  This paper interprets the stabilized finite element method via residual
minimization as a variational multiscale method. We approximate the solution to
the partial differential equations using two discrete spaces that we build on a
triangulation of the domain; we denote these spaces as coarse and enriched
spaces. Building on the adaptive stabilized finite element method via residual
minimization, we find a coarse-scale approximation in a continuous space by
minimizing the residual on a dual discontinuous Galerkin norm; this process
allows us to compute a robust error estimate to construct an on-the-fly
adaptive method. We reinterpret the residual projection using the variational
multiscale framework to derive a fine-scale approximation. As a result, on each
mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions
derived from a symmetric saddle-point formulation and an a-posteriori error
indicator to guide automatic adaptivity. We test our framework in several
challenging scenarios for linear and nonlinear convection-dominated diffusion
problems to demonstrate the framework's performance in providing stability in
the solution with optimal convergence rates in the asymptotic regime and robust
performance in the pre-asymptotic regime. Lastly, we introduce a heuristic
dual-term contribution in the variational form to improve the full-scale
approximation for symmetric formulations (e.g., diffusion problem).
