  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.
