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A Residual Based A Posteriori Error Estimators for AFC Schemes for Convection-Diffusion Equations

Abhinav Jha

TL;DR

This paper develops two a posteriori error estimators for AFC schemes solving stationary Convection-Diffusion-Reaction problems: a residual-based estimator giving a global energy-norm upper bound and an AFC-SUPG-based estimator that leverages the SUPG solution for diffusion-robust bounds. The authors prove a global bound in the energy norm with computable components and derive a formal local lower bound, highlighting limitations in diffusion-robustness and the influence of limiters. Numerical experiments in two dimensions demonstrate how the estimators drive adaptive refinement, compare Kuzmin and BJK limiters, and reveal that the AFC-SUPG-energy approach generally yields better effectivity while AFC-energy can offer superior refinement behavior in convection-dominated regimes. The work provides practical guidance for selecting limiters and adaptive strategies, while outlining directions for robustness improvements and extensions to hanging-node grids and 3D problems.

Abstract

In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter, which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the Streamline Upwind Petrov Galerkin (SUPG) estimator proposed in \cite{JN13}. Numerical examples study the effectivity index and the adaptive grid refinement for two limiters in two dimensions.

A Residual Based A Posteriori Error Estimators for AFC Schemes for Convection-Diffusion Equations

TL;DR

This paper develops two a posteriori error estimators for AFC schemes solving stationary Convection-Diffusion-Reaction problems: a residual-based estimator giving a global energy-norm upper bound and an AFC-SUPG-based estimator that leverages the SUPG solution for diffusion-robust bounds. The authors prove a global bound in the energy norm with computable components and derive a formal local lower bound, highlighting limitations in diffusion-robustness and the influence of limiters. Numerical experiments in two dimensions demonstrate how the estimators drive adaptive refinement, compare Kuzmin and BJK limiters, and reveal that the AFC-SUPG-energy approach generally yields better effectivity while AFC-energy can offer superior refinement behavior in convection-dominated regimes. The work provides practical guidance for selecting limiters and adaptive strategies, while outlining directions for robustness improvements and extensions to hanging-node grids and 3D problems.

Abstract

In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter, which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the Streamline Upwind Petrov Galerkin (SUPG) estimator proposed in \cite{JN13}. Numerical examples study the effectivity index and the adaptive grid refinement for two limiters in two dimensions.

Paper Structure

This paper contains 16 sections, 7 theorems, 86 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Limiters
  4. Kuzmin Limiter
  5. BJK Limiter
  6. Auxiliary Results
  7. A Posteriori Error Estimator
  8. Residual-Based Estimator
  9. Global Upper Bound
  10. Formal Local Lower Bound
  11. AFC-SUPG Estimator
  12. Numerical Studies
  13. A Known 2d Solution with a Boundary Layer
  14. Example with Interior and Boundary Layer
  15. Summary
  16. ...and 1 more sections

Key Result

Lemma 2

(Inverse estimate) (BS08) Let $C_{\mathrm{shrg}} h\leq h_K\leq h$, where $0<h\leq 1$, and $\mathcal{P}_h$ be a polynomial subspace of $H^m(K)$. Then for $0\leq l\leq m$ there exists a constant $C_{\mathrm{inv}}$ such that for all $v\in \mathcal{P}_h$ and $K\in \mathcal{T}_h$, we have

Figures (10)

  • Figure 1: 2d Boundary layer example. Solution (computed with the BJK limiter, level 7).
  • Figure 2: Example \ref{['ex:known_2d_boundary']}: Effectivity index in the energy norm with AFC-energy technique defined in Sec. \ref{['sec:upper_bound']} (left) and AFC-SUPG-energy technique defined in Sec. \ref{['sec:afc_supg_est']} (right).
  • Figure 3: Example \ref{['ex:known_2d_boundary']}: Comparison of $\eta_{\mathrm{SUPG}}$ and $\eta_{\mathrm{AFC-SUPG}}$ for AFC-SUPG-energy technique. Kuzmin limiter (left) and BJK limiter (right).
  • Figure 4: Example \ref{['ex:known_2d_boundary']}: Error in energy norm with AFC-energy technique defined in Sec. \ref{['sec:upper_bound']}. The line corresponding to $\eta$ (Kuzmin) is below $\eta_{d_h}$ (Kuzmin) in the left figure.
  • Figure 5: Example \ref{['ex:known_2d_boundary']}: Error in energy norm with AFC-SUPG-energy technique defined in Sec. \ref{['sec:afc_supg_est']}.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Remark 1: Consequences of the shape regularity assumption \ref{['eq:shape_regu_00']}
  • Lemma 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8: Estimate of the trace on an edge by the norm on the mesh cell
  • proof
  • ...and 11 more