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Analysis of a stabilized penalty free nitsche method for the brinkman, stokes, and darcy problems

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(1)See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/330787099. Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems Article in ESAIM Mathematical Modelling and Numerical Analysis · November 2018 DOI: 10.1051/m2an/2018063. CITATIONS. READS. 0. 70. 5 authors, including: Laura Blank. Alfonso Caiazzo. Weierstrass Institute for Applied Analysis and Stochastics. Weierstrass Institute for Applied Analysis and Stochastics. 3 PUBLICATIONS 7 CITATIONS. 47 PUBLICATIONS 655 CITATIONS. SEE PROFILE. SEE PROFILE. Franz Chouly. Alexei Lozinski. University of Burgundy. University of Franche-Comté. 63 PUBLICATIONS 507 CITATIONS. 62 PUBLICATIONS 732 CITATIONS. SEE PROFILE. Some of the authors of this publication are also working on these related projects:. Flow boundary conditions / backflow stabilization View project. Numerical methods for contact and friction View project. All content following this page was uploaded by Franz Chouly on 26 February 2019. The user has requested enhancement of the downloaded file.. SEE PROFILE.

(2) Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes and Darcy problems Laura Blank, Alfonso Caiazzo, Franz Chouly, Alexei Lozinski, Joaquin Mura. To cite this version: Laura Blank, Alfonso Caiazzo, Franz Chouly, Alexei Lozinski, Joaquin Mura. Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes and Darcy problems . 2018. <hal-01725387>. HAL Id: hal-01725387 https://hal.archives-ouvertes.fr/hal-01725387 Submitted on 7 Mar 2018. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(3) ANALYSIS OF A STABILIZED PENALTY-FREE NITSCHE METHOD FOR THE BRINKMAN, STOKES, AND DARCY PROBLEMS. Laura Blank 1 , Alfonso Caiazzo 1 , Franz Chouly 2 , Alexei Lozinski 2 and Joaquin Mura 3 Abstract. In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (T. Boiveau & E. Burman. IMA J. Numer. Anal. 36 (2016), no.2, 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy flows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.. 1991 Mathematics Subject Classification. 65N30, 65N12, 65N15. .. 1. Introduction The Brinkman problem [10], originally proposed as an alternative model approach for the flow in porous media, is obtained as a modification of the Darcy model by equipping Darcy’s law with a resistance term proportional to the fluid viscous stresses, targeting on a better handling of high permeability regions. In order to introduce the model problem of interest, let us consider a connected domain Ω ⊂ Rn , n = 2, 3, with boundary Γ := ∂Ω, and let us denote by n the outer unit normal vector on Γ. Our model problem is described by the following system of partial differential equations −∇ · ( µeff ∇u) + σu + ∇p = f ,. in Ω,. ∇ · u = g,. in Ω,. (1.1). Keywords and phrases: Brinkman problem, penalty-free Nitsche method, weak boundary conditions, stabilized finite elements 1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany. e-mail: laura.blank@wias-berlin.de & alfonso.caiazzo@wias-berlin.de 2 Laboratoire de Mathématiques de Besançon UMR CNRS 6623, Université Bourgogne Franche Comté, 16 route de Gray, 25030 Besançon Cedex, France. e-mail: franz.chouly@univ-fcomte.fr & alexei.lozinski@univ-fcomte.fr 3 Biomedical Imaging Center, School of Engineering and School of Medicine, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Santiago, Chile. e-mail: jamura@uc.cl.

(4) 2 where u : Ω → Rn represents the fluid velocity field, p : Ω → R is the fluid pressure and f : Ω → Rn , g : Ω → R are given data. In (1.1), the parameter µeff is called effective viscosity, while σ is given by the ratio between the fluid viscosity and the permeability of the porous medium. Depending on the values of the aforementioned parameters, the system (1.1) describes a whole range of problems between the Stokes (σ = 0) and the Darcy (µeff = 0) models. However, this transition does not depend continuously on the physical parameters. In particular, the standard boundary condition for µeff > 0 is u = 0,. on Γ,. (1.2). (essential boundary condition on the velocity u), whereas for µeff = 0, it has to be replaced by the condition. u · n = 0,. on Γ,. (1.3). which is appropriate for the Darcy problem. Likewise, when focusing on the weak counterpart of (1.1), one has to consider different natural functional settings for the Stokes/Brinkman (µeff > 0) and Darcy (µeff = 0) problems. These aspects affect also the discrete formulation of (1.1) and the strategies for its numerical solution. In the context of finite element methods, the different regularity properties of the limit problems (Stokes and Darcy) are reflected in the choice of the finite element spaces used for the velocity and the pressure: stable and efficient elements for the Stokes problem might not provide accurate or stable approximations in the Darcy case, and vice versa (see, e.g., [9, 12, 27]). Moreover, the discrepancy between the boundary conditions in the limit cases at the continuous level implies that imposing essential boundary conditions on the velocity space does not allow a smooth, parameter-dependent transition between (1.2) and (1.3), in particular at the discrete level. Our work is motivated by the solution of direct and inverse problems in clinical applications involving flows in porous media. Hence, the numerical method shall be robust with respect to different flow regimes, in order to handle unknown physical parameters, and, at the same time, require relatively low computational cost, allowing for the numerical solution in a reasonable time. One strategy to achieve a common discretization for both, the Stokes and the Darcy problems, which will be adopted in this paper, consists in using finite element pairs suited for both cases, possibly including stabilization terms. Among the different possibilities, we focus on equal-order (linear) finite elements, combined with a Galerkin Least Squares (GLS) and a Grad-Div stabilization, that guarantee stability for the pressure and control on the divergence of the velocity. This setting, with a particular focus on the Brinkman, Stokes and Darcy problems, has been deeply analyzed, e.g., in [3], considering different choices for the scaling of the stabilization terms. Other options, which have been proposed in the literature, are based on P1 /P0 (stabilized) finite elements (analyzed in [12] for the Stokes-Darcy coupling and discussed in [22] for the Brinkman problem), Taylor-Hood, MINI, and Pk /Pk (stabilized) elements [21, 26], as well as Pk /Pdisc k−1 [15]. In order to tackle the issue of the need of different boundary conditions depending on the (Stokes or Darcy) regime, we focus on the weak imposition of essential boundary conditions via a Nitsche method. This approach, originally introduced in [29], has been extended, applied, and analyzed in several contexts, including coupled Stokes-Darcy problems (see, e.g., [12,15] among others), and the general Brinkman problem (see, e.g., [21,22,26]), demonstrating that it is able to yield a robust transition between the two different flow regimes. In its pioneer version [29], the Nitsche approach was formulated as a consistent symmetric penalty method, for which stability was guaranteed choosing the penalty parameter sufficiently large. This assumption was relaxed considering a non-symmetric version proposed in [19], for which stability was proven for any strictly positive value of the penalty parameter. We investigate the so-called non-symmetric penalty-free Nitsche method, i.e., assessing the stability of the approach even without the presence of a penalty parameter. In this case, the method can be interpreted as a Lagrange multiplier method [31], where for the Brinkman problem the normal fluxes at the boundary and the.

(5) 3 pressure play the role of Lagrange multipliers. The stability of the Nitsche method without penalty was first shown in [11] for convection-diffusion problems, and more recently extended to compressible and incompressible elasticity [5] and to domain decomposition problems with discontinuous material parameters [4]. The unconditional applicability in presence of variable physical parameters is our main motivation for addressing and investigating the properties of the penalty-free Nitsche method for the Brinkman problem. In this case, the main challenges are related to the fact that stability has to be shown for the pressure (in the case of equal-order finite elements) and the velocity at the boundary. For the latter, it is important to observe that due to the differences in the limit problems (e.g., in the boundary conditions (1.2) and (1.3)), the natural norms to be controlled depend on the physical range. Our main result concerns the stability, the robustness, and the optimal convergence in a natural norm of the formulation obtained by combining a penalty-free Nitsche method and a stabilized equal-order finite element method. We show that the proposed finite element method is inf-sup stable in the whole range of physical parameters, including the limit values µeff = 0 or σ = 0. Moreover, our analysis shows that the inf-sup constant does neither depend on µeff nor on σ, but only on the regularity properties of the mesh and on the stabilization parameters. These results thus extend available estimates recently provided in [21, 26] using a similar discrete setting (stabilized finite elements), where the symmetric Nitsche method was analyzed focusing on an adimensional version of (1.1) which does not allow to control the divergence of the velocity and excludes the case σ = 0. To establish the stability of the Nitsche method, we follow a path inspired by the analysis in [5] for the incompressible elasticity, but proposing a simpler argument. As next, we discuss the stability estimate in the Darcy limit µeff = 0 (or in the more general case µσeff → 0), in which only the control on the boundary normal velocity is required. We show that, focusing on the case of two-dimensional polygonal boundaries, an additional stabilization to control the discontinuities of the normal velocity along the boundary is required. To tackle this issue, we introduce a corner stabilization, which penalizes the jump of the normal velocity solely on the corners of the discrete domain and allows to obtain the aforementioned robust stability estimates and optimal a priori error estimates in a mesh-dependent norm. The paper is organized as follows. In Section 2 we introduce the problem setting, the finite element formulation, and enunciate the main stability and convergence results. Section 3 is dedicated to the technical proofs, while numerical experiments are presented in Section 4. Section 5 draws the conclusive remarks.. 2. A Penalty-free Nitsche Method for the Brinkman Problem The purpose of this section is to introduce the stabilized finite element method for the Brinkman problem, the penalty-free Nitsche method for imposing essential boundary conditions, and to state the related stability and convergence results.. 2.1. The weak formulation In what follows, we will assume to deal with a two-dimensional domain Ω ⊂ R2 (i.e., setting n = 2) with polygonal boundary Γ. In this setting, let us consider the Sobolev spaces (see, e.g., [1, 20]):  H 10 (Ω) := v ∈ H 1 (Ω) : v|Γ = 0 ,  H div,0 (Ω) := v ∈ L2 (Ω) : ∇ · v ∈ L2 (Ω) , (v · n) |Γ = 0 ,   Z 2 2 L0 (Ω) := q ∈ L (Ω) : q=0 . Ω. We will denote by curved brackets (·, ·)A the L2 -scalar product on A ⊆ Ω, while h·, ·iE , will be used for integrals evaluated on the boundary, i.e., for any E ⊆ Γ. For the ease of notation, the subscripts Ω and Γ will be omitted, 2 simply denoting with (·, ·) and h·, ·i the scalar products in L2 (Ω) and  1L (Γ), n respectively. Furthermore, a bold 1 faced letter will indicate the n-th Cartesian power, e.g., H (R) = H (R) . Finally, we will denote with k·k0.

