Subscales on the element boundaries in the variational two-scale finite element method

In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. In particular, we consider the scalar convection–diffusion–reaction equation, the Stokes problem and Darcy’s problem. For these problems the transmission conditions are the continuity of the unknown and its fluxes through element boundaries. The former is automatically achieved by introducing a single valued subscale on the boundaries (for the conforming approximations we consider), whereas the latter provides the effective condition for approximating these values. The final result is that the subscale on the interelement boundaries must be proportional to the jump of the flux of the finite element component and the average of the subscale calculated in the element interiors.