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

2025/10/30 by Hun, Darith, Moës, Nicolas, Olbermann, Heiner
#49J45 #49Q10 #65N30 #65N50 #74S05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2510.26375

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.

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