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Asymptotic meshes from $r$-variational adaptation methods for static problems in one dimension

Darith Hun, Nicolas Moës, Heiner Olbermann

TL;DR

The paper develops a rigorous framework for mesh optimization in one-dimensional variational problems by embedding the mesh as a design variable in the energy functional. It proves that the renormalized discrete energies $\mathcal{F}_n$ $\Gamma$-converge to a limit functional $\mathcal{F}^*$, yielding an asymptotic description of optimal meshes through a balance between the second variation of the energy and a mesh-density term. The resulting limit problem identifies the asymptotically optimal mesh via a density proportional to $|f|^{2/3}$ in the Dirichlet-type example, and the authors provide numerical experiments showing close agreement between the asymptotic mesh and finite-element meshes obtained with large $n$. The framework offers a principled path to mesh optimization based on configurational equilibrium and suggests potential extensions to higher dimensions. Overall, the work connects $\Gamma$-convergence theory with practical adaptive meshing for nonlinear variational problems.

Abstract

We consider the minimization of integral functionals in one dimension and their approximation by $r$-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of $Γ$-convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the $Γ$-limit to the optimal finite meshes.

Asymptotic meshes from $r$-variational adaptation methods for static problems in one dimension

TL;DR

The paper develops a rigorous framework for mesh optimization in one-dimensional variational problems by embedding the mesh as a design variable in the energy functional. It proves that the renormalized discrete energies -converge to a limit functional , yielding an asymptotic description of optimal meshes through a balance between the second variation of the energy and a mesh-density term. The resulting limit problem identifies the asymptotically optimal mesh via a density proportional to in the Dirichlet-type example, and the authors provide numerical experiments showing close agreement between the asymptotic mesh and finite-element meshes obtained with large . The framework offers a principled path to mesh optimization based on configurational equilibrium and suggests potential extensions to higher dimensions. Overall, the work connects -convergence theory with practical adaptive meshing for nonlinear variational problems.

Abstract

We consider the minimization of integral functionals in one dimension and their approximation by -adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of -convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the -limit to the optimal finite meshes.

Paper Structure

This paper contains 13 sections, 4 theorems, 110 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline of the article
  4. Notation
  5. Setting and statement of the main theorem
  6. Tools, notation and preparatory lemmata
  7. Auxiliary notation and lemmata
  8. Proof of compactness and lower bound
  9. Proof of the upper bound
  10. Numerical experiments
  11. Application of AMF
  12. Comparison of the AMF algorithm with gradient descent
  13. An auxiliary lemma

Key Result

Theorem 2.2

Let $\mathcal{L}:I\times \mathbb{R}^N\times\mathbb{R}^N\to\mathbb{R}$ satisfy condition (A1).

Figures (5)

  • Figure 1: Optimal position of nodes for $f(x)=x^2$ and $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp( - \frac{(x-\mu)^2}{2\sigma^2})$, (with $\mu=0.5$ and $\sigma =0.05$). The dashed graph is the optimal piecewise affine function. For comparison, we display the exact solution $u_*$ (in color).
  • Figure 2: Relative error of the solution $u$ and $u'$ using an equal distributed mesh (black) with AMF (red), and descent gradient meshing method (blue).
  • Figure 3: Mesh distribution error between GD and AMF method for different f (left) quadratic and (right) gaussian ($\mu=0.5,\sigma=0.03$).
  • Figure 4: $L^1$-error between position of nodes obtained by GD and AMF method for different rational functions.
  • Figure 5: $L^1$-error between position of nodes obtained by GD and AMF method for Gaussian functions with different variances.

Theorems & Definitions (11)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:Fjgamma']} (o) and (i)
  • proof : Proof of Theorem \ref{['thm:Fjgamma']} (ii)
  • Lemma A.1
  • ...and 1 more