(6) 4 2. the norm on L2 (Ω), and with k·kk and | · |2k the norm and semi-norm, respectively, on the Sobolev space H k (Ω) (see, e.g., [6, Def. 1.3.1, 1.3.7]). With the above notations, we now introduce the bilinear forms A [(u, p) ; (v, q)] := µeff (∇u, ∇v) + σ (u, v) − (p, ∇ · v) + (∇ · u, q) , L (v, q) := (f , v) + (g, q) .. (2.1). In the case µeff > 0, the weak formulation of problem (1.1), (1.2) reads as: Find (u, p) ∈ H 10 (Ω) × L20 (Ω) such that A [(u, p) ; (v, q)] = L (v, q) ,. ∀ (v, q) ∈ H 10 (Ω) × L20 (Ω) .. (2.2). Detailed proofs of the well-posedness of problem (2.2) for f ∈ L2 (Ω) and g ∈ L20 (Ω) and the corresponding basic theory can be found in, e.g., [2, 7, 9, 20]. Depending on the given boundary conditions and the regularity of the given data, in the case µeff = 0 2 (Darcy limit), the weak solution to the mixed form of problem (1.1) can be sought  either in H div,0 (Ω) × L0 (Ω) 2 1 2 corresponding to the boundary condition (1.3) or in L (Ω) × H (Ω) ∩ L0 (Ω) .. 2.2. The discrete formulation Let us assume that the polygonal computational domain Ω admits a boundary conforming (fitted) family of triangulations {Th }h>0 , i.e., that the discrete domain and the original domain coincide for all h. The parameter h denotes a characteristic length of the finite element mesh Th , defined as h := maxT ∈Th hT , hT being the diameter of the cell T ∈ Th . Furthermore, we will denote by Gh the set of edges belonging to the boundary Γ and with hE the length of E ∈ Gh . Since we assume that Ω is polygonal, it can be decomposed as the union of NP straight boundary segments and we denote by C the set of corner nodes of Th . We will assume that all considered triangulations are nondegenerate, i.e., there exists a constants CSR > 0 independent from h, such that ∀h > 0,. ∀T ∈ Th :. hT 6 CSR , ρT. (2.3). where ρT is the radius of the largest inscribed sphere in T . This property is also known as (shape-) regularity, see [13, p. 124], [17, Def. 1.107]. In particular, it is assumed that there exists a constant η0 > 1 such that hE 0 hE 0 6 η0 , for any pair of adjacent edges E, E ∈ Gh . For the validity of the arguments discussed in this paper, we assume that the mesh satisfies also the condition √ η0 ≤ 7 + 4 3 ≈ 13.9 . (2.4) Moreover, we require that the triangulation is such that the inner triangles cover an area larger than the S boundary ones. Formally, let Bh := T : T ∩Γ6=∅ T be the union of all triangles which have at least a node on the boundary. We assume that |Bh | 6 ω|Ω| (2.5) with ω < 1 and independent from h. In order to define the discrete problem, let us introduce the quantity `Ω > 0, representing a typical physical length scale of the problem, and the parameter ν := µeff + σ`Ω 2. (2.6). (which has the units of a viscosity). The length `Ω has been introduced mainly for the purpose of consistency of physical units (see, e.g., the discussion in [3]) and it is assumed to satisfy `Ω > hT , for all T ∈ {Th }h>0 ..

(7) 5 Let us now introduce the finite element pair  V h := v h ∈ H 1 (Ω) ∩ C 0 (Ω) : v h |T ∈ P1 (T ) , ∀ T ∈ Th ,  Qh := qh ∈ L20 (Ω) ∩ C 0 (Ω) : qh |T ∈ P1 (T ) , ∀ T ∈ Th , and consider the problem: Find (uh , ph ) ∈ V h × Qh such that Ah [(uh , ph ) ; (v h , qh )] = Lh (v h , qh ) ,. ∀ (v h , qh ) ∈ V h × Qh ,. (2.7). with GLSns ,lhs GD,lhs Ah [(u, p) ; (v, q)] : = A [(u, p) ; (v, q)] + Sh,α [(u, p) , (v, q)] + Sh,δ [(u, v)] C,lhs − hµeff ∇u · n, vi + hpn, vi + hµeff ∇v · n, ui − hqn, ui + Sh,ρ (u, v) , GLSns ,rhs GD,rhs C,rhs Lh (v, q) := L (v, q) + Sh,α [(v, q)] + Sh,δ [v] + Sh,ρ [v] .. (2.8a) (2.8b). In (2.8a)-(2.8b), A [(u, p) ; (v, q)] is defined in (2.1) and we introduce a stabilization belonging to the nonsymmetric GLS method GLSns ,lhs Sh,α [(u, p) , (v, q)] := α. X h2 T (σu + ∇p, σv + ∇q)T , ν. T ∈Th GLSns ,rhs Sh,α. X h2 T [(v, q)] := α (f , σv + ∇q)T , ν. (2.9). T ∈Th. as well as a Grad-Div stabilization GD,lhs Sh,δ [(u, v)] := δ ν (∇ · u, ∇ · v) , GD,rhs Sh,δ [v] := δ ν (g, ∇ · v) .. Additionally, we employ a stabilization term, later referred to as corner stabilization, given by X C,lhs Sh,ρ [(u, v)] := ρ ν Ju · nK (x) · Jv · nK (x) ,. (2.10). (2.11). x∈C. where Ju · nK (x) := u (x) · nE − u (x) · nE 0 = u (x) · (nE − nE 0 ) denotes the jump of u · n at a corner node x ∈ C := {xc : ∃ E, E 0 ∈ Γ such that xc = E ∩ E 0 and nE 6= nE 0 }, with E, E 0 being the two boundary edges adjacent to x. In the above definitions, α, δ, and ρ are non-negative dimensionless stabilization parameters, which will be assumed to be independent from the mesh size and constant in space. Notice that, in the case of nonhomogeneous boundary conditions, the corresponding right hand sides, consistent with the boundary terms introduced in (2.8a) and with the corner stabilization (2.11), have to be included. The stabilized formulation (2.7) can be regarded as a consistent extension of the pressure stabilizing Petrov– Galerkin method (PSPG, [8]) that was introduced in [16] as an unconditionally stable (α > 0), non-symmetric formulation of the Stokes problem. This method can be interpreted as a non-symmetric modification of the method proposed in [23], known as Galerkin Least Squares (GLS) method, and for this reason we will refer to it as the ’non-symmetric GLS method’..

(8) 6 As it will be shown in Section 3, the stabilization (2.11) is one of the main ingredients of our method. This additional term is used in order to prove robust stability estimates, in particular in the Darcy limit (µeff = 0 or σ  1), in absence of the Nitsche penalty term. Remark 2.1 (On the GLS stabilization terms). Note that for (2.9) since the velocity is approximated using a first order Lagrange finite element space, it holds −∇ · (µeff ∇uh ) = 0, which allows the simplified expression we are using. The formulation can be analogously extended to the general case of equal finite element pairs Pk /Pk (k > 1). In these cases, the aforementioned term has to be included in the residuum of the momentum balance equation. Remark 2.2 (On the Grad-Div stabilization terms). The usage of the Grad-Div stabilization (2.10), originally proposed in [18] (see also, e.g., [24] for further detailed more recent discussions), is motivated here by the need of controlling the L2 -norm of the divergence of the velocity in the Darcy limit. However this term is also necessary in order to provide stability with respect to the normal velocity on the boundary (see Section 3, Lemma 3.6 for details). Remark 2.3 (On the discrete setting). The main focus of this paper is the analysis of the penalty-free Nitsche method for the Brinkman model. The non-symmetric GLS, the Grad-Div, as well as the corner stabilization are motivated by our choice of the discrete setting (P1 /P1 stabilized finite elements) valid for both the Stokes and the Darcy problems. However, it is worth noticing that the stability estimates which will be proven for the penalty-free Nitsche method and are based on the usage of linear finite elements for velocity and pressure, do not rely on this particular choice of the bulk stabilization, and they can be straightforwardly extended to other approaches (e.g., PSPG or symmetric GLS). We conclude this section by introducing the norms considered in our analysis: X ν X X µeff 2 2 2 2 2 kuk0,E + ku · nE k0,E + ρ ν |Ju · nK (x)| |||(u, p)|||h := |||u||| + θ hE hE E∈Gh. E∈Gh. + with θ :=. µeff ν. 2 kpk0. ν. x∈C. (2.12). X h2 2 T +α k∇pk0,T , ν T ∈Th. ∈ [0, 1] and 2. 2. 2. 2. |||u||| := µeff k∇uk0 + σ kuk0 + δ ν k∇ · uk0 . As it will be shown in the next section, the scaling by θ is necessary in order to obtain robust estimates also in the Darcy limit (µeff = 0). We also observe that, if σ = 0 (Stokes limit), the scaling factor is equal to one. In this case, the velocity norm is analogous to the norm used (for the displacement) in the context of the penalty-free Nitsche method for incompressible elasticity [5].. 2.3. Stability and convergence results This Section enunciates the main results of this paper, concerning stability and convergence of the proposed penalty-free Nitsche method (2.7). The technical proofs will be discussed in detail in Section 3. Theorem (Inf-sup stability) Let Ah [(uh , ph ) ; (v h , qh )] be the bilinear form defined in (2.8a), α, δ, ρ > 0, and µeff , σ > 0 with µeff + σ > 0. Then there exists a constant β > 0, independent from h and from the physical parameters, such that  ! Ah [(uh , ph ) ; (v h , qh )] > β. inf sup (uh ,ph )∈V h ×Qh \{(0,0)} (v h ,qh )∈V h ×Qh \{(0,0)} |||(uh , ph )|||h |||(v h , qh )|||h The inf-sup constant β depends only on the stabilization parameters and on the shape regularity of the mesh..

