A residual-free bubble formulation for nonlinear elliptic problems with oscillatory coefficients

ABSTRACT

We present an investigation of the Residual-Free Bubble (RFB) 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. All estimates are independent of eventual oscillations in the coefficients, and as far as we are aware, this is the first time that an analysis for the nonlinear RFB method is considered.

KEYWORDS:

• Multiscale PDE
• finite element method
• nonlinear elliptic PDE
• numerical analysis
• residual-free bubbles

2010 MSC SUBJECT CLASSIFICATIONS:

• 65N12
• 65N30

Disclosure statement

No potential conflict of interest was reported by the authors.