Stabilization Techniques and Subgrid Scales Capturing

Abstract

Abstract We present an overview of stabilized finite element methods and of the standard Galerkin method enriched with residual-free bubble functions. The inadequacy of the standard Galerkin method using piecewise polynomials is discussed for different applications; the treatment using stabilized methods in their different versions is reviewed; and the connection to the standard Galerkin method with richer subspaces follows using the subgrid method or the residual-free-bubbles viewpoint. We close with a discussion on how to approximate the exact problem suggested by residual-free bubbles. The standard Galerkin method can be roughly described as being an approximation of the variational formulation of a PDE (or system of PDE’s) in a space of functions that is spanned by piecewise polynomials. This simple idea presents several advantages: first, the discrete system of equations that arise from such an approximation is going to be “banded” since the piecewise polynomials can be constructed to have a “small” support, and therefore the matrices involved are sparse.