(9) 7 This statement assesses unconditional stability with respect to the physical parameters, including the limit cases σ = 0 or µeff = 0. Moreover, we also show that, for small values of the stabilization parameters, it holds  β −1 = O α−1 ρ−1 + δ −1 . Theorem (A priori error estimate) Let α, δ, ρ > 0 and µeff , σ > 0 with µeff + σ > 0. Moreover, let (u, p) be the solution of (1.1) with the appropriate boundary conditions and (uh , ph ) be the solution of problem (2.7). Assuming (u, p) ∈ H 2 (Ω) × H 1 (Ω), it holds |||(u − uh , p − ph )|||h 6 h (Cu kuk2 + Cp kpk1 ) ,. (2.13). where Cu and Cp are independent from h and for small respectively moderate values of stabilization parameters it holds !   1 1 ν2 , Cp = O . Cu = O 1 1 β ν2δ2β With respect to the a priori estimate (2.13), let us observe that it reduces to standard estimates in the Stokes and in the Darcy limits, for σ = 0 and µeff = 0, respectively (see, e.g., [3]). One of the main implications of the above Theorems is therefore the fact that the penalty-free Nitsche formulation possesses a convergence and stability behavior that is comparable to the standard formulation (where essential boundary conditions are imposed in a strong sense) and to the classical (penalty) Nitsche method (see, e.g., [22, 26]). For the sake of completeness, it is worth observing that, in order to obtain the robust convergence estimate (2.13), the scaling of the stabilization terms with the viscosity ν defined in (2.6) is a necessary requirement. Alternative formulations were analyzed, e.g., in [3], for the Brinkman problem with strong imposition of boundary conditions. There, it was shown that stability and optimal error estimates can also be obtained by scaling the stabilization of the Darcy terms with respect to the mesh, replacing ν by νT := µeff + σh2T on each triangle T ∈ Th . An analogous scaling as well has been analyzed in [28] (stabilized finite elements for the Darcy equation) and in [26] in the context of a rescaled Brinkman problem with a symmetric Nitsche penalty method (limited to the case σ > 0). However, as it will be shown in the next section, the scaling (2.6) is used in order to uniformly control the boundary velocity for µeff , σ > 0.. 3. Proof In this section, the proofs of the aforementioned Theorems, claiming inf-sup stability and convergence of the proposed method, will be discussed in detail.. 3.1. Preliminaries Let us begin by introducing some basic notation and stating a few results that will be utilized in the upcoming analysis. Let E be an edge of the mesh, and let us denote by T E a triangle attached to E. Then, the following discrete trace in ity is valid (see, e.g., [6, (10.3.8)], [14, Lemma 4], [32, pp. 28])   2 2 2 −2 h−1 kvk 6 c h kvk + k∇vk ∀v ∈ H 1 (T E ) , (3.1) 2 2 2 DT E TE L (E) L (T E ) L (T E ) , where cDT > 0 is a constant, only depending on the shape regularity (2.3) of the mesh. Under the assumption of shape regularity (and assuming h 6 1), there exists a constant cI > 0, independent of h and T , such that for all vh ∈ Pk (T ), k > 0, and for all T ∈ Th it holds the following inverse inequality [17, Lemma 1.138] k∇vh kL2 (T ) 6 cI h−1 T kvh kL2 (T ) .. (3.2).

(10) 8 Combining (3.1) and (3.2) one can conclude also that there exist a constant cDTI such that, for any element-wise linear (on the mesh Th ) function vh it holds (see, e.g., [25, Lemma 3.1], [32, Lemma 2.1]) 2. X. hE k∇vh ·. 2. kvh kL2 (E) 6 cDTI h−1 T E kvh kL2 (T E ) ,. (3.3a). 2 nE kL2 (E). (3.3b). 6. 2 cDTI k∇vh kL2 (T E ). .. E∈Gh. Let us denote with IhSZ the Scott–Zhang interpolator onto the finite element space V h [17,30], which preserves essential boundary conditions on Γ. Then, for l, m ∈ N0 with 1 6 l < ∞, there exists a constant cSZ > 0, depending on the geometry and on the mesh regularity, such that the following approximation properties hold:. ∀ 0 6 m 6 1: ∀ l 6 2:. l X. IhSZ (v). m,Ω. ∀v ∈ H l (Ω) , ∀h,. 6 cSZ kvkl,Ω ,. SZ hm T v − Ih (v). m,T. 6 cSZ hlT |v|l,S(T ) ,. (3.4a). ∀v ∈ H l (S (T )) , ∀h, ∀T ∈ Th ,. (3.4b). m=0. where S(T ) denotes the union of all cells in Th which have a vertex in common with T . Finally, let IhL be the Lagrange interpolator onto V h . Then, there exists a constant cLa > 0 such that, there holds (see, e.g., [17, Theorem 1.103]): v − IhL (v). 0,T. + hT v − IhL (v). 1,T. 6 cLa h2T kvk2,T ,. ∀v ∈ H 2 (T ) .. (3.5). 3.2. Stability As next, we will focus on the inf-sup stability of the discrete bilinear form (2.8a) with respect to the meshdependent norm (2.12). Throughout the proofs, the introduced constants depending on the physical parameters or on discretization parameters (mesh size, mesh topology, finite element spaces, stabilization parameters) will be explicated and discussed, in order to allow the reader to follow the derivation in detail and, eventually, to clearly assess the role of the physical parameters within the derived estimates (especially in the limit cases). The first result concerns the coercivity of the bilinear form (2.8a) in a norm which is weaker than (2.12). Lemma 3.1 (Coercivity in a weaker norm). Let α > 0, δ, ρ > 0 and µeff , σ > 0 with µeff + σ > 0. Then there exists a constant C0 = C0 (α) > 0, independent from the physical parameters and h, such that. Ah [(v h , qh ) ; (v h , qh )] > C0. 2. |||v h ||| + ρ ν. X x∈C. X h2 2 T k∇qh k0,T |Jv h · nK (x)| + α ν 2. ! ,. T ∈Th. for all (v h , qh ) ∈ V h × Qh . Proof: Let (v h , qh ) ∈ V h × Qh , then it is 2. 2. Ah [(v h , qh ) ; (v h , qh )] = µeff k∇v h k0 + σ kv h k0 + α. X h2 X h2 2 2 T T k∇qh k0,T + ασ 2 kv h k0,T ν ν. T ∈Th. T ∈Th. X h2 X 2 2 T + 2ασ (∇qh , v h ) + δ ν k∇ · v h k0 + ρ ν |Jv h · nK (x)| . ν T ∈Th. x∈C. (3.6).

(11) 9 Using the Cauchy–Schwarz and Young inequalities we obtain   1 X σh2T 2 2 2 kv h k0,T Ah [(v h , qh ) ; (v h , qh )] > µeff k∇v h k0 + σ kv h k0 + ασ 1 − ε ν T ∈Th. 2. + δ ν k∇ · v h k0 + ρ ν. X. 2. |Jv h · nK (x)| + (1 − ε) α. T ∈Th. x∈C. and since. σh2T ν. X h2 2 T k∇qh k0,T , ν. σ`Ω 2 ν. 6 1 we get with  < 1 the bound  α 2 2 2 Ah [(v h , qh ) ; (v h , qh )] > µeff k∇v h k0 + 1 + α − σ kv h k0 + δ ν k∇ · v h k0 ε X X h2 2 2 T + ρν |Jv h · nK (x)| + (1 − ε) α k∇qh k0,T . ν. 6. T ∈Th. x∈C.  α < ε < 1, so that 1 + α − αε and (1 − ε) In order to obtain the stability estimate, we choose ε such that α+1 q  α2 α are strictly positive. Taking ε := + α − these two coefficients coincide, and we obtain 4 2 2. 2. Ah [(v h , qh ) ; (v h , qh )] > µeff k∇v h k0 + δ ν k∇ · v h k0 + ρ ν. X. 2. |Jv h · nK (x)|. x∈C. X h2 2 2 T k∇qh k0,T + (1 − ε) σ kv h k0 + α ν. ! .. T ∈Th. The proof is concluded defining r C0 := 1 − ε = 1 −. α α2 +α+ . 4 2. (3.7) . Remark 3.1 (On the behavior of C0 ). Notice that the constant C0 introduced in (3.7) is a decreasing function of α satisfying C0 (0) = 1. In particular, C0 = O(1) for small and moderate values of α. Note that the estimate (3.6) holds also (trivially) for α = 0. The following Lemma provides stability in the L2 -norm of the pressure. Lemma 3.2 (Pressure control). Let α > 0, δ, ρ > 0, and µeff , σ > 0 with µeff + σ > 0. Then, there exists a constant C1 = C1 (α, δ) > 0, independent from the physical parameters and h, such that, for all (uh , ph ) ∈ V h × Qh , we can find a function v h ∈ V h that satisfies ! 2 X h2 X µeff 1 kph k0 2 2 2 T Ah [(uh , ph ) ; (v h , 0)] > − C1 |||uh ||| + α k∇ph k0,T + θ kuh k0,E . (3.8) 2 ν ν hE T ∈Th. E∈Gh. Proof: Let (uh , ph ) ∈ V h × Qh . Since ph ∈ Qh ⊂ L20 (Ω) (due to conformity), there exists exactly one v ph ∈ H 10 (Ω) and a dimensionless constant ĉΩ (that only depends on Ω) such that [20, Corollary 2.4] 1 ∇ · v ph = − p h , ν ĉΩ k∇v ph k0 6 kph k0 . ν. (3.9a) (3.9b).

