# Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

## TL;DR

This paper introduces two innovative methods using parabolic and elliptic cell problems to compute homogenized coefficients in multiscale elliptic PDEs, achieving exponential decay of resonance error and improving accuracy over traditional techniques.

## Contribution

The paper proposes novel approaches based on parabolic and elliptic cell problems that significantly reduce resonance error in homogenization computations.

## Key findings

- Resonance error decays exponentially with the new methods.
- New approaches outperform standard techniques in accuracy.
- Methods are applicable to multiscale elliptic PDEs.

## Abstract

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error.