Rodolfo Araya, Alfonso Caiazzo, Franz Chouly:
Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche
Slip boundary conditions arise naturally for Stokes or Navier-Stokes equations, for instance when modelling biological surfaces , in slide coating  or in the context of turbulence modeling . These are essential boundary conditions, and can be in fact considered as generalized Dirichlet conditions. They are not straightforward to implement into standard finite element libraries, with standard techniques such as a discrete lifting or a partitioning of the global matrix. As a result, many works were devoted to study alternative approaches. This work presents a simple approach based on Nitsche’s technique combined with a stabilized equal- order finite element method. To simplify the presentation, we focus on the Stokes equation on a polygonal boundary and without any specific law that involve the tangential components of the velocity, such as a Navier law. We consider both symmetric and non-symmetric variants of Nitsche, since they have different advantages, particularly to enforce accurately the boundary condition, see, e.g. [10, 17] and references therein. We take advantage of the stabilization terms to carry out the analysis. Notably we are able to prove the stability with a constant independent of the fluid viscosity. The overall method is consistent, introduces no extra unknown and can be implemented easily. To assess the properties of the method, we propose an implementation in the FEniCS environment  and present several numerical experiments. Let us put our work in a general perspective. The first methods to enforce slip conditions were based on Lagrange multipliers: see, e.g., [22, 27, 28]. In  the condition was enforced pointwise at nodal values of the velocity. Many studies have been devoted to the study of penalty methods, to enforce approximately the slip condition with a regularization term. These methods are not consistent, but remain popular and very easy to implement. Moreover, penalty can be interpreted as a penetration condition with a given resistance . A first work has been focused on the Navier-Stokes equation , and followed by [12, 13], with emphasis on the case of a curved boundary, where a Babuska-type paradox may appear. Other recent works have been devoted to the usage of penalty terms combined with Lagrange finite elements [20, 30, 31] or Crouzeix-Raviart finite elements [21, 32]. To our knowledge, Nitsche’s method has been first considered in , as a simple, consistent and primal technique to take into account the slip condition. Notably, it has been noticed that the skew-symmetric variant of Nitsche remains operational even when the Nitsche parameter vanishes (penalty-free variants), a result which opened the path to further research on this topic [6, 7, 8]. Later on, in , different variants of Nitsche have been proposed and linked, as usual, with stabilized mixed methods (following ). Emphasis has been once again made on the curved boundary and a possibly related Babuska-type paradox. More recently, a specific treatment of the Navier boundary condition has been studied in , building on the specific Nitsche-type method proposed by Juntunen & Stenberg  to discretize robustly Robin-type boundary conditions (see also ), and a symmetric Nitsche method with specific, accurate, discretization of the curved boundary, has been designed and studied in . In conclusion, we observe that almost all the aforementioned works have considered inf-sup stable pairs to discretize the Stokes equation, except  where the penalty method combined with a P1/P1 finite element pair with pressure stabilization is taken into account. This paper is structured as follows. Section 2 describes the model equations in strong form. The weak formulation and the corresponding functional setting is object of Section 3. Section 4 presents the discretization with finite elements, stabilization and Nitsche. Section 5 details the stability and convergence analysis. Numerical experiments are provided in Section 6.