Unification of variational multiscale analysis and Nitsche's method, 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

arXiv · 2009.14571

Unification of variational multiscale analysis and Nitsche's method, 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

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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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Stein K. F. Stoter, Marco F. P. ten Eikelder, Frits de Prenter, Ido Akkerman, E. Harald van Brummelen, Clemens V. Verhoosel, Dominik Schillinger. 2020-09-30. Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model. https://arxiv.org/abs/2009.14571

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