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 H¹₀‐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 L²‐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 H¹₀‐optimality and the streamline‐upwind Petrov‐Galerkin (SUPG) method is described.