-FEM and integral-equation methods on complex domains.

Abstract

We prove best approximation error estimates for discontinuous piecewise polynomial approximation in fractional Sobolev spaces on non-Lipschitz meshes of non-Lipschitz domains. In particular, the boundary of the domain, and the boundaries of the mesh elements, can be fractal.

Piecewise polynomial approximation on non-Lipschitz domains

TL;DR

This work develops best-approximation error estimates for discontinuous piecewise polynomial spaces on non-Lipschitz meshes of non-Lipschitz domains with fractal boundaries. The authors introduce an intrinsic/extrinsic Sobolev-space framework and prove local hp-approximation results (Lemma \ref{lem:hp}) that lead to global broken- estimates (Theorem \ref{thm:Approx}) valid for arbitrary meshes and domains. They also derive corollaries for -estimates in fractional spaces, dual-norm estimates, and intrinsic-space estimates, under varying regularity assumptions on the domain (e.g., Koch snowflake) and interpolation-scale conditions. The results extend hp-type approximation theory to fractal geometries, with potential applications to dG-FEM and integral-equation methods on complex domains.

Abstract

We prove best approximation error estimates for discontinuous piecewise polynomial approximation in fractional Sobolev spaces on non-Lipschitz meshes of non-Lipschitz domains. In particular, the boundary of the domain, and the boundaries of the mesh elements, can be fractal.

Paper Structure

This paper contains 11 sections, 6 theorems, 20 equations, 1 figure.

Table of Contents

  1. MSC2020 classifications:
  2. Introduction
  3. Definitions and results
  4. Proofs
  5. Proof of Lemma \ref{['lem:hp']}
  6. Proof of Theorem \ref{['thm:Approx']}
  7. Proof of Corollary \ref{['cor:int']}
  8. Proof of Corollary \ref{['cor:Duality']}
  9. Proof of Corollary \ref{['cor:Duality2']}
  10. Proof of Corollary \ref{['cor:Intrinsic']}
  11. Conclusion

Key Result

Lemma 2.3

Let $\Omega\subset\mathbb{R}^n$ be a domain, $\mathcal{T}$ a mesh of $\Omega$ with covering $\mathcal{T}^\#$, and $\kappa:\mathcal{T}\to \mathcal{T}^\#$ a covering choice function. Let $h_0>0$ and suppose that $h_\mathcal{K}\leq h_0$ for each $\mathcal{K}\in\mathcal{T}^\#$. Let $u\in L^2(\Omega)$ an

Figures (1)

  • Figure 1: (a) The Koch snowflake domain $\Omega$, an open set with fractal boundary of Hausdorff dimension $\log4/\log3$. (b) A mesh of $\Omega$ comprising 13 elements with fractal boundary (7 large and 6 small), each a scaled/rotated/translated version of $\Omega$ (see CaChHe25 for details). (c) A mesh of $\Omega$ comprising 12 elements, 6 of which are triangles and 6 of which have a boundary that is the union of a line segment and a fractal curve.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Remark 2.10