Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equilibrated Averaging Residual Method: A General Approach to Conservative Flux Recovery

Cuiyu He

TL;DR

This work presents the Equilibrated Averaging Residual Method (EARM), a unified, residual-based flux recovery framework that guarantees locally conservative fluxes for conforming, nonconforming, and discontinuous Galerkin FEMs, enabling reliable and robust a posteriori error estimation. EARM constructs a weighted averaged flux in Raviart–Thomas spaces and adds a correction either via an orthogonal-complement constraint (for NC and CG) or via Partial-PoU localization to produce explicit, local flux recoveries in 2D and simple local problems in 3D. The paper demonstrates that the recovered fluxes reproduce known equilibrated fluxes while also furnishing novel recoveries, with automatic global reliability and robust local efficiency independent of coefficient jumps. It also provides practical localization strategies, algorithmic outlines, and detailed analyses for odd/even NC orders and conforming methods, offering a flexible and efficient toolkit for flux recovery across FEM families.

Abstract

Many equilibrated flux recovery methods for finite element solutions rely on ad hoc or method-specific techniques, limiting their generalizability and efficiency. In this work, we introduce the Equilibrated Averaging Residual Method (EARM), a unified framework for flux recovery that not only reproduces state-of-the-art locally conservative fluxes but also enables the derivation of new equilibrated fluxes with improved properties. In this paper, EARM is applied to conforming, nonconforming, and discontinuous Galerkin methods, ensuring local conservation and robust a posteriori error estimation. Despite the unified nature of the variational problem, the framework retains the flexibility to fully leverage the inherent properties of finite element spaces. Moreover, EARM offers explicit and computationally efficient flux reconstructions for all methods in two dimensions. In three dimensions, only simple local problems need to be solved for the conforming finite element methods.

Equilibrated Averaging Residual Method: A General Approach to Conservative Flux Recovery

TL;DR

This work presents the Equilibrated Averaging Residual Method (EARM), a unified, residual-based flux recovery framework that guarantees locally conservative fluxes for conforming, nonconforming, and discontinuous Galerkin FEMs, enabling reliable and robust a posteriori error estimation. EARM constructs a weighted averaged flux in Raviart–Thomas spaces and adds a correction either via an orthogonal-complement constraint (for NC and CG) or via Partial-PoU localization to produce explicit, local flux recoveries in 2D and simple local problems in 3D. The paper demonstrates that the recovered fluxes reproduce known equilibrated fluxes while also furnishing novel recoveries, with automatic global reliability and robust local efficiency independent of coefficient jumps. It also provides practical localization strategies, algorithmic outlines, and detailed analyses for odd/even NC orders and conforming methods, offering a flexible and efficient toolkit for flux recovery across FEM families.

Abstract

Many equilibrated flux recovery methods for finite element solutions rely on ad hoc or method-specific techniques, limiting their generalizability and efficiency. In this work, we introduce the Equilibrated Averaging Residual Method (EARM), a unified framework for flux recovery that not only reproduces state-of-the-art locally conservative fluxes but also enables the derivation of new equilibrated fluxes with improved properties. In this paper, EARM is applied to conforming, nonconforming, and discontinuous Galerkin methods, ensuring local conservation and robust a posteriori error estimation. Despite the unified nature of the variational problem, the framework retains the flexibility to fully leverage the inherent properties of finite element spaces. Moreover, EARM offers explicit and computationally efficient flux reconstructions for all methods in two dimensions. In three dimensions, only simple local problems need to be solved for the conforming finite element methods.

Paper Structure

This paper contains 20 sections, 23 theorems, 181 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem
  3. Notations
  4. Applying the EARM for DG FEM
  5. Applying EARM for NC FEMs
  6. NC FEM of arbitrary odd order
  7. Case 1: $u_{h} \in NC({\@fontswitch{}{\mathcal{}} T}_{h}, k = 2n+1), \hat{\hbox{\boldmath$\sigma$}}_{h} \in RT_f({\@fontswitch{}{\mathcal{}} T}_h, k-1, k-1)$
  8. Case 2: $u_{h} \in NC({\@fontswitch{}{\mathcal{}} T}_{h}, k = 2n+1), n\ge 0, \hat{\hbox{\boldmath$\sigma$}}_{h} \in RT_f({\@fontswitch{}{\mathcal{}} T}_h, 0)$
  9. NC FEM of order $2$: ($u_{h} \in NC({\@fontswitch{}{\mathcal{}} T}_{h}, 2), \hat{\hbox{\boldmath$\sigma$}}_{h}^{nc} \in RT_f({\@fontswitch{}{\mathcal{}} T}_h, 0)$)
  10. Applying EARM for Conforming FEM
  11. Deriving the equilibrated residual method
  12. Restricting $\hbox{\boldmath$\sigma$}_{s}^{\Delta}$ to the orthogonal complement of divergence-free space
  13. Localization via Partial POU
  14. Automatic Global Reliability
  15. Robust Efficiency
  16. ...and 5 more sections

Key Result

Lemma 3.1

The recovered flux $\hat{\hbox{\boldmath$\sigma$}}_{h}^{dg}$ defined in DG-final- where $\hbox{\boldmath$\sigma$}_{s}^{\Delta }$ is defined in rt:-cg-rt::dg-cg belongs to $RT({\@fontswitch{}{\mathcal{}} T}_{h},s)$ for some $(0 \le s \le k)$. Futhermore, it is locally conservative, satisfying $\hat{\

Figures (1)

  • Figure :

Theorems & Definitions (56)

  • Lemma 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Proof 2
  • Lemma 4.2
  • Proof 3
  • Lemma 4.3
  • Proof 4
  • ...and 46 more