Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods

We derive an explicit formula for the fine-scale Green’s function arising in variational multiscale analysis. The formula is expressed in terms of the classical Green’s function and a projector which defines the decomposition of the solution into coarse and fine scales. The theory is presented in an abstract operator format and subsequently specialized for the advection-diffusion equation. It is shown that different projectors lead to fine-scale Green’s functions with very different properties. For example, in the advection-dominated case, the projector induced by the H^1_0-seminorm produces a fine-scale Green’s function which is highly attenuated and localized. These are very desirable properties in a multiscale method and ones that are not shared by the L^2-projector. By design, the coarse-scale solution attains optimality in the norm associated with the projector. This property, combined with a localized fine-scale Green’s function, indicates the possibility of effective methods with local character for dominantly hyperbolic problems. The constructs lead to a new class of stabilized methods, and the relationship between H^1_0-optimality and the streamline-upwind Petrov-Galerkin (SUPG) method is described.