A novel Lagrange multipliers-based formulation of the Stokes system involving pressure boundary condition on a part of the boundary is proposed.

Discretization in a finite element framework and complete analysis, from the continuous to the discrete level, in two and three dimensions are provided.

Optimal convergence rates are obtained for standard inf–sup stable finite element spaces, such as Taylor–Hood elements.

Different algebraic solution strategies are proposed, including block factorization based preconditioners.

Verification of the convergence properties and three-dimensional simulations of blood flow in the cerebral venous network are displayed.