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Note on a weakly over-penalised symmetric interior penalty method on anisotropic meshes for the Poisson equation, Ver. 1

Hiroki Ishizaka

TL;DR

The work addresses stable and accurate discretization of the Poisson equation $-\Delta u=f$ on anisotropic meshes using a weakly over-penalised symmetric interior penalty (WOPSIP) DG method. It develops a comprehensive anisotropic DG framework, defining CR and RT finite element spaces, Piola mappings, and precise directional interpolation estimates, and proves an energy-norm error bound $|u-u_h^{wop}|_{wop} \le \inf_{v_h} |u-v_h|_{wop} + E_h(u)$ along with an $L^2$-error bound via a duality argument, with geometry-dependent refinements for Type I/II elements. The results rely on trace inequalities, tailored penalty parameters $\kappa_F$, and a discrete Poincaré inequality to handle anisotropy, establishing stability and convergence under semi-regular mesh conditions. Practically, these contributions advance DG methods for Poisson problems in settings with strong mesh anisotropy, offering theoretical guarantees and guidance for accurate computations in challenging geometries.

Abstract

The purpose is to make an easy-to-understand note of "Special Topics in Finite Element Methods." There might be typos and mistakes. Therefore, I do not take any responsibility for unauthorised use.

Note on a weakly over-penalised symmetric interior penalty method on anisotropic meshes for the Poisson equation, Ver. 1

TL;DR

The work addresses stable and accurate discretization of the Poisson equation on anisotropic meshes using a weakly over-penalised symmetric interior penalty (WOPSIP) DG method. It develops a comprehensive anisotropic DG framework, defining CR and RT finite element spaces, Piola mappings, and precise directional interpolation estimates, and proves an energy-norm error bound along with an -error bound via a duality argument, with geometry-dependent refinements for Type I/II elements. The results rely on trace inequalities, tailored penalty parameters , and a discrete Poincaré inequality to handle anisotropy, establishing stability and convergence under semi-regular mesh conditions. Practically, these contributions advance DG methods for Poisson problems in settings with strong mesh anisotropy, offering theoretical guarantees and guidance for accurate computations in challenging geometries.

Abstract

The purpose is to make an easy-to-understand note of "Special Topics in Finite Element Methods." There might be typos and mistakes. Therefore, I do not take any responsibility for unauthorised use.
Paper Structure (27 sections, 15 theorems, 121 equations)

This paper contains 27 sections, 15 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continuous problems
  4. Meshes, mesh faces, averages and jumps
  5. Penalty parameters and energy norms
  6. Edge characterisation on a simplex, a geometric parameter, and a condition
  7. Reference elements
  8. Affine mappings
  9. Construct mapping $\Phi_{\widetilde{T}}: \widehat{T} \to \widetilde{T}$
  10. Construct mapping $\Phi_{T}: \widetilde{T} \to T$
  11. Additional notation and assumption
  12. Piola transformations
  13. Finite element spaces and anisotropic interpolation error estimates
  14. Finite element spaces
  15. Discontinuous space and the $L^2$-orthogonal projection
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let $T \subset \mathbb{R}^d$ be a simplex. There exists a positive constant $c$ such that for any $\varphi \in H^{1}(T)$, $F \in \mathcal{F}_{T}$, and $h$, where $\ell_{T,F} := \frac{d! |T|_d}{|F|_{d-1}}$ denotes the distance of the vertex of $T$ opposite to $F$ to the face. Furthermore, there exists a positive constant $c$ such that for any $v = (v^{(1)}, \ldots,v^{(d)})^T \in H^{1}(T)^d$, $F \i

Theorems & Definitions (31)

  • Lemma 2.1: Trace inequality
  • Proof
  • Definition 2.4
  • Theorem 2.5
  • Proof
  • Theorem 2.6
  • Proof
  • Theorem 2.7
  • Proof
  • Theorem 2.8
  • ...and 21 more