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Optimal convergence rates of an adaptive finite element method for unbounded domains

Théophile Chaumont-Frelet, Gregor Gantner

TL;DR

This work develops and analyzes an adaptive finite element method for linear reaction-diffusion problems posed on unbounded domains. By truncating the domain to a bounded subdomain with an artificial boundary and employing a residual-based a posteriori estimator that accounts for truncation, the authors prove reliability and efficiency under standard mesh-regularity assumptions and then design an adaptive loop that simultaneously refines the mesh and pushes the truncation boundary toward infinity. They establish R-linear convergence and optimal rates with respect to a problem-dependent approximation class $\mathbb{A}_s$, leveraging axioms of adaptivity adapted to the unbounded setting. Numerical experiments on smooth and singular solutions confirm the theory, demonstrating optimal convergence under the proposed push-refine strategy and illustrating how the truncation boundary expands where needed to maintain accuracy.

Abstract

We consider linear reaction-diffusion equations posed on unbounded domains, and discretized by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, it is necessary to introduce a truncation boundary, whereby only a bounded computational subdomain is meshed, leading to an approximation of the solution by zero in the remainder of the domain. We propose a residual-based error estimator that accounts for both the standard discretization error as well as the effect of the truncation boundary. This estimator is shown to be reliable and efficient under appropriate assumptions on the triangulation. Based on this estimator, we devise an adaptive algorithm that automatically refines the mesh and pushes the truncation boundary towards infinity. We prove that this algorithm converges and even achieves optimal rates in terms of the number of degrees of freedom. We finally provide numerical examples illustrating our key theoretical findings.

Optimal convergence rates of an adaptive finite element method for unbounded domains

TL;DR

This work develops and analyzes an adaptive finite element method for linear reaction-diffusion problems posed on unbounded domains. By truncating the domain to a bounded subdomain with an artificial boundary and employing a residual-based a posteriori estimator that accounts for truncation, the authors prove reliability and efficiency under standard mesh-regularity assumptions and then design an adaptive loop that simultaneously refines the mesh and pushes the truncation boundary toward infinity. They establish R-linear convergence and optimal rates with respect to a problem-dependent approximation class , leveraging axioms of adaptivity adapted to the unbounded setting. Numerical experiments on smooth and singular solutions confirm the theory, demonstrating optimal convergence under the proposed push-refine strategy and illustrating how the truncation boundary expands where needed to maintain accuracy.

Abstract

We consider linear reaction-diffusion equations posed on unbounded domains, and discretized by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, it is necessary to introduce a truncation boundary, whereby only a bounded computational subdomain is meshed, leading to an approximation of the solution by zero in the remainder of the domain. We propose a residual-based error estimator that accounts for both the standard discretization error as well as the effect of the truncation boundary. This estimator is shown to be reliable and efficient under appropriate assumptions on the triangulation. Based on this estimator, we devise an adaptive algorithm that automatically refines the mesh and pushes the truncation boundary towards infinity. We prove that this algorithm converges and even achieves optimal rates in terms of the number of degrees of freedom. We finally provide numerical examples illustrating our key theoretical findings.

Paper Structure

This paper contains 13 sections, 12 theorems, 94 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A posteriori error estimation
  3. Refinement of infinite meshes
  4. Adaptive algorithm and optimal convergence
  5. Axioms of adaptivity for the estimator
  6. Proof of R-linear and optimal convergence
  7. Numerical experiments
  8. Settings
  9. Meshes
  10. Dörfler marking
  11. Iteration counts
  12. Smoothed fundamental solution
  13. Singular unknown solution

Key Result

Theorem 2.1

There exists a constant $C_{\text{\rmrel}}>0$ which depends only on the dimension $d$, shape regularity of $\widehat{\mathcal{T}}_H$, and $\min\{\kappa_T^- h_T\,:\,T\in\mathcal{T}_H, T\cap \Gamma_H \neq \emptyset\}$ such that In particular, coercivity of $a(\cdot,\cdot)$ yields reliability

Figures (9)

  • Figure 1: Convergence history for the smoothed fundamental solution and $p=1$.
  • Figure 2: Convergence history for the smoothed fundamental solution and $p=2$.
  • Figure 3: Convergence history for the smoothed fundamental solution and $p=3$.
  • Figure 4: Effectivity indices for the smoothed fundamental solution and $p=1$.
  • Figure 5: Effectivity indices for the smoothed fundamental solution and $p=2$.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Theorem 2.1: reliability
  • Proof 1
  • Remark 2.2: reliability on full triangulation
  • Theorem 2.3: efficiency
  • Proof 2
  • Remark 2.4: reliability on full triangulation
  • Proposition 3.1: mesh-closure estimate stevenson08
  • Proposition 3.2: overlay estimate
  • Proof 3
  • Remark 3.3: alternative proof of closure estimate
  • ...and 21 more