Nitsche’s method as a variational multiscale formulation and a resulting boundary layer fine-scale model

• Published in: Computer methods in applied mechanics and engineering 2021-08, Vol.382, p.113878, Article 113878
• Main Authors: Stoter, Stein K.F., ten Eikelder, Marco F.P., de Prenter, Frits, Akkerman, Ido, van Brummelen, E. Harald, Verhoosel, Clemens V., Schillinger, Dominik
• Format: Article
• Language: English
• Subjects: Advection-diffusion equation; Basis functions; Boundary conditions; Boundary layer accuracy; Boundary layers; Dirichlet problem; Fine-scale Green’s function; Higher-order basis functions; Mathematical models; Nitsche’s method; Parameter estimation; Scale models; Variational multiscale method; Weak boundary conditions
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2021.113878

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. •Nitsche’s method corresponds directly to a scale decomposition in the VMS framework.•This interpretation reveals the exact fine-scale terms in the finite element formulation.•The (SUPG type) fine-scale model requires an additional boundary term.•We provide parameter estimations based on fine-scale Green’s functions.•These parameter estimations are suitable for higher-order discretizations.