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.

R. Codina, J. Principe, S. Badia. Dissipative Structure and Long Term Behavior of a Finite Element Approximation of Incompressible Flows with Numerical Subgrid Scale Modeling. (2011) DOI 10.1007/978-90-481-9809-2_5

J. Baiges, R. Codina. A variational multiscale method with subscales on the element boundaries for the Helmholtz equation. Int. J. Numer. Meth. Engng 93(6) (2012) DOI 10.1002/nme.4406

N. Vega Reyes, L. Cavaller, J. Marco de la Rosa, J. Baiges, A. Pont, D. Pérez-Sánchez, R. Codina, C. Grivel, M. Collados. Local seeing determination by thermal-CFD analysis to optimize the European Solar Telescope image quality. DOI 10.1117/12.2230564

R. Codina, J. Baiges. Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales. Int. J. Numer. Meth. Engng. 87(1-5) (2011) DOI 10.1002/nme.3111

R. Codina, J. Principe, M. Ávila. Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub‐grid scale modelling. Int Jnl of Num Meth for HFF 20(5) DOI 10.1108/09615531011048213

S. Komala Sheshachala, R. Codina. Finite element modeling of nonlinear reaction–diffusion–advection systems of equations. Int Jnl of Num Meth for HFF 28(11) DOI 10.1108/hff-02-2018-0077

V. Jazarević, B. Rašuo. Numerical Calculation of Aerodynamic Noise Generated from an Aircraft in Low Mach Number Flight. (2017) DOI 10.1007/978-3-319-67202-1_9

O. Colomes, G. Scovazzi, I. Sraj, O. Knio, O. Le Maître. A Finite Volume Error Estimator Inspired by the Variational Multiscale Approach. (2018) DOI 10.2514/6.2018-1178

R. Codina. On hp convergence of stabilized finite element methods for the convection–diffusion equation. SeMA 75(4) (2018) DOI 10.1007/s40324-018-0154-4

R. Codina. Finite Element Approximation of the Convection-Diffusion Equation: Subgrid-Scale Spaces, Local Instabilities and Anisotropic Space-Time Discretizations. (2011) DOI 10.1007/978-3-642-19665-2_10

L. Yang, S. Badia, R. Codina. A pseudo-compressible variational multiscale solver for turbulent incompressible flows. Comput Mech 58(6) (2016) DOI 10.1007/s00466-016-1332-9