Alternative Variational Formulations

In general, the finite element methods based on Galerkin and mixed-Galerkin variational principles described in Section 1.3 do not possess all of the theoretical and practical advantages held by finite element methods in the Rayleigh–Ritz setting. As a result, many alternative variational formulations have been proposed with the goal of recovering at least some of these advantages in more general settings. Generally speaking, there are two classes of alternative variational formulations that have been introduced for this purpose. First, for a given partial differential equation (PDE) problem, one may modify the “naturally” occurring variational principle with the goal of defining better quasi-projections. This approach usually allows for the recovery of some, but not all, of the advantages possessed by finite element methods in the Rayleigh–Ritz setting. Methods based on stabilized, penalty, and augmented Lagrangian variational formulations are members of this class. Although they are not within the focus of this book, in Section 2.1, we provide a concise summary of the corresponding finite element methods. There are several reasons for doing so. First, such formulations lead to important classes of methods that are often used in practice. Second, these methods sometimes use least-squares type terms to improve an existing quasi-projection scheme and thus provide additional examples of applications of least-squares notions. Third, our catalogue of modified variational formulations also assists in drawing comparisons between least-squares finite element methods (LSFEMs) and other methods.