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.

我们表明，在变分多尺度框架中，通过Nitsche方法对基本边界条件的弱执行直接对应于投影算子的特定选择。Nitsche方法的一致性，对称性和惩罚项都源自相应规模分解所决定的精细规模封闭。由于这种形式主义，我们能够确定Nitsche型配方的确切的小规模贡献。在对流扩散方程的背景下，我们开发了一个基于残差的模型，该模型将不消失的精细尺度纳入了Dirichlet边界。这会导致带有新模型参数的附加边界项。然后，我们为所有涉及的参数提出了一种参数估计策略，该策略对于高阶基函数也是一致的。我们通过数值实验说明，我们的新增强模型减轻了经典残差精细模型在边界层中表现出来的过度扩散行为，该边界层具有弱强制性的基本条件。