(12) 10 Let now v h := IhSZ (v ph ) be the Scott–Zhang interpolator of the function v ph onto V h . Due to the H 1 -stability of the Scott–Zhang interpolator (3.4a), property (3.9b), and the Poincaré inequality [17, Lemma B.61], there also holds k∇v h k0 6. cΩ kph k0 ν. and. kv h k0 6. `Ω cΩ kph k0 , ν. (3.10). with a constant cΩ that only depends on the domain and on the regularity of the mesh. Moreover, according to (3.4b), it holds X 1 X 1 2 2 2 2 kv ph − v h k0,T 6 c2 h2 k∇v ph k0,S(T ) 6 cf SZ k∇v ph k0 , 2 hT h2T SZ T. (3.11). T ∈Th. T ∈Th. 2. 2 with cf SZ := cSZ (maxT ∈Th #S (T )). Here, #S (T ) denotes the number of triangles contained in S (T ) which depends on the regularity of the mesh. Since the Scott–Zhang interpolator preserves essential boundary conditions, it holds v h ∈ H 10 (Ω) ∩ V h such that the boundary terms involving v h vanish. Using the decomposition v h = v ph − (v ph − v h ) and integration by parts for the term (v ph − v h ) ∈ H 10 (Ω) we get. Ah [(uh , ph ) ; (v h , 0)] = µeff (∇uh , ∇v h ) + σ (uh , v h ) − (ph , ∇ · v h ) X h2 T +α (σuh + ∇ph , σv h )T + δ ν (∇ · uh , ∇ · v h ) ν T ∈Th. + hµeff ∇v h · n, uh i = µeff (∇uh , ∇v h ) + σ (uh , v h ) − (ph , ∇ · v ph ) − (∇ph , v ph − v h ) X h2 T (σuh + ∇ph , σv h )T + δ ν (∇ · uh , ∇ · v h ) +α ν T ∈Th. + hµeff ∇v h · n, uh i . 1. Using the Cauchy–Schwarz inequality, the equality (3.9a), the inequality (3.11) and k∇ · v h k0 6 n 2 k∇v h k0 , we obtain 2  1   1  1 kph k0 1 2 2 Ah [(uh , ph ) ; (v h , 0)] > −µeff µeff k∇uh k0 k∇v h k0 − σ 2 σ 2 kuh k0 kv h k0 + ν ! 21 2 X X σh 2 T − cSZ h2T k∇ph k0,T k∇v ph k0 − ασ kuh k0,T kv h k0,T ν T ∈Th T ∈Th | {z } =:T1. X σh2 1 1 1 T k∇ph k0,T kv h k0,T −δ ν 2 k∇ · uh k0 n 2 ν 2 k∇v h k0 ν T ∈Th | {z }. −α. =:T2. + hµeff ∇v h · n, uh i .. (3.12).

(13) 11 The terms T1 and T2 introduced above can be estimated using the Cauchy–Schwarz inequality and the inequalities (3.10), yielding T1 = ασ. X σh2 kph k0 1 1 1 T kuh k0,T kv h k0,T 6 ασ 2 kuh k0 σ 2 kv h k0 6 cΩ ασ 2 kuh k0 1 ν ν2 T ∈Th | {z }. (3.13). 61. and. T2 = α. X. σ. 1 2. . T ∈Th. h2T ν.  12 .  12. σh2T ν | {z. k∇ph k0,T kv h k0,T 6 α. 6α. X h2 2 T k∇ph k0,T ν. ! 12. X h2 2 T k∇ph k0,T ν. `Ω cΩ kph k0 6 cΩ α σ ν 1 2. T ∈Th. ! 21 1. σ 2 kv h k0. T ∈Th. }. 61. X h2 2 T k∇ph k0,T ν ! 12. kph k0 1. ν2. T ∈Th. .. (3.14). For the boundary term we apply the Cauchy–Schwarz inequality, the inequality (3.3b), and the estimate (3.10) to derive ! 21 X. hµeff ∇v h · n, uh i 6. µeff hE k∇v h ·. X µeff 2 kuh k0,E hE. 2 nE k0,E. E∈Gh. E∈Gh. ! 12. 1 2. 6 cDTI. ! 21. X. µeff. X. 2 k∇v h k0,T E. E∈Gh 1 2 6 cΩ cDTI. E∈Gh. kph k0  µeff  12 1 ν 2 | ν{z }. µeff 2 kuh k0,E hE ! 21. X µeff 2 kuh k0,E hE. ! 21. .. (3.15). E∈Gh. 1. =θ 2. Inserting (3.13), (3.14), and (3.15) into (3.12), using the estimates (3.10) and (3.9b), and rearranging the terms one obtains Ah [(uh , ph ) ; (v h , 0)] >. 2  1  kp k kph k0 kph k0 1 h 0 2 − cΩ µeff k∇uh k0 − cΩ (1 + α) σ 2 kuh k0 1 1 ν ν2 ν2 1 !   2 X h2 √ kph k0 cSZ ĉΩ 2 T √ α − + cΩ α k∇ph k0,T 1 ν α ν2 T ∈Th. − cΩ (nδ). 1 2. . δ ν k∇ ·. 2 uh k0.  21 kp k h 0 ν. 1 2. 1 2. − cΩ cDTI. X E∈Gh. µeff 2 θ kuh k0,E hE. We now define ( C10. := max. 2. c2Ω. (cSZ ĉΩ + cΩ α) (1 + α) , , c2Ω nδ, c2Ω cDTI α 2. ). ! 21. kph k0 1. ν2. ..

(14) 12 and using the Young inequality we obtain the estimate 2. 1 kph k0 − C1 Ah [(uh , ph ) ; (v h , 0)] > 2 ν. X h2 X µeff 2 2 T |||uh ||| + α k∇ph k0,T + θ kuh k0,E ν hE 2. T ∈Th. ! ,. E∈Gh. with C1 := 3C10 .. . Remark 3.2 (On the behavior of C1 ). The constant C1 in (3.8) depends only on the stabilization parameters α and δ, on the domain Ω, and on the discretization (through the constants n, cΩ , ĉΩ , cSZ , and cDTI ). In particular C1 ∼ α1 for α  1. The next step concerns the stability of the proposed formulation with respect to the boundary velocity. To this aim, we will show that the skew-symmetric Nitsche terms in (2.8a) yield a stable formulation by defining two particular test functions that provide control of the boundary norms of the velocity. The construction of the first test function and its main properties are stated in the following Lemma. h Lemma 3.3. For any uh ∈ V h we define wu h ∈ V h such that, for each mesh node x, it holds  uh (x) , for x ∈ Γ, h wu (x) := h 0, for x ∈ Ω \ Γ.. (3.16). h Then the function wu h satisfies the following properties: (1) There exist two positive constants c0 and c1 , depending only on the regularity of the mesh, such that X µeff 2 2 h kuh k0,E − c1 µeff k∇uh k0 . (3.17) hµeff ∇wu · n, u i > c h 0 h hE. E∈Gh. (2) There exists a constant c2 > 0, depending only on the regularity of the mesh, such that X µeff 2 h 2 µeff k∇wu kuh k0,E . h k 0 6 c2 hE. (3.18). E∈Gh. (3) There exists a constant c3 > 0, depending on the mesh regularity, such that h kwu h k0 6 c3 kuh k0 ,. (3.19). (4) There exists a constant ce3 > 0, depending on the mesh regularity, such that 2. 2. uh 2 h kwu h k0,T 6 ce3 hT k∇w h k0,T ,. ∀T ∈ Th .. (3.20). h Proof: Let us consider uh ∈ V h and let wu h ∈ V h be defined as in (3.16). In the following proof, for an edge E ∈ Gh with vertices x1 and x2 we will denote the (unique) attached triangle by T E = conv {x0 , x1 , x2 }. h (1) In order to prove (3.17), let us introduce wE : R2 → R2 as the linear function that coincides with wu h in uh 2 T E and extends it everywhere in R . Since wh (x0 ) = 0, it holds. h (∇wu h · nE )|E =. wE (x⊥ ) , hE,⊥. where x⊥ is the perpendicular foot of the vertex x0 and hE,⊥ is the height of the triangle T E with respect to the edge E. Depending on the shape of T E , x⊥ might fall inside or outside the edge E. Formally, there exists an a ∈ R, such that x⊥ = ax1 + (1 − a) x2 ,. |a| + |1 − a| 6 M,. (3.21).

(15) 13 where M > 0 depends only on the mesh regularity constant. Hence, by adding and subtracting uh we can reformulate 1 1 hwE (x⊥ ) , uh iE = (huh , uh iE − huh − wE (x⊥ ) , uh iE ) hE,⊥ hE,⊥   1 hE 1 2 kuh k0,E − huh − wE (x⊥ ) , uh iE . = hE,⊥ hE hE,⊥. h h∇wu h · n, uh iE =. (3.22). Exploiting (3.21), the linearity of wE , and the fact that wE coincindes with uh on E, we get, for all x ∈ E, |uh (x) − wE (x⊥ )| = |uh (x) − (auh (x1 ) + (1 − a) uh (x2 ))| 6 (|a| |uh (x) − uh (x1 )| + |1 − a| |uh (x) − uh (x2 )|) 6 M (|uh (x) − uh (x1 )| + |uh (x) − uh (x2 )|) 6 2M hE |(∇uh ) |T E | , where | · | stands for the Euclidean norm. Since ∇uh is constant on T E , it holds also 1. k∇uh k0,T E = |T E | 2 |(∇uh ) |T E | ,. (3.23). from which we deduce |(∇uh ) |T E | 6 ch−1 E k∇uh k0,T E , where the constant c > 0 only depends the regularity of the mesh. The above arguments allow to conclude 1. 3. 1. kuh − wE (x⊥ )k0,E 6 hE2 max |uh (x) − wE (x⊥ )| 6 2M hE2 |(∇uh ) |T E | 6 cΓ hE2 k∇uh k0,T E , x∈E. (3.24). with cΓ := 2M c. Thus, applying the Cauchy–Schwarz inequality, the inequality (3.24), and the Young inequality yields 1 1 hE 2 2 huh − wE (x⊥ ) , uh iE 6 cΓ hE2 k∇uh k0,T E kuh k0,E 6 kuh k0,E + c2Γ k∇uh k0,T E . 2 2 Combining this inequality with (3.22) leads to h hµeff ∇wu h. 1 hE · n, uh iE > 2 hE,⊥. . µeff 2 kuh k0,E hE. . − c2Γ.  hE  2 µeff k∇uh k0,T E . 2hE,⊥. The proof is concluded taking the sum over all boundary edges and defining c0 :=. 1 min 2 E∈Gh. . hE hE,⊥.  c1 :=. c2Γ max 2 E∈Gh. . hE hE,⊥.  ,. (3.25). which are only dependent on the shape regularity of the mesh. 2 uh 2 h (2) First of all, since wu h and uh coincide on E, it holds kw h k0,E = kuh k0,E . Moreover, let us consider a triangle T = conv {x0 , x1 , x2 } such that T ∩ Γ 6= ∅, assuming (without loss of generality) that x0 6∈ Γ, x1 ∈ Γ and denoting with N ∈ {1, 2} the number of vertices T has on the boundary. Using the linearity of uh , it holds, for an appropriate c > 0 depending only on mesh regularity, 2. h k∇wu h k0,T.  N c P |u (x )|2 , h i 6  i=1 0,. if T ∩ Γ 6= ∅, otherwise..

