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

TL;DR

This paper studies the asymptotic behavior of optimal meshes generated by $r$-adaptive finite element methods for one-dimensional static problems, demonstrating convergence of the meshes as the number of nodes increases.

Contribution

It introduces a framework using $ ext{Gamma}$-convergence to analyze the limit of optimal $r$-adaptive meshes in one dimension, linking finite element approximation to asymptotic mesh configurations.

Findings

Optimal meshes have a well-defined limit as nodes increase.

Numerical examples show the asymptotic mesh closely approximates finite optimal meshes.

The $ ext{Gamma}$-limit provides a theoretical foundation for mesh asymptotics.

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 $\Gamma$-convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the $\Gamma$-limit to the optimal finite meshes.