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$d$-dimensional extension of a penalization method for Neumann or Robin boundary conditions: a boundary layer approach and numerical experiments

Bouchra Bensiali, Jacques Liandrat

TL;DR

This work extends a fictitious-domain penalization technique to the $d$-dimensional setting for Neumann and Robin boundary conditions. It leverages J. Droniou's noncoercive elliptic framework and develops a boundary-layer approach to prove penalization convergence, including a formal dual problem and construction of supersolutions. The authors establish existence/uniqueness of the penalized problem, prove $O(\varepsilon)$ convergence inside the fluid and reveal a localized boundary layer in the obstacle, supported by 2D numerical experiments using upwind schemes. The results provide a rigorous mechanism for penalization-based boundary-condition imposition in multidimensions and guide numerical treatment of the resulting advection-dominated regimes, with potential extensions to Stokes/Navier–Stokes and moving-domain problems.

Abstract

This paper studies the $d$-dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions. We apply Droniou's approach for non-coercive linear elliptic problems to obtain the existence and uniqueness of the solution of the penalized problem, and we derive a boundary layer approach to establish the convergence of the penalization method. The developed boundary layer approach is adapted from the one used for Dirichlet boundary conditions, but in contrast to the latter where coercivity enables a straightforward estimate of the remainders, we reduce the convergence of the penalization method to the existence of suitable supersolutions of a dual problem. These supersolutions are then constructed as approximate solutions of the dual problem using an additional formal boundary layer approach. The proposed approach results in an advection-dominated problem, requiring the use of appropriate numerical methods suitable for singular perturbation problems. Numerical experiments, using upwind finite differences, validate both the convergence rate and the boundary layer thickness, illuminating the theoretical results.

$d$-dimensional extension of a penalization method for Neumann or Robin boundary conditions: a boundary layer approach and numerical experiments

TL;DR

This work extends a fictitious-domain penalization technique to the -dimensional setting for Neumann and Robin boundary conditions. It leverages J. Droniou's noncoercive elliptic framework and develops a boundary-layer approach to prove penalization convergence, including a formal dual problem and construction of supersolutions. The authors establish existence/uniqueness of the penalized problem, prove convergence inside the fluid and reveal a localized boundary layer in the obstacle, supported by 2D numerical experiments using upwind schemes. The results provide a rigorous mechanism for penalization-based boundary-condition imposition in multidimensions and guide numerical treatment of the resulting advection-dominated regimes, with potential extensions to Stokes/Navier–Stokes and moving-domain problems.

Abstract

This paper studies the -dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions. We apply Droniou's approach for non-coercive linear elliptic problems to obtain the existence and uniqueness of the solution of the penalized problem, and we derive a boundary layer approach to establish the convergence of the penalization method. The developed boundary layer approach is adapted from the one used for Dirichlet boundary conditions, but in contrast to the latter where coercivity enables a straightforward estimate of the remainders, we reduce the convergence of the penalization method to the existence of suitable supersolutions of a dual problem. These supersolutions are then constructed as approximate solutions of the dual problem using an additional formal boundary layer approach. The proposed approach results in an advection-dominated problem, requiring the use of appropriate numerical methods suitable for singular perturbation problems. Numerical experiments, using upwind finite differences, validate both the convergence rate and the boundary layer thickness, illuminating the theoretical results.
Paper Structure (68 sections, 19 theorems, 252 equations, 12 figures, 7 tables)

This paper contains 68 sections, 19 theorems, 252 equations, 12 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem analysis and useful tools
  3. Reaction-diffusion equation with Neumann or Robin boundary conditions; some analysis
  4. Existence, uniqueness and stability
  5. Elliptic regularity
  6. Maximum principle
  7. Mathematical tools
  8. The space $H_{\mathrm{div}}$
  9. Sobolev embeddings
  10. Boundary regularity and distance to the boundary
  11. Solution to advection-reaction equation via the method of characteristics
  12. Existence and uniqueness
  13. Weak formulation
  14. Lack of coercivity
  15. Existence and uniqueness using Droniou's approach
  16. ...and 53 more sections

Key Result

Theorem 1

(see for instance E10) If $\mathcal{U}$ is a connected, bounded, Lipschitz domain of ${\mathbb R}^d$, if $f\in L^2(\mathcal{U})$, $\tilde{g}\in H^{1/2}(\partial\mathcal{U})$ and $\alpha\ge 0$, then, there exists a unique solution $u\in H^1(\mathcal{U})$ of eq:reactiondiffusion, and it satisfies, for

Figures (12)

  • Figure 1: Domains and notations
  • Figure 2: Comparison between the numerical solution of the penalized problem \ref{['eq:reactiondiffusionpenalized']} and the numerical solution of the initial problem \ref{['eq:reactiondiffusion']} for $\alpha = 0$, $f(x,y)=\cos(x)\sin(y)$ and $\tilde{g}(x,y)=0$. We took the penalization parameter $\varepsilon=10^{-2}$ and the mesh size $h =0.04$ (fitted mesh) and $h=0.02$ (unfitted mesh).
  • Figure 3: Cartesian mesh of the fictitious domain $\Omega={]}0,1[^2$.
  • Figure 4: Condition number $\kappa(A_h)$ in the 2-norm versus $\varepsilon$ for $\alpha = 2$, $R=0.3$ and $N = 100$.
  • Figure 5: Comparison between the numerical solution of the penalized problem and the exact solution of the initial problem for $\alpha = 2$, $\varepsilon=10^{-10}$ and $N = 150$.
  • ...and 7 more figures

Theorems & Definitions (36)

  • Theorem 1: Well-posedness
  • Theorem 2: Elliptic regularity for Neumann or Robin boundary conditions
  • proof
  • Theorem 3: Elliptic regularity for the Laplace-Neumann problem
  • Theorem 4: Maximum principle for Neumann or Robin boundary conditions
  • proof
  • Corollary 1
  • proof
  • Definition 1
  • Theorem 5
  • ...and 26 more