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Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations

Abhinav Jha, Volker John, Petr Knobloch

TL;DR

This work analyzes three algebraically stabilized finite element schemes—AFC with Kuzmin limiter, AFC with BJK limiter, and MUAS—for steady-state convection-diffusion-reaction equations on adaptively refined grids, including grids with hanging nodes. It develops a crucial algorithmic step to transform the non-conforming system to conforming test/ansatz spaces for reliable limiter computation and DMP satisfaction. Numerical experiments show that the BJK limiter and MUAS reliably satisfy the global discrete maximum principle across grid types, while the Kuzmin limiter may fail on some conforming closures but remains robust on hanging-node grids. Overall, MUAS emerges as the most robust and efficient approach among the three, with hanging-node transformations enabling stable application of algebraic stabilization on non-conforming meshes.

Abstract

Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter, the AFC scheme with BJK limiter, and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method. Both, conforming closure of the refined grids and grids with hanging vertices are considered. A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.

Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations

TL;DR

This work analyzes three algebraically stabilized finite element schemes—AFC with Kuzmin limiter, AFC with BJK limiter, and MUAS—for steady-state convection-diffusion-reaction equations on adaptively refined grids, including grids with hanging nodes. It develops a crucial algorithmic step to transform the non-conforming system to conforming test/ansatz spaces for reliable limiter computation and DMP satisfaction. Numerical experiments show that the BJK limiter and MUAS reliably satisfy the global discrete maximum principle across grid types, while the Kuzmin limiter may fail on some conforming closures but remains robust on hanging-node grids. Overall, MUAS emerges as the most robust and efficient approach among the three, with hanging-node transformations enabling stable application of algebraic stabilization on non-conforming meshes.

Abstract

Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter, the AFC scheme with BJK limiter, and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method. Both, conforming closure of the refined grids and grids with hanging vertices are considered. A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.

Paper Structure

This paper contains 13 sections, 2 theorems, 27 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Triangulations and Finite Element Spaces
  3. Algebraically Stabilized Schemes
  4. AFC Scheme with Kuzmin Limiter
  5. AFC Scheme with BJK Limiter
  6. Monotone Upwind-type Algebraically Stabilized (MUAS) Meth-od
  7. Hanging Nodes and Algebraically Stabilized Schemes
  8. Numerical Studies
  9. Solution Becoming Locally Diffusion-Dominated under Adaptive Grid Refinement
  10. A Convection-Dominated Problem with Interior and Boundary Layers
  11. Hemker Problem
  12. Summary of the Numerical Studies
  13. Summary

Key Result

Lemma 4

(Car11) Let $\left\lbrace\mathcal{T}_0, \cdots, \mathcal{T}_j\right\rbrace$ be a grid hierarchy on $\Omega$. Let us denote $\mathcal{T}_h=\mathcal{T}_j$, i.e., the final refinement level. Then, for all $q\in\mathcal{H}_h$ there are coefficients $a_{qp}$ with $p\in\mathcal{N}_h\setminus\mathcal{H}_h$

Figures (13)

  • Figure 1: Patch considered in Examples \ref{['ex:ImplementationP1']} and \ref{['ex:ImplementationP1_0']}.
  • Figure 2: Example \ref{['ex:boundary_layer']}: Solution computed with AFC scheme and Kuzmin limiter, level 7 with uniform refinement.
  • Figure 3: Example \ref{['ex:boundary_layer']}: $L^2(\Omega)$ error (top) and $L^2(\Omega)$ error of the gradient (bottom); grids with conforming closure (left) and grids with hanging nodes (right).
  • Figure 4: Example \ref{['ex:boundary_layer']}: Number of iterations and rejections on grids with conforming closure (left) and on grids with hanging nodes (right).
  • Figure 5: Example \ref{['ex:hmm86']}: Solution to the interior and boundary layer example, computed with the BJK limiter, level 9.
  • ...and 8 more figures

Theorems & Definitions (11)

  • Definition 1: Refinement, Car11, Def. 3.3
  • Definition 2: Grid hierarchy, Car11, Def. 3.4
  • Definition 3: Hanging vertex, Car11, Def. 3.6
  • Lemma 4
  • Definition 5: Non-conforming nodal basis functions
  • Theorem 6
  • Example 7: System corresponding to non-conforming ansatz and conforming test functions
  • Example 8: System corresponding to conforming ansatz and test functions
  • Remark 9
  • Remark 10
  • ...and 1 more