The variational multiscale method—a paradigm for computational mechanics

TLDR

The paper presents a general treatment of the variational multiscale method for an abstract Dirichlet problem. The authors analyze hierarchical p‑methods and bubble functions to approximate the fine‑scale Green’s function, and review connections among residual‑free bubbles, element Green’s functions, and stabilized methods. The study demonstrates that the exact theory provides a paradigm for subgrid‑scale modeling and a posteriori error estimation, and indicates its applicability to fluid mechanics, acoustics, and electromagnetics.

Abstract

We present a general treatment of the variational multiscale method in the context of an abstract Dirichlet problem. We show how the exact theory represents a paradigm for subgrid-scale models and a posteriori error estimation. We examine hierarchical p-methods and bubbles in order to understand and, ultimately, approximate the 'fine-scale Green's function' which appears in the theory. We review relationships between residual-free bubbles, element Green's functions and stabilized methods. These suggest the applicability of the methodology to physically interesting problems in fluid mechanics, acoustics and electromagnetics.