Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

TL;DR

This paper unifies variational multiscale analysis with Nitsche's method, deriving a boundary layer fine-scale model that improves boundary condition enforcement and reduces diffusion in advection-diffusion problems.

Contribution

It establishes a formal connection between Nitsche's method and fine-scale projections, deriving an exact fine-scale contribution model for boundary layers.

Findings

The new model accurately captures fine-scale effects at boundaries.

It mitigates excessive diffusion in boundary layer simulations.

Parameter estimation ensures model robustness for higher-order bases.

Abstract

We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection-diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.