# A Residual-Free Bubble Formulation for nonlinear elliptic problems with oscillatory coefficients

## TL;DR

This paper introduces a novel nonlinear Residual Free Bubble finite element method for multiscale nonlinear elliptic PDEs, providing analysis on existence, uniqueness, and approximation, marking the first such study.

## Contribution

It develops and analyzes the first nonlinear Residual Free Bubble method for multiscale elliptic problems, including solution existence, uniqueness, and approximation properties.

## Key findings

- Established existence and uniqueness of solutions.
- Derived a best approximation result.
- Explored different linearizations of the method.

## Abstract

We present an investigation of the Residual Free Bubble finite element method for a class of multiscale nonlinear elliptic partial differential equations. After proposing a nonlinear version for the method, we address fundamental questions as existence and uniqueness of solutions. We also obtain a best approximation result, and investigate possible linearizations that generate different versions for the method. As far as we are aware, this is the first time that an analysis for the nonlinear Residual Free Bubble method is considered.