A Residual-Free Bubble Formulation for nonlinear elliptic problems with oscillatory coefficients
This work provides the first theoretical analysis of the Residual Free Bubble method for nonlinear problems, benefiting researchers in numerical analysis and multiscale modeling.
The paper proposes and analyzes a nonlinear version of the Residual Free Bubble finite element method for multiscale nonlinear elliptic PDEs, establishing existence, uniqueness, and best approximation results for the first time.
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.