(16) 14 Hence, denoting by NΓ the total number of boundary nodes, and by cNB the maximum number of triangles adjacent to a boundary node (which can be bounded, e.g., depending on the smallest angle of the triangulation Th ), one can write NΓ X X µeff 2 2 h 2 kuh kE . µeff k∇wu k 6 µ c c |uh (xi )| 6 c2 eff NB h 0 h E i=1 E∈Gh. where c2 depends only on the regularity of the mesh. (3) The inequality (3.19) can be proven using scaling arguments similar to the previous ones, observing that h wu h and uh coincide on each boundary edge. h (4) Also the inequality (3.20) follows by standard scaling arguments, exploiting that wu h is a component-wise linear function that vanishes on interior nodes of the mesh.  Remark 3.3 (Extension to higher order finite elements). It is worth noticing that an analogous of this Lemma h can be also proven for higher order finite elements, using the same definition of the function wu h with different definitions of the constants c0 , c1 , c2 , and c3 . In particular, some of the equalities (due to the fact that both uh h and wu h are linear), e.g., (3.23) have to be replaced with inequalities obtained by proper scaling arguments. h Using the above defined function wu h , the next lemma allows to state stability of the boundary velocity.. Lemma 3.4 (Boundary control - I). Let α, δ, ρ, µeff , σ > 0 with µeff + σ > 0. For any (uh , ph ) ∈ V h × Qh , there exist a function wh ∈ V h and a constant C2 = C2 (α, δ) > 0 which is independent from the physical parameters, from uh , and from h, such that c0 X µeff 2 θ kuh k0,E − C2 Ah [(uh , ph ) ; (wh , 0)] > 4 hE E∈Gh. X h2 2 T k∇ph k0,T |||uh ||| + α ν 2. ! ,. T ∈Th. where c0 is the constant defined in Lemma 3.3. uh h Proof: For a given pair (uh , ph ) ∈ V h × Qh , let wh := θwu h , where w h is the function defined in Lemma 3.3. Then, we get uh uh uh h Ah [(uh , ph ) ; (wh , 0)] = θµeff (∇uh , ∇wu h ) + θ (σuh , w h ) − θ hµeff ∇uh · n, w h i + θ hµeff ∇w h · n, uh i X h2 uh T h h (σuh + ∇ph , σwu − θ (ph , ∇ · wu h )T h ) + θ hph n, w h i + αθ ν T ∈Th X 2 uh + δ νθ (∇ · uh , ∇ · wh ) + ρ νθ |Juh · nK (x)| . x∈C. Observing that the corner stabilization is always positive and that θ 6 1, using the Cauchy–Schwarz inequality, the inequalities (3.17), (3.18), and (3.19) leads to ! 12 µeff 2 2 θ θ kuh k0,E − σc3 kuh k0 Ah [(uh , ph ) ; (wh , 0)] > −c2 µeff k∇uh k0 |{z} hE E∈Gh 61 X µeff 2 2 h − θ hµeff ∇uh · n, wu i + c θ kuh k0,E − c1 µeff k∇uh k0 0 h hE 1 2. 1 2. 1 2. X. E∈Gh. uh h − θ (ph , ∇ · wu h ) + θ hph n, w h i + αθ. X T ∈Th. h + δ νθ (∇ · uh , ∇ · wu h ).. h2T ν. h (σuh + ∇ph , σwu h )T. (3.26).

(17) 15 h Combining the Cauchy–Schwarz inequality, the trace inequality (3.3b), and the fact kwu h k0,E = kuh k0,E we obtain. h n, wu h i. θ hµeff ∇uh ·. 1 2. 1 2. X. θ 6 (cDTI µeff ) k∇uh k0 |{z}. E∈Gh. 61. µeff 2 θ kuh k0,E hE. ! 12 ,. which, inserted in (3.26), yields 1 2. . 1 2. . 1 2. Ah [(uh , ph ) ; (wh , 0)] > − c2 + cDTI µeff k∇uh k0. X E∈Gh. X. + c0. E∈Gh. µeff 2 kuh k0,E θ hE. ! 21 2. − σc3 kuh k0. µeff 2 2 θ kuh k0,E − c1 µeff k∇uh k0 hE. X h2 uh T h h + αθ (σuh + ∇ph , σwu −θ (ph , ∇ · wu ) + θ hp n, w i h h )T h h ν {z } | T ∈Th =:Q1 {z } |. (3.27). =:Q2. h + δ νθ (∇ · uh , ∇ · wu h ). {z } |. =:Q3. In order to bound the term Q1 , we use the integration by parts formula, the Cauchy–Schwarz inequality, the inequalities (3.20), (3.18), and θν = µeff to obtain. Q1 =. X T ∈Th. α. 6. X h2 2 T h θ (∇ph , wu k∇ph k0,T h )T 6 θ α ν T ∈Th. X h2 2 T k∇ph k0,T ν. ! 12 . θ2 ν ce3. T ∈Th.  6. ! 12. c2 ce3 α.  21. X h2 2 T α k∇ph k0,T ν. 1 h 2 k∇wu h k0 α. ! 21 X. T ∈Th. E∈Gh. ν h 2 kwu h k0,T αh2T. X T ∈Th. ! 12.  21. µeff 2 θ kuh k0,E hE. ! 12 .. (3.28). Next, we observe that the term Q2 , coming from the pressure stabilization, can be bounded using the Cauchy– Schwarz inequality, Young’s inequality, and (3.19) as. 1. Q2 > −θα 2. α. X h2 2 T k∇ph k0,T ν. T ∈Th. ! 21.  21. .   X σh2T 2 h 2   σ kwu h k0,T  − θασc3 kuh k0  ν T ∈Th | {z } 61. >− α. X h2 2 T k∇ph k0,T ν. ! 21. 1. 2. (ασ) 2 c3 kuh k0 − ασc3 kuh k0. T ∈Th.  1 X h2T 2 2 >− α k∇ph k0,T − c23 + c3 ασ kuh k0 . 4 ν T ∈Th. (3.29).

(18) 16 1. 1. 2 allow also to conclude Finally, (3.18) and (νθ) 2 = µeff. 1 2. Q3 > −δ (νθn) k∇ · uh k0 (νθ). 1 2. h k∇wu h k0. > − (δnc2 ). 1 2. . δ ν k∇ ·. 2 uh k 0.  21. X E∈Gh. µeff 2 θ kuh k0,E hE. ! 12 .. (3.30). We observe that scaling the test function by θ allows to, on the one hand, assure coercivity in the chosen norm, and, on the other hand, to obtain a parameter independent estimate for the terms involving ν∇ · wh . Notice as well that the scaling by θ implies that the test function vanishes in the Darcy limit (µeff = 0). Inserting (3.28), (3.29), and (3.30) into (3.27), and reordering the terms yields  µeff 2 2 2 kuh k0,E − c1 µeff k∇uh k0 − c3 + α c23 + c3 σ kuh k0 hE E∈Gh ! 21  1  1 X µeff 1 1 X h2T 2 2 2 2 µeff kuh k0,E k∇ph k0,T k∇uh k0 θ − α − c22 + cDTI hE 4 ν X. Ah [(uh , ph ) ; (wh , 0)] > c0. θ. T ∈Th. E∈Gh.  −. c2 ce3 α.  21 α. X h2. T. T. ν. ! 12 X. 2. k∇ph k0,T. E. 1  1 2 2 − (δnc2 ) 2 δ ν k∇ · uh k0. X. θ. E∈Gh. θ. ν 2 kuh k0,E hE ! 21. µeff 2 kuh k0,E hE. ! 12. .. Applying three times the Young inequality yields, for any ε > 0,  µeff 2 2 2 kuh k0,E − c1 µeff k∇uh k0 − c3 + α c23 + c3 σ kuh k0 hE E∈Gh  1 2 1 2 c22 + cDTI ε X µeff 1 X h2T 2 2 2 µeff k∇uh k0 − θ kuh k0,E − α k∇ph k0,T − 2ε 2 hE 4 ν E∈Gh T ∈Th ! ! 2 Xh c2 ce3 ε X µeff 2 2 T − α k∇ph k0,T − θ kuh k0,E 2αε ν 2 hE E T !   X δnc2 ε µeff 2 2 − δ ν k∇ · uh k0 − θ kuh k0,E . 2ε 2 hE. Ah [(uh , ph ) ; (wh , 0)] > c0. X. θ. E∈Gh. Choosing ε =. c0 2. allows to conclude . Ah [(uh , ph ) ; (wh , 0)] >. . 1 2. 1 2. c2 + cDTI c0 X µeff  2 θ kuh k0,E − c1 + 4 hE c0. 2   2  µeff k∇uh k0. E∈Gh. − c3 + α c23 + c3. . 2. σ kuh k0 −.  δnc2  2 − δ ν k∇ · uh k0 . c0. . 1 c2 ce3 + 4 αc0.  α. X h2 2 T k∇ph k0,T ν. T ∈Th.

(19) 17 The proof is completed defining. C2 := max.   .  c1 +. 1. 1. 2 c22 + cDTI. 2. c0.  .    1 c2 ce3 δnc2  2 , c3 + α c3 + c3 , + , . 4 αc0 c0   . Remark 3.4 (On the behavior of C2 ). It is worth noticing that the constant C2 is bounded for any choice of the stabilization parameters. In particular, C2 grows like c−1 0 , which is related to the anisotropy of the mesh near the boundary (see (3.25)). Moreover, as it has been stated in Remark 2.3, the stability estimate does not rely on the particular technique chosen for the stabilization of the equal-order finite element (i.e., α > 0 is not strictly required). The last step needed to show the fulfillment of the inf-sup condition is related to the control of the normal velocity at the boundary, which is particularly relevant in order to guarantee stability towards the Darcy limit, i.e., for σ`Ω 2  µeff and especially µeff = 0. It is worth recalling our assumption (2.4) on the mesh, stating that for any two adjacent boundary edges E, E 0 ∈ Gh it holds √ hE (3.31) 6 η0 < 7 + 4 3 (≈ 13.9) . hE 0 We observe that this assumption is weaker than quasi-uniformity of the mesh, as it only restricts the ratio between the lengths of adjacent boundary edges. Moreover, let us also recall that the mesh is assumed to satisfy (2.5), i.e., that the area of inner triangles is larger than the area of the boundary triangles. Lemma 3.5. Let us assume that the family of triangulations {Th }h satisfies (3.31) and (2.5). For a given uh ∈ V h , let us define qhuh ∈ Qh as the function whose values at the boundary nodes are uniquely defined to satisfy the L2 -projection property hqhuh , ϕh i = −. X 1 huh · nE , ϕh iE , hE. ∀ϕh ∈ Qh .. (3.32). E∈Gh. and its value at the interior nodes is given by a constant cq , chosen in order to satisfy Then the function qhuh has the following properties: (1) There exists a constant c4 > 0, depending only on η0 , such that X. 2. hE kqhuh k0,E 6 c4. E∈Gh. q uh Ω h. R. = 0.. X 1 2 kuh · nE k0,E . hE. (3.33). E∈Gh. (2) There exists a constant c5 > 0, depending only on η0 , such that − hqhuh , uh · ni >. X 1 X 1 2 2 kuh · nE k0,E − c5 |Juh · nK (x)| . 2 hE E∈Gh. (3.34). x∈C. (3) There exists a constant c6 > 0, depending only on the properties of the mesh, such that 2. kqhuh k0 6 c6. X 1 2 kuh · nE k0,E hE. E∈Gh. (3.35).

