  In this paper, we present a stabilized mixed formulation for unsteady
Brinkman equation. The formulation is systematically derived based on the
variational multiscale formalism and the method of horizontal lines. The
derivation does not need the assumption that the fine-scale variables do not
depend on the time, which is the case with the conventional derivation of
multiscale stabilized formulations for transient mixed problems. An expression
for the stabilization parameter is obtained in terms of a bubble function, and
appropriate bubble functions for various finite elements are also presented.
Under the proposed formulation, equal-order interpolation for the velocity and
pressure (which is computationally the most convenient) is stable.
Representative numerical results are presented to illustrate the performance of
the proposed formulation. Spatial and temporal convergence studies are also
performed, and the proposed formulation performed well.
