Multiscale and stabilized methods

This article presents an introduction to multiscale and stabilized methods, which represent unied approaches to modeling and numerical solution of uid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of nite dierence, nite volume, and spectral methods, in addition to nite element methods.) The analytical ideas are rst illustrated for time-harmonic wave-propagation problems in unbounded uid domains governed by the Helmholtz equation. This leads to the well-known Dirichlet-to-Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented which is applicable to advective-diusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid-scale models and a posteriori error estimation. Hierarchical p-methods and bubble function methods are examined in order to understand and, ultimately, approximate the \ ne-scale Green's function " which appears in the theory. Relationships among so-called residual-free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of non-symmetric, linear evolution operators formulated in space