(20) 18 and X. 2. h2T k∇qhuh k0,T 6 c6 c2I. T ∈Th. X 1 2 kuh · nE k0,E . hE. (3.36). E∈Gh. Proof: (1) In order to prove (3.33), let us restrict for simplicity, and without loss of generality, to the case of a boundary with a single connected component. In this case, let us number the boundary nodes as x1 , . . . , xN and the boundary edges as E1 , . . . , EN such that the edge Ei connects the nodes xi and xi+1 , for all i ∈ {1, . . . , N }. Moreover, we identify xN +1 with x1 , so that the above defined convention is well-defined also for i = N . To simplify the notations, let us abbreviate hi = hEi and qi = qhuh (xi ). We now consider a function ϕh ∈ Qh defined at each node x of the mesh by ( hi qi , x ∈ Γ, ϕh (x) := cϕ , otherwise, R where cϕ is a constant defined in order to have Ω ϕ = 0. On any boundary edge Ei , by the linearity of qhuh and ϕh , application of the Simpson rule yields Z  hi 2 2hi qi2 + (hi + hi+1 ) qi qi+1 + 2hi+1 qi+1 qhuh ϕh = 6 Ei and. Z.  hi 2 2 qi + qi qi+1 + qi+1 . 3 Ei We will first prove that there exists a constant ĉ > 0, independent from qhuh and hi , such that Z Z hi |qhuh |2 6 ĉ qhuh ϕh . |qhuh |2 =. Ei. (3.37). Ei. i 6 ĉ, i.e., with ĉ > η0 . Assume now that qi 6= 0 and set η := If qi = 0, (3.37) holds if hhi+1 The inequality (3.37) then reduces to. 1 + t + t2 6. hi+1 hi. and t :=. qi+1 qi ..  ĉ 2 + (1 + η) t + 2ηt2 . 2. Since η > 0, η ∈ [1/η0 , η0 ], and 1 + t + t2 > 0, for all t ∈ R, the above condition is equivalent to. 1 η0. inf 6η6η0 t∈R. 2 + (1 + η) t + 2ηt2 > 0. 1 + t + t2. Since the polynomial in the denominator is always strictly positive, the whole infimum is positive if the numerator as a polynomial in t is strictly positive for all t ∈ R and for all η ∈ [1/η0 , η0 ]. This is the case if and only if its discriminant does not have real roots for the selected range of η. The discriminant of this √ √ polynomial √  is 2 D (η) := (1 + η) − 16η, which vanishes if η = 7 ± 4 3 and is strictly negative if η ∈ 7 − 4 3, 7 + 4 3 =  √  √ 1√ , 7 + 4 3 . Hence, D(η) is negative if η0 < 7+4 3 (i.e., assumption (3.31)). Estimate (3.37) is therefore 7+4 3 proven by setting   −1     2 2 + (1 + η) t + 2ηt    ĉ = ĉ(η0 ) := max η0 , 2  1 inf .    1 + t + t2   η0 6η6η0 t∈R.

(21) 19 The assumption. hi+1 hi. 6 η0 yields also, on any boundary edge Ei. 1 2 kϕh k0,Ei 6 hi hi. hi+1 1+ + hi. . hi+1 hi. 2 !.  2 2 kqhuh k0,Ei 6 1 + η0 + η02 hi kqhuh k0,Ei .. (3.38). Summing (3.37) over all boundary edges and using (3.32), the Cauchy–Schwarz inequality, (3.38), and the Young inequality yields X. X 1 huh · nE , ϕh iE hE E∈Gh ! 12 X 1 2 kuh · nE k0,E 6 ĉ hE. 2. hE kqhuh k0,E 6 ĉ hqhuh , ϕh i = −ĉ. E∈Gh. E∈Gh. X 1 2 kϕh k0,E hE. ! 12. E∈Gh.  X ε X 1 1 2 2 kuh · nE k0,E + 1 + η0 + η02 hE kqhuh k0,E 2 hE 2ε. 6 ĉ. E∈Gh. ! .. E∈Gh.  Choosing ε := ĉ 1 + η0 + η02 leads to X. 2. hE kqhuh k0,E 6 ĉ2 1 + η0 + η02. E∈Gh.  X 1 2 kuh · nE k0,E . hE E∈Gh.  2. Estimate (3.33) is obtained defining c4 := ĉ2 1 + η0 + η0 , which only depends on η0 . (2) To prove the second inequality, let us consider the function ϕh ∈ Qh such that, at the mesh nodes x, it holds  for x ∈ Γ \ C,  uh · n (x) , 1 0 (x)) , ϕh (x) := (u · n (x) + u · n for x ∈ C, with E, E 0 two adjacent boundary edges, h E h E  2 cϕ , otherwise, R with a constant cϕ defined in order to have Ω ϕh = 0. Remember that C is the set of corner nodes at the boundary. Using (3.32), the Cauchy–Schwarz, and the Young inequalities we obtain, for any ε > 0, X 1 X u huh · nE , ϕh iE − hqh h , uh · nE − ϕh iE hE E∈Gh E∈Gh X X u 1 2 kuh · nE k0,E − huh · nE , uh · nE − ϕh iE − hqh h , uh · nE − ϕh iE hE E∈Gh E∈Gh   ε X 1 ε 2 2 2 1− kuh · nE k0,E − kuh · nE − ϕh k0,E − hE kqhuh k0,E . (3.39) 2 ε 2. − hqhuh , uh · ni = − hqhuh , ϕh i − hqhuh , uh · n − ϕh i = =. X 1 hE. E∈Gh. >. X 1 hE. E∈Gh. E∈Gh. The function (uh · n − ϕh ) |E∈Gh is different from zero only on boundary edges that are adjacent to a corner. In particular, let us consider a corner node xc ∈ C with an adjacent edge E = xi xc . It holds |(uh · nE − ϕh ) (xc )| =. 1 |Juh · nE K (xc )| 2. and. (uh · nE − ϕh ) (xi ) = 0,. which yields (Simpson rule) 2. kuh · nE − ϕh k0,E =. 1 2 hE |Juh · nE K (xc )| . 12. (3.40).

(22) 20 Thus, inserting (3.33) and (3.40) into (3.39) and choosing ε := − hqhuh , uh · ni >. 1 c4 +1. we obtain. X 1 X 1 2 2 kuh · nE k0,E − c5 |Juh · nK (x)| 2 hE x∈C. E∈Gh. +1 , which only depends on η0 . with c5 := c412 (3) To prove (3.35), let us first introduce a continuous, element-wise linear function q0uh , which coincides with. qhuh on the boundary Γ and vanishes at all the interior nodes, and a continuous, element-wise linear function ψh vanishing on the boundary and equal to 1 at all the interior nodes. Let us observe that, using an argument analogous to the one of Lemma 3.3 (omitted here), it holds 2. X. kq0uh k0 6 cNB. 2. hE kqhuh k0,E ,. (3.41). E∈Gh. where cNB is the maximum number of triangles adjacent to a boundary node. Moreover, we have qhuh = q0uh + cq ψh . From. q uh Ω h. R. = 0 we obtain 1. R uh q cq = − RΩ 0 ψ Ω h. kq uh k |Bh | 2 |cq | 6 0 R 0 , ψ Ω h. and. since q0uh is different from 0 only on Bh . Hence, using the assumption (2.5), 1. kqhuh k0. 6. kψh k0 |Bh | 2 R 1+ ψ Ω h. !. 1. kq0uh k0. 6. 1. |Ω| 2 |Bh | 2 1+ |Ω| − |Bh |. !. 1. kq0uh k0. 6. ω2 1+ 1−ω. ! kq0uh k0 .. (3.42). Since 0 < ω < 1, the coefficient inside the parentheses is always strictly larger than one (and it approaches one on fine meshes). Inserting (3.41) into (3.42) and using (3.33) we obtain (3.35) with a constant c6 depending on cNB , η0 , and ω. Finally, (3.36) can be obtained combining the inverse inequality (3.2) on each triangle and (3.35): X. 2. h2T k∇qhuh k0,T 6 c2I. T ∈Th. X. 2. kqhuh k0,T 6 c6 c2I. T ∈Th. X 1 2 kuh · nE k0,E . hE. E∈Gh.  Finally, we are able to show control of the normal velocity for arbitrary values of physical parameters. Lemma 3.6 (Boundary control - II). Let δ, ρ > 0, α > 0, and µeff , σ > 0 with µeff + σ > 0 and let us assume that the family of triangulations {Th }h satisfies (3.31) and (2.5). Then, for any (uh , ph ) ∈ V h × Qh , there exists a function qh ∈ Qh and a constant C3 = C3 (α, δ, ρ) > 0 independent from the physical parameters, from uh , and from h, such that 1 X ν 2 kuh · nE k0,E − C3 Ah [(uh , ph ) ; (0, qh )] > 4 hE E∈Gh. |||uh ||| + ρ ν. X x∈C. X h2 2 T |Juh · nK (x)| + α k∇ph k0,T ν 2. T ∈Th. ! ..

