In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. We define an anisotropic time–space discretization as a family of time–space partitions that does not satisfy the condition $h^{2}\leq C\delta t$ with $C$ uniform with respect to $h$ and $\delta t$. Standard residual-based stabilization techniques are motivated by a multiscale approach, approximating the effect of the subscales onto the large scales. One of the approximations is to consider the subscales quasi-static (neglecting their time derivative). It is well known that these techniques are unstable for anisotropic time–space discretizations. We show that the use of dynamic subscales (where the subscales time derivatives are not neglected) solves the problem, and prove optimal convergence and stability results that are valid for anisotropic time–space discretizations. Also the improvements related to the use of orthogonal subscales are addressed.

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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

E. Hachem, S. Feghali, R. Codina, T. Coupez. Immersed stress method for fluid-structure interaction using anisotropic mesh adaptation. Int. J. Numer. Meth. Engng 94(9) (2013) DOI 10.1002/nme.4481

E. Hachem, S. Feghali, T. Coupez, R. Codina. A three-field stabilized finite element method for fluid-structure interaction: elastic solid and rigid body limit. Int. J. Numer. Meth. Engng 104(7) (2015) DOI 10.1002/nme.4972

S. Badia, R. Codina, J. Gutiérrez-Santacreu. Long-Term Stability Estimates and Existence of a Global Attractor in a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling. SIAM J. Numer. Anal. 48(3) DOI 10.1137/090766681

E. Oñate, S. Idelsohn, C. Felippa. Consistent pressure Laplacian stabilization for incompressible continua via higher-order finite calculus. Int. J. Numer. Meth. Engng. 87(1-5) (2010) DOI 10.1002/nme.3021

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

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

S. Badia, F. Guillén-Gónzalez, J. Gutiérrez-Santacreu. An Overview on Numerical Analyses of Nematic Liquid Crystal Flows. Arch Computat Methods Eng 18(3) (2011) DOI 10.1007/s11831-011-9061-x

E. Oñate, M. Celigueta, S. Idelsohn, F. Salazar, B. Suárez. Possibilities of the particle finite element method for fluid–soil–structure interaction problems. Comput Mech 48(3) (2011) DOI 10.1007/s00466-011-0617-2