A unified analysis of stabilized mixed finite element methods for the generalized Stokes problem

Abstract

In this paper, we present a unified analytical framework for stabilized finite element methods applied to the generalized Stokes problem using P₁-P₀ and Q₁-Q₀ element pairs. Stability is attained by including discrete pressure stabilization terms within the variational formulation. Specifically, we investigate the global jump stabilization method, the local jump stabilization method, and their reduced variants. The global approach penalizes pressure jumps across all interior element interfaces, whereas the local and reduced techniques confine these penalties to macro-element partitions through localized jump operators. We establish a unified theoretical framework that encompasses all stabilization variants within a single analytical setting. This framework enables systematic and rigorous proofs of stability and convergence.