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Robust hybrid finite element methods for reaction-dominated diffusion problems

Thomas Führer, Diego Paredes

TL;DR

This paper tackles reaction-dominated diffusion problems of the form $-\ abla\cdot(\varepsilon^2\nabla u) + u = f$ with small $\varepsilon$, proposing two robust hybrid finite element methods (PHFEM and DHFEM) that employ Lagrange multipliers and enriched spaces. Stability independent of $\varepsilon$ and $h$ is achieved by incorporating modified face bubble functions that decay exponentially inside elements, enabling a Fortin-operator-based analysis and a reduced number of global degrees of freedom. Robust a posteriori error estimators are derived for both methods and proven reliable and locally efficient, allowing reliable adaptive mesh refinement that captures boundary and interior layers. Numerical experiments confirm significantly reduced oscillations compared with standard continuous Galerkin methods and demonstrate effective adaptivity driven by the estimators. The work has potential to impact the numerical treatment of singularly perturbed and multiscale diffusion problems by providing uniform-stability HFEMs with efficient solvers and robust error control.

Abstract

For a reaction-dominated diffusion problem we study a primal and a dual hybrid finite element method where weak continuity conditions are enforced by Lagrange multipliers. Uniform robustness of the discrete methods is achieved by enriching the local discretization spaces with modified face bubble functions which decay exponentially in the interior of an element depending on the ratio of the singular perturbation parameter and the local mesh-size. A posteriori error estimators are derived using Fortin operators. They are robust with respect to the singular perturbation parameter. Numerical experiments are presented that show that oscillations, if present, are significantly smaller then those observed in common finite element methods.

Robust hybrid finite element methods for reaction-dominated diffusion problems

TL;DR

This paper tackles reaction-dominated diffusion problems of the form with small , proposing two robust hybrid finite element methods (PHFEM and DHFEM) that employ Lagrange multipliers and enriched spaces. Stability independent of and is achieved by incorporating modified face bubble functions that decay exponentially inside elements, enabling a Fortin-operator-based analysis and a reduced number of global degrees of freedom. Robust a posteriori error estimators are derived for both methods and proven reliable and locally efficient, allowing reliable adaptive mesh refinement that captures boundary and interior layers. Numerical experiments confirm significantly reduced oscillations compared with standard continuous Galerkin methods and demonstrate effective adaptivity driven by the estimators. The work has potential to impact the numerical treatment of singularly perturbed and multiscale diffusion problems by providing uniform-stability HFEMs with efficient solvers and robust error control.

Abstract

For a reaction-dominated diffusion problem we study a primal and a dual hybrid finite element method where weak continuity conditions are enforced by Lagrange multipliers. Uniform robustness of the discrete methods is achieved by enriching the local discretization spaces with modified face bubble functions which decay exponentially in the interior of an element depending on the ratio of the singular perturbation parameter and the local mesh-size. A posteriori error estimators are derived using Fortin operators. They are robust with respect to the singular perturbation parameter. Numerical experiments are presented that show that oscillations, if present, are significantly smaller then those observed in common finite element methods.
Paper Structure (24 sections, 22 theorems, 113 equations, 7 figures)

This paper contains 24 sections, 22 theorems, 113 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Novel contributions
  3. Overview
  4. Derivation and abstract analysis of the hybrid methods
  5. Sobolev spaces and traces
  6. Primal hybrid variational formulation
  7. PHFEM
  8. Dual hybrid variational formulation
  9. DHFEM
  10. Reducing degrees of freedom and parallel computations
  11. Formal comparison between the two methods
  12. Uniform stability of discrete methods
  13. Face bubble functions with exponential layer
  14. Conditions on $U_h$ and Fortin operator
  15. Conditions on $\Sigma_h$ and Fortin operator
  16. ...and 9 more sections

Key Result

Lemma 1

For all $\lambda\in \Lambda$ and all $w\in W$ we have that

Figures (7)

  • Figure 1: Visualization of the exponential layer functions for different values of $\varepsilon$.
  • Figure 2: Comparison of face bubble functions with exponential resp. polynomial layers defined in Section \ref{['sec:expbubble']} resp. \ref{['sec:expbubble:alt']} for different values of $\varepsilon$.
  • Figure 3: Comparison of the projected PHFEM solution component $\Pi_h^0u_h$ and the lowest-order continuous Galerkin solution $u_h^\mathrm{cG}$ on meshes with $\#\mathcal{T} = 64$ (upper row) and $\#\mathcal{T}=1024$ (lower row).
  • Figure 4: Postprocessed solution $\Pi_h^0u_h^\mathrm{dual}$ of the DHFEM on two meshes and $\varepsilon=10^{-4}$.
  • Figure 5: Comparison between $L_2$ errors of the PHFEM, DHFEM solution and continuous Galerkin method for the example from Section \ref{['sec:numeric:ex1']} with $\varepsilon=10^{-4}$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 21 more