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

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)
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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
Publication statusPublished - 2021

Bibliographical note

Accepted Author Manuscript


  • 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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