Stabilization Techniques for the Finite Element Method

Abstract

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. Second, taking derivatives and integrating polynomials is a very attractive task for any first year calculus student, and the simplicity of the implementation of the method for the most cumbersome PDE or system of PDE’s seems straightforward. Third, the mathematical analysis seems to be possible without a lot of sophistication (at least if we have an elliptic problem, and we disregard technicalities referring to domain shape, etc.).