On some new results on anisotropic singular perturbations of second order elliptic operators

TL;DR

This paper investigates the asymptotic behavior of singularly perturbed elliptic operators, demonstrating the asymptotic preserving property of Galerkin methods and analyzing convergence rates and semigroup limits.

Contribution

It introduces new results on the asymptotic analysis of anisotropic singular perturbations and validates a Galerkin approach for semilinear elliptic problems.

Findings

Galerkin method is asymptotic preserving for the problem

Convergence rate estimates are provided for the linear case

Asymptotic behavior of the generated semigroup is characterized

Abstract

In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behavior of the semigroup generated.