Nitsche's method as a variational multiscale formulation and a resulting boundary layer fine-scale model
Stein K.F. Stoter (Leibniz University of Hannover)
M.F.P. ten Eikelder (TU Delft - Ship Hydromechanics and Structures)
F. de Prenter (Eindhoven University of Technology)
I. Akkerman (TU Delft - Ship Hydromechanics and Structures)
E. Harald van Brummelen (Eindhoven University of Technology)
Clemens Verhoosel (Eindhoven University of Technology)
Dominik Schillinger (Leibniz University of Hannover)
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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.
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