Multiscale and Stabilized Methods
This chapter presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid 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 finite difference, finite volume, and spectral methods, in addition to finite-element methods.) The analytical ideas are first illustrated for time-harmonic wave-propagation problems in unbounded fluid 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-diffusive 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 “fine-scale Green's function” that appears in the theory. Relationships among the so-called residual-free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space-time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods [e.g., SUPG (streamline upwind Petrov-Galerkin) and GLS (Galerkin least squares)) that have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective-diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence, is also described. This represents an alternative formulation of large-eddy simulation, which provides a simplified theoretical framework of LES with potential for improved modeling.
