$hp$-FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains

TL;DR

The paper develops $hp$-FEM on geometric boundary layer meshes to solve linear, singularly perturbed reaction-diffusion problems in curvilinear polygons with multiple length scales, achieving robust exponential convergence in the energy norm $\|\cdot\|_{\varepsilon,\Omega}$ under a scale-resolution condition. It builds a patch-based framework using a macro-triangulation and a catalog of reference refinement patterns, proving exponential error bounds for corner, boundary layer, and corner-layer components, and then aggregates them into a global estimate with explicit dimension growth. Numerical experiments on polygons confirm the theoretical rates and illustrate two convergence regimes, with Netgen enabling automatic mesh generation in practice. The results extend the applicability of $hp$-FEM to multi-scale, nonsmooth geometries, providing a principled approach for accurate, efficient simulations of reaction-diffusion systems across multiple length scales.

Abstract

In bounded, polygonal domains $Ω\subset \mathbb{R}^2$ with Lipschitz boundary $\partialΩ$ consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze $hp$-FEM discretizations of linear, second order, singularly perturbed reaction diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these $hp$-FEM afford exponential convergence in the natural "energy" norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the robust exponential convergence of the proposed $hp$-FEM.

Figures (10)

• Figure 1: \newlabelfig:curvilinear-polygon0 Example of a curvilinear polygon $\Omega$.
• Figure 1: \newlabelfig:patches0 Catalog ${\mathfrak P}$ of mesh patches of geometric boundary layer meshes $\tilde{{\mathcal{T}}}_{geo}$. Top row: boundary layer patch $\widetilde{{\mathcal{T}}}^{{\sf BL},L}_{geo,\sigma}$ with $L$ layers of geometric refinement towards $\{{\widetilde{y}}=0\}$; corner patch $\widetilde{{\mathcal{T}}}^{{\sf C},n}_{geo,\sigma}$ with $n$ layers of geometric refinement towards $(0,0)$; trivial patch. Bottom row: tensor patch $\widetilde{{\mathcal{T}}}^{{\sf T},L,n}_{geo,\sigma}$ with $n$ layers of isotropic geometric refinement towards $(0,0)$ and $L$ layers of anisotropic geometric refinement towards $\{{\widetilde{x}} = 0\}$ and $\{{\widetilde{y}}=0\}$; mixed patch $\widetilde{{\mathcal{T}}}^{{\sf M},L,n}_{geo,\sigma}$ with $L$ layers of refinement towards $\{y=0\}$ and $n$ layers of refinement towards $(0,0)$. Geometric entities shown in boldface indicate parts of $\partial \widetilde{S}$ that are mapped to $\partial\Omega$. Patch meshes are transported into the curvilinear polygon $\Omega$ shown in Fig. \ref{['fig:curvilinear-polygon']} via analytic patch maps $F_{K^{\mathcal{M}}}$.
• Figure 1: \newlabelfig:bdyfitted-coord0 Left: boundary fitted coordinates $\psi_j: (\rho_j, \theta_j) \mapsto (x,y)$. Right: typical situation at a reentrant corner: boundary fitted coordinates $(\rho_j,\theta_j)$ and $(\rho_{j+1},\theta_{j+1})$ are valid in the regions $\Omega_j$, $\Omega_{j+1}$, respectively. $\widetilde{\Gamma}_j$ and $\widetilde{\Gamma}_{j+1}$ are analytic continuations of $\Gamma_j$, $\Gamma_{j+1}$. The analytic arc $\Gamma_j^\prime$ is such that the angles $\angle(\Gamma_j^\prime,\Gamma_j)$ and $\angle(\Gamma_{j+1},\Gamma_j^\prime)$ are both less than $\pi$.
• Figure 1: Right panel: Convergence in the energy norm (\ref{['eq:Erreh']}) for the square domain for different values of $\varepsilon$ and $q = L = n = p$. Left panel: a Netgen-generated mesh used for the computations.
• Figure 2: \newlabelfig:half-figures0 From left to right: half-patches $\widetilde{{\mathcal{T}}}^{{\sf M},\text{\rm half},L,n}_{geo,\sigma}$, $\widetilde{{\mathcal{T}}}^{{\sf C},\text{\rm half},n}_{geo,\sigma}$, and $\widetilde{{\mathcal{T}}}^{{\sf C},\text{\rm half},\text{\rm flip},n}_{geo,\sigma}$. They are given by the elements of $\widetilde{{\mathcal{T}}}^{{\sf C},n}_{geo,\sigma}$ and $\widetilde{{\mathcal{T}}}^{{\sf M},L,n}_{geo,\sigma}$ below the diagonal $\{{\widetilde{y}} = {\widetilde{x}}\}$ and the mirror image of $\widetilde{{\mathcal{T}}}^{{\sf C},\text{\rm half},n}_{geo,\sigma}$ at the diagonal $\{{\widetilde{y}} = {\widetilde{x}}\}$.

Theorems & Definitions (33)

• Definition 2.1: catalog ${\mathfrak P}$ of refinement patterns
• Remark 2.2
• Definition 2.3: geometric boundary layer mesh $\mathcal{T}^{L,n}_{geo,\sigma}$ in $\Omega$
• Remark 2.4
• Remark 2.5
• Example 2.6
• Definition 2.7: half-patches, cf. Fig. \ref{['fig:half-figures']}
• Lemma 2.8: properties of mesh patches
• Proof 1
• Lemma 3.1: element-by-element approximation on triangles