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A Primal Staggered Discontinuous Galerkin Method on Polytopal Meshes

L. Chen, X. Huang, E. Park, R. Wang

Abstract

This paper introduces a novel staggered discontinuous Galerkin (SDG) method tailored for solving elliptic equations on polytopal meshes. Our approach utilizes a primal-dual grid framework to ensure local conservation of fluxes, significantly improving stability and accuracy. The method is hybridizable and reduces the degrees of freedom compared to existing approaches. It also bridges connections to other numerical methods on polytopal meshes. Numerical experiments validate the method's optimal convergence rates and computational efficiency.

A Primal Staggered Discontinuous Galerkin Method on Polytopal Meshes

Abstract

This paper introduces a novel staggered discontinuous Galerkin (SDG) method tailored for solving elliptic equations on polytopal meshes. Our approach utilizes a primal-dual grid framework to ensure local conservation of fluxes, significantly improving stability and accuracy. The method is hybridizable and reduces the degrees of freedom compared to existing approaches. It also bridges connections to other numerical methods on polytopal meshes. Numerical experiments validate the method's optimal convergence rates and computational efficiency.

Paper Structure

This paper contains 19 sections, 21 theorems, 109 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Staggered Discontinuous Galerkin Methods
  3. Primal and dual grids
  4. Finite element spaces
  5. Staggered discontinuous Galerkin method
  6. Inf-sup condition
  7. Hybridization
  8. Spaces and weak differential operators
  9. Hybridized variational form
  10. Error analysis
  11. Connections to Other Methods
  12. Nonconforming linear element method on simplices
  13. Stabilization free weak Galerkin methods
  14. Stabilization free non-conforming virtual element methods
  15. Numerical Examples
  16. ...and 4 more sections

Key Result

Lemma 2.1

Define the following two functionals where $Q_b$ is the $L^2$-projection operator onto space $\mathbb P_{k}(F)$, on each edge $F\in \mathcal{F}_h^{K}$. They are norms on space $U_h$ and $\Sigma_h$ respectively.

Figures (10)

  • Figure 1: The polygonal and triangle partitions.
  • Figure 1: Refine an element $K$ into simplices.
  • Figure 1: Partitions of $\Omega$ for $h=0.25$.
  • Figure 2: Different continuity of $\sigma_h$ and $u_h$.
  • Figure 2: Connection of $P_0$-$P_1$ SDG and CR non-conforming linear finite elements.
  • ...and 5 more figures

Theorems & Definitions (38)

  • Lemma 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Proof 2
  • Remark 2.7
  • Lemma 2.8
  • ...and 28 more