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

Stein K.F. Stoter, Marco F.P. ten Eikelder, Frits de Prenter, Ido Akkerman, E. Harald van Brummelen, Clemens V. Verhoosel, Dominik Schillinger

Research output: Contribution to journalArticleScientificpeer-review

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.

Original languageEnglish
Article number113878
Number of pages26
JournalComputer Methods in Applied Mechanics and Engineering
Volume382
DOIs
Publication statusPublished - 2021

Keywords

  • Boundary layer accuracy
  • Fine-scale Green's function
  • Higher-order basis functions
  • Nitsche's method
  • Variational multiscale method
  • Weak boundary conditions

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