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Adaptive Finite Elements with Algebraic Stabilization for Convection-Dominated Transport

Naveed Ahmed, Abhinav Jha

Abstract

We present a numerical investigation of residual-based a posteriori error estimation for finite element discretizations of convection--diffusion equations stabilized by algebraic flux correction and related algebraic stabilization techniques. In particular, we consider AFC schemes employing the BJK and Monolithic Convex (MC) limiters and algebraically stabilized methods including MUAS, SMUAS, and the BBK approach. The performance of the estimators and limiters are studied on adaptively refined meshes for several two-dimensional test problems, including boundary layers, interior layers, and a nonlinear convection problem with solution-dependent transport. The experiments assess accuracy, preservation of the discrete maximum principle, adaptive mesh behaviour, and computational efficiency. The results show that the interaction between stabilization and a posteriori error estimation depends strongly on mesh alignment and on the character of the convection field. In particular, for problems with moving or curved layers, the behaviour of the limiters differs significantly: strongly upwind-biased limiters provide the most accurate solutions, while smoother algebraic stabilizations lead to more efficient nonlinear iterations. The study also indicates that residual-based estimators remain reliable for both linear and nonlinear problems but may react to changes in limiter activation during adaptive refinement. Overall, the numerical results clarify the practical behaviour of several widely used stabilization techniques within an adaptive framework and highlight aspects that are not yet fully explained by the current theory, particularly for nonlinear transport problems.

Adaptive Finite Elements with Algebraic Stabilization for Convection-Dominated Transport

Abstract

We present a numerical investigation of residual-based a posteriori error estimation for finite element discretizations of convection--diffusion equations stabilized by algebraic flux correction and related algebraic stabilization techniques. In particular, we consider AFC schemes employing the BJK and Monolithic Convex (MC) limiters and algebraically stabilized methods including MUAS, SMUAS, and the BBK approach. The performance of the estimators and limiters are studied on adaptively refined meshes for several two-dimensional test problems, including boundary layers, interior layers, and a nonlinear convection problem with solution-dependent transport. The experiments assess accuracy, preservation of the discrete maximum principle, adaptive mesh behaviour, and computational efficiency. The results show that the interaction between stabilization and a posteriori error estimation depends strongly on mesh alignment and on the character of the convection field. In particular, for problems with moving or curved layers, the behaviour of the limiters differs significantly: strongly upwind-biased limiters provide the most accurate solutions, while smoother algebraic stabilizations lead to more efficient nonlinear iterations. The study also indicates that residual-based estimators remain reliable for both linear and nonlinear problems but may react to changes in limiter activation during adaptive refinement. Overall, the numerical results clarify the practical behaviour of several widely used stabilization techniques within an adaptive framework and highlight aspects that are not yet fully explained by the current theory, particularly for nonlinear transport problems.
Paper Structure (15 sections, 1 theorem, 41 equations, 28 figures, 3 tables)

This paper contains 15 sections, 1 theorem, 41 equations, 28 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Algebraic Stabilization Schemes
  3. BJK Limiter
  4. Monolithic Convex Limiter
  5. MUAS Method
  6. SMUAS Method
  7. BBK Method
  8. A Posteriori Error Estimators
  9. Numerical Studies
  10. Solution with Regular Boundary Layer
  11. Solution with Corner Singularity on L-shaped Domain
  12. Solution with Different Regimes
  13. Hemker Problem
  14. Solution with Inner Layer, Non-Linear Convection and Linear Reaction
  15. Summary

Key Result

Theorem 4

Let $u_h \in W_h$ be a solution of Eq. eq:as_scheme. Then the global a posteriori error estimate in the energy norm is given by where and the element and face residuals are defined as follows: where $\left[\![ \cdot ]\!\right]$ denotes the jump across the face $F$, and $\boldsymbol n_F$ is the outward-pointing unit normal to $F$.

Figures (28)

  • Figure 1: Coarsest grids for the numerical studies, Grid 1,2,3,4 (left to right)
  • Figure 2: Example \ref{['ex:boundary_layer']}: Numerical solution with the MUAS method on Grid 1 with $16641$ degrees of freedom.
  • Figure 3: Example \ref{['ex:boundary_layer']}: Effectivity index for Grids 1–3 (left to right).
  • Figure 4: Example \ref{['ex:boundary_layer']}: Error in the ${\mathrm L}^2(\Omega)$ norm for Grids 1–3 (left to right).
  • Figure 5: Example \ref{['ex:boundary_layer']}: Error of the gradient in the ${\mathrm L}^2(\Omega)$ norm for Grids 1–3 (left to right).
  • ...and 23 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6