A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

TL;DR

This paper develops a variational multiscale framework from an adaptive stabilized finite element method, enabling robust, stable solutions with optimal convergence for convection-dominated problems through residual minimization and dual norms.

Contribution

It introduces a novel variational multiscale interpretation of residual minimization-based stabilized finite element methods, enhancing adaptivity and stability in solving challenging PDEs.

Findings

Achieves stable solutions with optimal convergence rates.

Demonstrates robustness in convection-dominated regimes.

Provides an adaptive framework guided by a posteriori error indicators.

Abstract

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).