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Finite element spaces by Whitney k-forms on cubical meshes

Shuo Zhang

TL;DR

The paper addresses constructing finite element spaces for $H$CusΛ^k$ on cubical meshes using Whitney $k$-forms to realize minimal-degree, locally supported spaces that still admit commuting interpolators and compatible discretizations. A nonconforming Whitney-based space $\boldsymbol{W}^{\rm def}_h\Lambda^k$ is introduced on cubical grids via a global constraint against the conforming tensor-product space and an adjoint projection $\mathbb{I}^{\mathbf{d}^k}_h$. The authors establish capacity and approximation properties, present a compatible discretization for $H\Lambda^k$ elliptic problems, and derive discrete de Rham complexes with commuting diagrams for $H\Lambda^k$. The framework is implementable using standard routines and sets the stage for extensions to adjoint problems, $H^*\Lambda^k$, and mixed cubical-simplicial meshes.

Abstract

Finite element spaces by Whitney $k$-forms on cubical meshes in $\mathbb{R}^n$ are presented. Based on the spaces, compatible discretizations to $HΛ^k$ problems are provided, and discrete de Rham complexes and commutative diagrams are constructed.

Finite element spaces by Whitney k-forms on cubical meshes

TL;DR

The paper addresses constructing finite element spaces for CusΛ^kk\boldsymbol{W}^{\rm def}_h\Lambda^k\mathbb{I}^{\mathbf{d}^k}_hH\Lambda^kH\Lambda^kH^*\Lambda^k$, and mixed cubical-simplicial meshes.

Abstract

Finite element spaces by Whitney -forms on cubical meshes in are presented. Based on the spaces, compatible discretizations to problems are provided, and discrete de Rham complexes and commutative diagrams are constructed.

Paper Structure

This paper contains 12 sections, 12 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^2$ theory of exterior calculus
  4. Basics of Whitney $k$-forms
  5. Tensor product space $\mathcal{Q}_1^-\Lambda^k$
  6. Notations and symbols
  7. Structure of $\mathcal{Q}^-_1\Lambda^k$ revisited
  8. Finite element spaces for $H\Lambda^k$ by Whitney forms
  9. Capacity of the finite element spaces
  10. Compatible scheme to $H\Lambda^k$ problem
  11. Discrete complex and commutative diagram
  12. Concluding remarks

Key Result

Lemma 2.1

Arnold.D2018feec The adjoint of

Theorems & Definitions (24)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Commutative diagrams
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • ...and 14 more