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Brezzi--Douglas--Marini interpolation on anisotropic simplices and prisms

Volker Kempf

TL;DR

This work addresses the need for interpolation error estimates of the Brezzi–Douglas–Marini (BDM) operator on anisotropic finite elements for $H_{\mathrm{div}}$ discretizations. It extends known $L^2$ simplex results and $L^p$ parallelotope results to general $L^p$ bounds on anisotropic simplices ($1\le p\le \infty$), and derives new triangular-prism estimates by establishing stability on a reference element and transferring via diagonal affine maps. The main contributions are (i) $L^p$ anisotropic interpolation bounds on simplices under regular vertex and maximum angle conditions, and (ii) stability and $L^p$ interpolation bounds for anisotropic triangular prisms with explicit dependence on edge directions and facet geometry, including an additional term involving higher derivatives of the prism side component. These results facilitate accurate $H_{\mathrm{div}}$-conforming discretizations on highly stretched meshes, with potential impact on boundary-layer and edge-singularity problems in incompressible flow and related pressure-robust reconstructions.

Abstract

The Brezzi--Douglas--Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in $L^2$, the other considering parallelotopes with estimates in terms of $L^p$-norms. This contribution provides generalized estimates for anisotropic simplices for the $L^p$ case, $1\leq p\leq\infty$, and shows new estimates for anisotropic prisms with triangular base.

Brezzi--Douglas--Marini interpolation on anisotropic simplices and prisms

TL;DR

This work addresses the need for interpolation error estimates of the Brezzi–Douglas–Marini (BDM) operator on anisotropic finite elements for discretizations. It extends known simplex results and parallelotope results to general bounds on anisotropic simplices (), and derives new triangular-prism estimates by establishing stability on a reference element and transferring via diagonal affine maps. The main contributions are (i) anisotropic interpolation bounds on simplices under regular vertex and maximum angle conditions, and (ii) stability and interpolation bounds for anisotropic triangular prisms with explicit dependence on edge directions and facet geometry, including an additional term involving higher derivatives of the prism side component. These results facilitate accurate -conforming discretizations on highly stretched meshes, with potential impact on boundary-layer and edge-singularity problems in incompressible flow and related pressure-robust reconstructions.

Abstract

The Brezzi--Douglas--Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in , the other considering parallelotopes with estimates in terms of -norms. This contribution provides generalized estimates for anisotropic simplices for the case, , and shows new estimates for anisotropic prisms with triangular base.
Paper Structure (3 sections, 7 theorems, 24 equations, 1 figure)

This paper contains 3 sections, 7 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Error estimates on anisotropic simplices
  3. Error estimates on anisotropic prisms

Key Result

Theorem 1

Let a simplicial element $T$ satisfy the regular vertex property with constant $\bar{c}$. Then for $k\geq 1$, $0\leq m\leq k$ and $\boldsymbol{v}\in \boldsymbol{W}^{m+1,p}(T)$, $1\leq p\leq\infty$, the estimate holds, the constant only depends on $\bar{c}$ and $k$. Here $h_T={\operatorname{diam}\,} T$, $h^{\boldsymbol{\alpha}} = \prod_{j\in I_d}h_j^{\alpha_j}$ and $D_{\boldsymbol{l}}^{\boldsymbol

Figures (1)

  • Figure 1: Reference prism $\widehat{P}$ with vertex numbering, and a transformed prism of the reference family $\mathcal{R}_P$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • Theorem 4