Error estimates for a Neumann problem in highly oscillating thin domains

TL;DR

This paper investigates the convergence and error estimates of solutions to a Neumann problem for the Laplace operator in highly oscillating thin domains, using a corrector approach and multiple-scale method.

Contribution

It provides new error estimates and convergence analysis for Laplace problems in thin, oscillating domains with all parameters scaled by a small epsilon.

Findings

Established strong convergence properties of solutions.

Derived second-order error estimates using multiple-scale expansion.

Analyzed the impact of oscillation parameters on solution behavior.

Abstract

In this work we analyze convergence of solutions for the Laplace operator with Neumann boundary conditions in a two-dimensional highly oscillating domain which degenerates into a segment (thin domains) of the real line. We consider the case where the height of the thin domain, amplitude and period of the oscillations are all of the same order, given by a small parameter $\epsilon$. We investigate strong convergence properties of the solutions using an appropriate corrector approach. We also give error estimates when we replace the original solutions for the second-order expansion through the Multiple-Scale Method.