(23) 21 Proof: Let (uh , ph ) ∈ V h × Qh and let qh := νqhuh , where qhuh is the function defined in Lemma 3.5. Using the Cauchy–Schwarz inequality, Young inequality, (3.34), (3.35), and (3.36), we obtain X h2 T (σuh + ∇ph , ν∇qhuh )T − hνqhuh , uh · ni ν T ∈Th  0 X h2  εν uh 2 1 2 2 uh 2 2ε 2 1 T kuh k0,T + ν > − νδ k∇ · uh k0 − kq k − α σ k∇qh k0,T 2ε 2δ h 0 ν 2ε0 2 T ∈Th  0 X h2  1 2 uh 2 2ε T k∇ph k0,T + ν −α k∇qh k0,T − ν hqhuh , uh · ni ν 2ε0 2 T ∈Th   X X 1 ε ν c5 2 2 2 0 > − c6 − αcI c6 ε kuh · nE k0,E − ρ ν |Juh · nK (x)| 2 2δ hE ρ. Ah [(uh , ph ) ; (0, qh )] = (∇ · uh , νqhuh ) + α. E∈Gh. x∈C. 1 1 X σh2T 1 X h2T 2 2 2 − νδ k∇ · uh k0 − α 0 σ kuh k0,T − α 0 k∇ph k0,T , 2ε 2ε ν 2ε ν | {z } T ∈Th. T ∈Th. 61. for any ε, ε0 > 0. The proof is completed choosing ε =  C3 := max. δ 4c6. and ε0 =. 1 , 8αc6 c2I. c5 2c6 , , 4α2 c6 c2I , 4αc6 c2I ρ δ. and defining.  ,. which depends only on the shape-regularity of the mesh and on the three stabilization parameters.. . Remark 3.5. As stated in Section 2.3, the scaling of the stabilization terms by ν is a necessary requirement in order to obtain stability estimates independent from the physical parameters. In the argument used for the last proof, using an element-dependent scaling νT := µeff + σh2T (a suitable alternative for the case of essential boundary conditions, see, e.g., [3]) instead of ν for the Grad-Div stabilization does not allow to uniformly bound the term (∇ · uh , νqhuh ). Remark 3.6 (On the behavior of C3 ). Notice that, in order to assure the validity of Lemma 3.6,  both, Grad-Div and corner stabilization, are required (i.e., δ, ρ > 0). In particular, it holds C3 = O δ −1 + ρ−1 for small values of δ and ρ. Moreover, as already observed in Remark 3.4, α > 0 is not strictly required. The previously proven Lemmata allow to prove inf-sup stability of the considered formulation (2.7), which is stated in the following theorem. Theorem 3.1 (Inf-sup stability). Let α, δ, ρ > 0 and µeff , σ > 0 with µeff + σ > 0, and let us assume that the family of triangulations {Th }h fulfills the assumptions stated in Lemma 3.5. Then there exists a constant β > 0, independent from the physical parameters and from h, such that  inf. (uh ,ph )∈V h ×Qh \{(0,0)}. sup (v h ,qh )∈V h ×Qh \{(0,0)}.  Moreover, β −1 = O α−1 δ −1 + ρ−1 , for α, δ, ρ  1.. Ah [(uh , ph ) ; (v h , qh )] |||(uh , ph )|||h |||(v h , qh )|||h. ! > β..

(24) 22 Proof: Let (uh , ph ) ∈ V h × Qh . For the sake of simplicity, let us introduce the following notation: 2. ξ0 := |||(uh , ph )|||h −. 2 X µeff X ν kph k0 2 2 − θ kuh k0,E − kuh · nE k0,E , ν hE hE E∈Gh. ξ1 := ξ2 :=. 2 kph k0. ν X E∈Gh. ξ3 :=. E∈Gh. ,. θ. µeff 2 kuh k0,E , hE. X ν 2 kuh · nE k0,E , hE. E∈Gh 2. such that |||(uh , ph )|||h = ξ0 + ξ1 + ξ2 + ξ3 . Now we can rewrite and summarize the estimates proven in Lemmata 3.1, 3.2, 3.4, and 3.6 as   Ah (uh , ph ) ; C0−1 (uh , ph ) > ξ0 , Ah [(uh , ph ) ; 2 (v h , 0)] > ξ1 − 2C1 (ξ0 + ξ2 ) > ξ1 − 2C1 (ξ0 + ξ2 + ξ3 ) ,   4C2 uh Ah (uh , ph ) ; 4c−1 ξ0 , 0 (θw h , 0) > ξ2 − c0 uh Ah [(uh , ph ) ; 4 (0, νqh )] > ξ3 − 4C3 ξ0 . Summing up the last two inequalities leads to   uh uh Ah (uh , ph ) ; 4c−1 > (ξ2 + ξ3 ) − Ĉ2 ξ0 , 0 θw h , 4νqh b2 := where C. 4C2 c0.  + 4C3 . Consider first a test function z 1h , rh1 ∈ V h × Qh of the form   uh uh z 1h , rh1 := (1 − η1 ) C0−1 (uh , ph ) + η1 4c−1 , 0 θw h , 4νqh. depending on a parameter η1 ∈ (0, 1) which will be determined later. It holds     b2 η1 ξ0 + η1 (ξ2 + ξ3 ) . Ah (uh , ph ) ; z 1h , rh1 > 1 − η1 − C Hence, defining η1 := . 1 b2 + 2 C.  ∈. 1 0, 2. . . b2 η1 = η1 and thus gives 1 − η1 − C   Ah (uh , ph ) ; z 1h , rh1 >. 1 (ξ0 + ξ2 + ξ3 ) . b C2 + 2.  Next, consider a test function z 2h , rh2 ∈ V h × Qh of the form     b2 + 2 z 1h , rh1 + η2 (2v h , 0) , z 2h , rh2 := (1 − η2 ) C depending on a parameter η2 ∈ (0, 1) to be determined later. This yields   Ah (uh , ph ) ; z 2h , rh2 > (1 − η2 − 2C1 η2 ) (ξ0 + ξ2 + ξ3 ) + η2 ξ1 ..

(25) 23 Therefore, the choice η2 :=. 1 ∈ 2C1 + 2.  0,. 1 2. . leads to. 1 1 2 (ξ0 + ξ1 + ξ2 + ξ3 ) = |||(uh , ph )|||h . 2C1 + 2 2C1 + 2 It remains to control the norm of the above constructed test function z 2h , rh2 . From the properties of v h stated in (3.10) we have   Ah (uh , ph ) ; z 2h , rh2 >. 2. 2. 2. 2. 2. |||(v h , 0)|||h = µeff k∇v h k0 + σ kv h k0 + δ ν k∇ · v h k0 6 c2Ω (1 + nδ). kph k0 . ν. Moreover, from Lemmas 3.3, and Equations (3.35) and (3.36) we infer  2 2 2 h |||(θwu h , 0)|||h 6 c3 + c2 + δnc2 + 1 |||(uh , ph )|||h , 2. |||(0, νqhuh )|||h =. X h2  X ν 1 2 2 2 T kνqhuh k0 + α kν∇qhuh k0,T 6 c6 1 + αc2I kuh · nE k0,E . ν ν hE T ∈Th. Hence, since 0 < η1 < z 2h , rh2. . 2 h. 1 2. and 0 < η2 <. 1 2,. E∈Gh. we can estimate.  2  2 2 b2 + 2 6 η22 |||(2v h , 0)|||h + (1 − η2 )2 C z 1h , rh1 h 2    2 b2 + 2 η12 16c−2 |||(θwuh , 0)|||2 + 16|||(0, νq uh )|||2 6 |||(v h , 0)|||h + (1 − η2 )2 C 0 h h h h 2. + (1 − η1 )2 C0−2 |||(uh , ph )|||h  2 b2 + 2 < c2Ω (1 + nδ) + 4 C. ! !   c23 + c2 + δnc2 + 1 2 −2 2 + c6 1 + αcI + C0 |||(uh , ph )|||h c20. which allows to conclude   Ah (uh , ph ) ; z 2h , rh2 > β |||(uh , ph )|||h. z 2h , rh2.  h. with β = (2C1 + 2). −1. c2Ω. . b2 + 2 (1 + nδ) + 4 C. 2. ! !− 12   c23 + c2 + δnc2 + 1 −2 2 + c6 1 + αcI + C0 . c20.  The behavior for small values of stabilization parameters follows from C0 = O(1), C1 = O α−1 , C2 = O(1), and C3 = O δ −1 + ρ−1 . . 3.3. Convergence Firstly, let us observe that the discrete, stabilized, penalty-free, non-symmetric Nitsche formulation (2.7) is consistent with problem (1.1): Lemma 3.7 (Consistency and Galerkin Orthogonality). Assume that (u, p) ∈ H 2 (Ω) × H 1 (Ω) solves (1.1) satisfying either (1.2) (if µeff > 0) or (1.3) (if µeff = 0). Moreover, let (uh , ph ) ∈ V h × Qh be the solution of (2.8). Then, Ah [(u − uh , p − ph ) ; (v h , qh )] = 0,. ∀ (v h , qh ) ∈ V h × Qh ..

(26) 24 The result follows from the consistency of the discrete formulation and the conformity of the triangulation. The next lemma is related to the quality of the approximation with respect to the mesh-dependent norm. Lemma 3.8 (Approximability). Let σ, µeff > 0 with µeff + σ > 0. Let IhL (·) and MhSZ (·) be the Lagrange interpolation operator onto V h and the Scott-Zhang quasi-interpolation operator onto Qh , respectively. Moreover, let us assume that (u, p) ∈ H 2 (Ω) × H 1 (Ω). Then it holds    2 1 2 2 (3.43) u − IhL (u) , p − MhSZ (p) h 6 c2LSZ h2 ν (1 + nδ + 2cDT ) kuk2 + (1 + α) kpk1 , ν where cLSZ is a constant that depends only on (3.5) and (3.4b). Proof: We start by estimating the bulk terms of the triple norm using the properties (3.5) of the interpolation operators:  2  2 2 µeff ∇ u − IhL (u) 0 + σ u − IhL (u) 0 + δ ν ∇ · u − IhL (u) 0 X h2  2 1 2 T + p − MhSZ (p) 0 + α ∇ p − MhSZ (p) 0,T ν ν T ∈Th. 6. µeff c2La h2 +. 2 kuk2. +. σc2La h4. 2. 2. kuk2 + nδ νc2La h2 kuk2. 1 2 2 h2 2 2 cSZ h kpk1 + α c2SZ kpk1 ν ν  .   2 6 c2LSZ h2 µeff + σh2 +nδ ν  kuk2 + | {z }.  . . 1 2 (1 + α) kpk1  . ν. 6ν. For the additional boundary terms related to the penalty-free Nitsche method we get with (3.1): X µeff X ν  2 2 θ u − IhL (u) 0,E + u − IhL (u) · nE 0,E hE hE E∈Gh E∈Gh  X  2  2 L L 6 cDT (µeff + ν) h−2 T E u − Ih (u) 0,T + ∇ u − Ih (u) 0,T E. E. T E : E∈Gh 2. 6 cDT (µeff + ν) c2La h2 kuk2 . Finally, we observe that the interpolation error due to the corner stabilization term vanishes since the Lagrange interpolator is exact on mesh nodes. The inequality (3.43) is obtained summing up all the above estimated terms and observing that µeff , σh2 6 ν.  Theorem 3.2 (A priori error estimate). Let α, δ, ρ > 0 and µeff , σ > 0 with µeff + σ > 0. Let (u, p) be the solution of (1.1) and (uh , ph ) be the solution of problem (2.7). Let us assume (u, p) ∈ H 2 (Ω) × H 1 (Ω). Then it holds |||(u − uh , p − ph )|||h 6 cLSZ h (Cu kuk2 + Cp kpk1 ) with. and.   1   1 1 1 1 1 1 1 2 (2cDT cDTI ) 2 + n 2 δ 2 + 1 + 1 + (2cDT ) 2 + α 2 + α Cu := ν 2 (1 + nδ + 2cDTI ) 2 + β   1 1 1 1 1 1 Cp := ν − 2 (1 + α) 2 + (2cDT ) 2 + α 2 + α + δ − 2 . β.

