Localized orthogonal decomposition techniques for boundary value problems

TL;DR

This paper introduces a Local Orthogonal Decomposition (LOD) method for elliptic PDEs with complex boundary conditions, achieving accurate solutions efficiently without fine-scale resolution of boundary oscillations or conductivity channels.

Contribution

The paper develops new boundary correctors for LOD that maintain convergence rates even with rapidly oscillating boundary conditions and demonstrates the method's robustness for complex diffusion structures.

Findings

Achieves optimal convergence rates with boundary oscillations

Reliable in the presence of thin conductivity channels

Provides accurate results without fine-scale resolution

Abstract

In this paper we propose a Local Orthogonal Decomposition method (LOD) for elliptic partial differential equations with inhomogeneous Dirichlet- and Neumann boundary conditions. For this purpose, we present new boundary correctors which preserve the common convergence rates of the LOD, even if the boundary condition has a rapidly oscillating fine scale structure. We prove a corresponding a-priori error estimate and present numerical experiments. We also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix. Accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels.