Preprint 2026-26
Eider Aldana, Ricardo Oyarzúa:
A Lagrange multiplier approach for fluid-flow problem with inhomogeneous mixed boundary conditions
Abstract:
In this work, we propose a Lagrange multiplier-based numerical scheme for fluid-flow problems with inhomogeneous mixed boundary conditions. More precisely, we consider the Stokes and Navier--Stokes equations in a bounded polyhedral domain subject to an inhomogeneous Dirichlet condition on a portion $\Gamma_D$ of the boundary, together with a prescribed Neumann condition on the complementary portion $\Gamma_N$ for the Stokes problem, and a directional do-nothing condition on $\Gamma_N$ for the Navier--Stokes problem. The Dirichlet datum is imposed weakly by means of a Lagrange multiplier, which avoids the need for the discrete velocity space to conform exactly to the Dirichlet data. For the Stokes problem, we derive the corresponding mixed variational formulation and establish its well-posedness, for arbitrary finite element subspaces satisfying a set of abstract discrete inf-sup and stability hypotheses, via the classical Babuška--Brezzi theory. For the Navier--Stokes problem, the presence of the convective term requires a more delicate analysis: we identify an explicit correction of the convective trilinear form, involving the Dirichlet datum on $\Gamma_D$, and introduce an abstract hypothesis on the resulting discrete convective form guaranteeing coercivity. We prove well-posedness of both the Stokes and the Navier--Stokes discrete schemes and derive the corresponding Céa estimates, all stated in terms of the abstract hypotheses on the finite element subspaces. We then introduce concrete finite element subspaces satisfying these hypotheses and derive the corresponding theoretical rates of convergence. Finally, numerical experiments confirm the predicted convergence rates and illustrate the performance and robustness of the proposed method, including an application to a reverse osmosis membrane channel with spacer filaments in submerged and zigzag configurations.
