A stabilized mixed formulation for unsteady Brinkman equation based on the method of horizontal lines

TL;DR

This paper introduces a new stabilized mixed formulation for the unsteady Brinkman equation using the method of horizontal lines, enabling stable equal-order interpolation for velocity and pressure without additional assumptions.

Contribution

The paper develops a novel stabilized mixed formulation based on variational multiscale formalism and horizontal lines, allowing stable equal-order interpolation without assuming time independence of fine-scale variables.

Findings

Stable equal-order interpolation achieved for velocity and pressure.

The formulation performs well in spatial and temporal convergence tests.

Numerical results validate the effectiveness of the proposed method.

Abstract

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.