(27) 25 Proof: The proof is based on the combination of the inf-sup condition (Theorem 3.1), the Galerkin orthogonality (Lemma 3.7), Lemma 3.8, and the approximation properties of the (quasi-)interpolation operators. Let us consider the Lagrange interpolant v h := IhL (u) and the Scott-Zhang quasi-interpolant qh := MhSZ (p), and decompose the error as |||(u − uh , p − ph )|||h 6 |||(u − v h , p − qh )|||h + |||(v h − uh , qh − ph )|||h . Exploiting the inf-sup stability (Theorem 3.1) and the Galerkin orthogonality (Lemma 3.7), it holds:. |||(u − uh , p − ph )|||h 6 |||(u − v h , p − qh )|||h +. 1 β. |Ah [(u − v h , p − qh ) ; (wh , rh )]| . |||(wh , rh )|||h (wh ,rh )∈V h ×Qh \{(0,0)} sup. Next, we bound |Ah [(u − v h , p − qh ) ; (wh , rh )]|. For the bulk terms related to the weak formulation of the Brinkman problem we obtain 1. 1. 1. 2 2 2 k∇ (u − v h )k0 µeff k∇wh k0 6 cLa µeff h kuk2 |||(wh , rh )|||h , |µeff (∇ (u − v h ) , ∇wh )| 6 µeff 1. 1. 1. |σ (u − v h , wh )| 6 σ 2 ku − v h k0 σ 2 kwh k0 6 cLa σ 2 h2 kuk2 |||(wh , rh )|||h , |− (p − qh , ∇ · wh )| 6 (δ ν). − 21. 1. − 12. kp − qh k0 (δ ν) 2 k∇ · wh k0 6 cSZ (δ ν) 1 2. |(∇ · (u − v h ) , rh )| 6 ν k∇ · (u − v h )k0 ν. − 12. h kpk1 |||(wh , rh )|||h ,. 1 2. krh k0 6 cLa (νn) h kuk2 |||(wh , rh )|||h .. The GLS stabilization terms yield. α. X h2 1 T (σ (u − v h ) , σwh )T 6 cLa ασ 2 h2 kuk2 |||(wh , rh )|||h , ν. T ∈Th. α. X h2 1 T (σ (u − v h ) , ∇rh )T 6 cLa (ασ) 2 h2 kuk2 |||(wh , rh )|||h , ν. T ∈Th. α. X h2 1 T (∇ (p − qh ) , σwh )T 6 cSZ αν − 2 h kpk1 |||(wh , rh )|||h , ν. T ∈Th. X h2 1 1 T α (∇ (p − qh ) , ∇rh )T 6 cSZ α 2 ν − 2 h kpk1 |||(wh , rh )|||h , ν T ∈Th. while for the terms related to the Grad-Div stabilization, we obtain. δ. X T ∈Th. ν (∇ · (u − v h ) , ∇ · wh )T 6. X. 1. 1. (δ ν) 2 k∇ · (u − v h )k0,T (δν) 2 k∇ · wh k0,T. T ∈Th 1. 6 cLa (nδ ν) 2 h kuk2 |||(wh , rh )|||h ..

(28) 26 The additional terms related to the penalty-free Nitsche method can be controlled as follows. First we treat: ! 21. 1 2. X. |hµeff ∇ (u − v h ) · n, wh i| 6 µeff. hE k∇ (u − v h ) ·. E∈Gh 1 2. 6. µeff. ! 12. E∈Gh. . X. cDT k∇ (u −. 1. θ2. X µeff 2 kwh k0,E hE. 2 nE k0,E. 2 v h )k0,T E. +. h2T E. k∇ (∇ (u −. 2 v h ))k0,T E. . ! 21. T E : E∈Gh. |||(wh , rh )|||h 1. 6 cLa (2ν cDT ) 2 h kuk2 |||(wh , rh )|||h , and this implies X 1 2 ku − v h k0,E hE. 1 2. |hµeff ∇wh · n, u − v h i| 6 µeff. ! 12. ! 21 X. E∈Gh. 6 (µeff cDT cDTI ). 1 2. X. hE µeff k∇wh ·. 2 nE k0,E. E∈Gh. h−2 TE. . ku −. 2 v h k0,T E. +. h2T E. k∇ (u −. 2 v h )k0,T E. . ! 21. T E : E∈Gh. ! 12 X. ×. 2 µeff k∇wh k0,T E. T E : E∈Gh 1. 6 cLa (2µeff cDT cDTI ) 2 h kuk2 |||(wh , rh )|||h . Then we bound |hp − qh , wh · ni| 6. X hE 2 kp − qh k0,E ν. ! 21. X ν 2 kwh · nE k0,E hE. E∈Gh. ! 12. 1. 1. 6 cSZ (2cDT ) 2 ν − 2 h kpk1 |||(wh , rh )|||h ,. E∈Gh. and finally |h(u − v h ) · n, rh i| 6. X 1 2 ku − v h k0,E hE. ! 21. E∈Gh. 6 (cDT cDTI ν). X . 1 2. ! 12 X. 2 hE krh k0,E. E∈Gh. h−2 TE. ku −. 2 v h k0,T E. T E : E∈Gh. + k∇ (u −. 2 v h )k0,T E. . ! 12. 2. X. krh k0,T E. T E : E∈Gh. ! 12. ν. 1 2. 6cLa (2cDT cDTI ν) h kuk2 |||(wh , rh )|||h . The terms related to the corner stabilization vanish as the Lagrange interpolator is exact on mesh nodes. The proof is concluded summing up all the contributions and using Lemma 3.8. .

(29) 27. 4. Numerical Examples The goal of this Section is to validate the results of the analysis of the penalty-free Nitsche method (2.7) against numerical experiments, especially testing the robustness of the formulation with respect to the physical parameters µeff and σ. 2 To this aim, we consider two examples defined on the unit square, i.e., Ω := (0, 1) and discretized using uniform triangular meshes obtained by regular refinements (see Figure 1). In what follows, the four boundary components of Ω will be referred to as Γi , i = 0, 1, 2, 3, with Γ0 := {(x, 0) : x ∈ [0, 1]}, Γ1 := {(1, y) : y ∈ [0, 1]}, Γ2 := {(x, 1) : x ∈ [0, 1]}, Γ3 := {(0, y) : y ∈ [0, 1]}.. Γ2 level 0 1 2 3 4 5 6 7 8 9. hT 1.41421 0.707107 0.353553 0.176777 0.0883883 0.0441942 0.0220971 0.0110485 0.00552427 0.00276214. # Cells (Triangles) 2 8 32 128 512 2048 8192 32768 131072 524288. # Dofs (v, p) (8, 4) (18, 9) (50, 25) (162, 81) (578, 289) (2178, 1089) (8450, 4225) (33282, 16641) (132098, 66049) (526338, 263169). (0, 1). (1, 1). Γ3. Γ1. (0, 0). (1, 0). Γ0 Figure 1. Left: Characteristic element size, number of elements, amount of degrees of freedom for the uniform triangular meshes used for the numerical computations. Right: Meshes corresponding to level 0 (black) and level 1 (black/grey).. In both examples, we compare the results considering different values of the stabilization parameters. The common legend for all forthcoming plots is shown in Figure 2. In particular, line colors will denote different values of α (GLS stabilization), line marker will refer to δ (Grad-Div stabilization) and line style will be related to the value of the characteristic length L0 . The numerical solutions have been computed using the finite element library ParMooN [33].. x x x. α = 0.1, α = 0.1, α = 0.1, α = 0.1, α = 0.1, α = 0.1,. δ δ δ δ δ δ. = 0.1, = 0.1, = 1, = 1, = 10, = 10,. L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1. x x x. α = 1, α = 1, α = 1, α = 1, α = 1, α = 1,. δ δ δ δ δ δ. = 0.1, = 0.1, = 1, = 1, = 10, = 10,. L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1. x x x. α = 10, α = 10, α = 10, α = 10, α = 10, α = 10,. δ δ δ δ δ δ. = 0.1, = 0.1, = 1, = 1, = 10, = 10,. L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1 L0 = 0.1 L0 = 1. Figure 2. In each plot, we compare the errors varying the GLS stabilization parameter α (orange: 0.1, yellow: 1, purple: 10), the Grad-Div stabilization parameter δ (dashed line: 0.1, solid line: 1, dotted line: 10) and the characteristic length L0 ..

Figure

Figure 2. In each plot, we compare the errors varying the GLS stabilization parameter α (orange: 0.1, yellow: 1, purple: 10), the Grad-Div stabilization parameter δ (dashed line: 0.1, solid line: 1, dotted line: 10) and the characteristic length L 0 .
Figure 3 depicts the velocity profile (u 1 (y)) for a few values of µ eff and σ. Notice that, for smaller values of the ratio µ σ eff , the solution has a boundary layer near the Dirichlet boundaries
Figure 4. Example I: Error in the mesh dependent norm (2.12) against the mesh size (in double logarithmic scale), for the cases (µ eff , σ) = (1, 1) (left) and (µ eff , σ) = (0.001, 10) (right).
Figure 5. Example I: Velocity and pressure errors against the mesh size (in double logarithmic scale), for the case (µ eff , σ) = (1, 1), The lines with slope equal to 1 (dashed), 3 2 (dotted) and 2 (solid) are also shown.
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