A multiscale finite element method for oscillating Neumann problem on rough domain

TL;DR

This paper introduces a multiscale finite element method for solving Laplace equations with oscillating Neumann boundary conditions on rough domains, effectively capturing microscale boundary features.

Contribution

The paper presents a novel boundary condition that integrates geometric and physical information of rough boundaries within a multiscale finite element framework.

Findings

Optimal convergence rate in energy norm for periodic roughness

Effective handling of nonperiodic roughness

Numerical validation of the method's accuracy

Abstract

We develop a new multiscale finite element method for Laplace equation with oscillating Neumann boundary conditions on rough boundaries. The key point is the introduction of a new boundary condition that incorporates both the microscopically geometrical and physical information of the rough boundary. We prove the method has optimal convergence rate in the energy norm with a weak resonance term for periodic roughness. Numerical results are reported for both periodic and nonperiodic roughness.