A high-order approach to elliptic multiscale problems with general unstructured coefficients

TL;DR

This paper introduces a high-order multiscale method for elliptic problems with unstructured coefficients, achieving high convergence rates without requiring extra regularity assumptions, supported by rigorous error analysis and numerical validation.

Contribution

It presents a novel high-order multiscale approach that handles unstructured coefficients without additional regularity, with proven error estimates and numerical performance analysis.

Findings

Achieves high-order convergence rates in multiscale elliptic problems.

Does not rely on regularity assumptions on domain or coefficients.

Provides rigorous a priori error estimates and numerical validation.

Abstract

We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The method allows for suitable localization and does not rely on additional regularity assumptions on the domain, the diffusion coefficient, or the exact (weak) solution as typically required for high-order approaches. Rigorous a priori error estimates are presented with respect to the involved discretization parameters, and the interplay between these parameters as well as the performance of the method are studied numerically.