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On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation

Masaki Imagawa, Daisuke Kawagoe

TL;DR

The paper analyzes the strong L^2 convergence of an elliptic regularization of a stationary advection equation with Neumann-type boundary conditions, proving that solutions u_ε converge to the true solution u as ε → 0 and quantifying convergence rates that depend on the regularity of u and the boundary geometry relative to the advection field. The authors establish well-posedness in appropriate Sobolev-like spaces H_{β,±}(Ω), prove strong L^2 convergence (and trace convergence on Γ_+) under a density assumption, and derive a hierarchy of rates: r = 1/2 for H^1-class regularity, r = 3/4 for H^2 regularity with positive-measure Γ_0, and up to r = 1 in degenerate or favorable boundary configurations. They also provide α-dependent rates for general boundary degeneracy and validate the theory with comprehensive numerical experiments in 2D that corroborate the predicted rates. The results offer a quantitative understanding of boundary layer effects and trace behavior in elliptic regularizations of advection-dominated problems, with implications for error control in finite element simulations.

Abstract

We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $εΔ$, where $ε$ is a positive parameter and $Δ$ is the Laplacian. In this article, we show the $L^2$ strong convergence of solutions as the parameter $ε$ tends to $0$, and discuss its convergence rates assuming $H^1$ or $H^2$ regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates.

On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation

TL;DR

The paper analyzes the strong L^2 convergence of an elliptic regularization of a stationary advection equation with Neumann-type boundary conditions, proving that solutions u_ε converge to the true solution u as ε → 0 and quantifying convergence rates that depend on the regularity of u and the boundary geometry relative to the advection field. The authors establish well-posedness in appropriate Sobolev-like spaces H_{β,±}(Ω), prove strong L^2 convergence (and trace convergence on Γ_+) under a density assumption, and derive a hierarchy of rates: r = 1/2 for H^1-class regularity, r = 3/4 for H^2 regularity with positive-measure Γ_0, and up to r = 1 in degenerate or favorable boundary configurations. They also provide α-dependent rates for general boundary degeneracy and validate the theory with comprehensive numerical experiments in 2D that corroborate the predicted rates. The results offer a quantitative understanding of boundary layer effects and trace behavior in elliptic regularizations of advection-dominated problems, with implications for error control in finite element simulations.

Abstract

We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by , where is a positive parameter and is the Laplacian. In this article, we show the strong convergence of solutions as the parameter tends to , and discuss its convergence rates assuming or regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates.
Paper Structure (10 sections, 14 theorems, 107 equations, 9 figures)

This paper contains 10 sections, 14 theorems, 107 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^2$ strong convergence to $H_{\beta, -}$ solutions
  4. Convergence rate to $H^1$ solutions
  5. Convergence rates to $H^2$ solutions
  6. Numerical experiments
  7. Example 1: corresponding to Theorem \ref{['thm:1/2convergence']}
  8. Example 2: corresponding to Theorem $\ref{['thm:3/4convergence']}$
  9. Example 3: corresponding to Theorem $\ref{['thm:1convergence']}$
  10. Example 4: corresponding to Theorem $\ref{['thm:general_convergence1']}$

Key Result

Proposition 2.1

Suppose that the density density holds. Then, the boundary value problem BVP has the unique solution $u \in H_{\beta, -}(\Omega)$. Moreover, there exists a positive constant $C$ independent of $f$ such that

Figures (9)

  • Figure 6.1: Norms of $u - u_\epsilon$ vs $\epsilon$ in log-log scale. For the lease square fitting data corresponding to $\epsilon=1.6^{-k}$, $8 \leq k \leq 14$ are used.
  • Figure 6.6: $\| u-u_\epsilon \|_{L^2(\Omega)}$
  • Figure 6.7: $\| u-u_\epsilon \|_{L^2(\Gamma_{+};\beta \cdot n)}$
  • Figure 6.8: $\| u-u_\epsilon \|_{L^2(\Gamma_0)}$
  • Figure 6.10: Norms of $u - u_\epsilon$ vs $\epsilon$ in log-log scale. For the lease square fitting data corresponding to $\epsilon=1.6^{-k}$, $8 \leq k \leq 14$ are used.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Corollary 4.2
  • ...and 12 more