Stabilized nonconforming finite element methods for data assimilation in incompressible flows
SStabilized nonconforming finite element methodsfor data assimilation in incompressible flows
Erik Burman ∗ and Peter Hansbo † Abstract
We consider a stabilized nonconforming finite element method for data assimilation inincompressible flow subject to the Stokes’ equations. The method uses a primal dual structurethat allows for the inclusion of nonstandard data. Error estimates are obtained that areoptimal compared to the conditional stability of the ill-posed data assimilation problem.
The design of computational methods for the numerical approximation of the Stokes’ system ofequations modelling creeping incompressible flow is by and large well understood in the case wherethe underlying problem is well-posed. Indeed, provided suitable boundary conditions are set, thesystem of equations are known to satisfy the hypotheses of the Lax-Milgram lemma and Brezzi’stheorem ensuring well-posedness of velocities and pressure. These theoretical results then underpinmuch of the theory for the design of stable and accurate finite element methods for the Stokessystem [14, 4].In many cases of interest in applications, however, the necessary data for the theoretical resultsto hold are not known; this is the case for instance in data assimilation in atmospheric sciencesor oceanography. Instead of knowing the solution on the boundary, data in the form of measuredvalues of velocities may be known in some other set. It is then not obvious how best to applythe theory developed for the well-posed case. A classical approach is to rewrite the system as anoptimisation problem and add some regularization, making the problem well-posed on the contin-uous level and then approximate the well-posed problem using known techniques. For examplesof methods using this framework see [1] and [6].In this paper we advocate a different approach in the spirit of [8, 9]. The idea is to formulate theoptimization problem on the continuous level, but without any regularization. We then discretizethe ill-posed continuous problem and instead regularize the discrete solution. This leads to amethod in the spirit of stabilized finite element methods where the properties of the differentstabilizing operators are well studied. An important feature of this approach is that it eliminatesthe need for a perturbation analysis on the continuous level taking into account the Tikhonovregularization and perturbations in data, that the discretization error then has to match. In ourcase we are only interested in the discretization error and the perturbations in data. This allowsus to derive error estimates that are optimal in the case of unperturbed data in a similar fashionas for the well-posed case.We exemplify the theory in a model case for data assimilation where data is given in somesubset of the computational domain instead of the boundary, and we obtain error estimates usinga conditional stability result in the form of a three ball inequality due to Lin, Uhlmann, and Wang[19]. A particular feature of the method formulated for the integration of data in the bulk (andnot on the boundary), is that the dual adjoint problem does not require any regularization on ∗ Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom,[email protected] † Department of Mechanical Engineering, J¨onk¨oping University, SE-55111 J¨onk¨oping, Sweden, [email protected] a r X i v : . [ m a t h . NA ] S e p he discrete level. Indeed, the adjoint equation is inf–sup stable, similarly to the case of ellipticproblems on non-divergence form discussed in [21].The rest of the paper can be outlined as follows. First, in Section 2, we introduce the Stokes’problem that we are interested in and propose the continuous minimization problem. Then,in Section 3, we present the non-conforming finite element method and prove some preliminaryresults. In Section 4 we prove the fundamental stability and convergence results of the formulation.Finally we show the performance of the approach on some numerical examples. Let Ω be a polygonal (polyhedral) domain in R d , d = 2 or 3. We are interested in computingsolutions to the Stokes’ system − ∆ u + ∇ p = f in Ω ∇ · u = g in Ω . (2.1)Typically these equations are then equipped with suitable boundary conditions and are knownto be well-posed using the Lax-Milgram Lemma for the velocities and Brezzi’s theorem for thepressures. It is also known that the following continuous dependence estimate holds, here givenunder the assumption of homogeneous Dirichlet conditions on the boundary. (cid:107) u (cid:107) H (Ω) + (cid:107) p (cid:107) Ω (cid:46) (cid:107) f (cid:107) H − (Ω) + (cid:107) g (cid:107) Ω , (2.2)where we used the notation (cid:107) x (cid:107) Ω := (cid:107) x (cid:107) L (Ω) and a (cid:46) b for a ≤ Cb with C > B R ⊂ Ω there holds( u, p ) | B R ∈ [ H ( B R )] d × H ( B R ) . (2.3)Provided f ∈ [ L (Ω)] d and g ∈ H (Ω). See for instance [20, Proposition 3.2].We will in the following make the stronger assumption that ( u, p ) ∈ [ H (Ω)] d × H (Ω). Observethat this is not a strong assumption for the particular problem we will study below, since thedomain Ω here is somewhat arbitrary and not necessarily determined by a physical geometry.Indeed the only situation in which this assumption can fail is when the boundary of Ω coincideswith a physical boundary with a corner.Herein the main focus will be on methods that allow for the accurate approximation of thesolution under the much weaker stability estimates that remain valid in the case of ill-posedproblems where (2.2) fails.A situation of particular interest is the case where the boundary data g D is known only ona portion Γ D of ∂ Ω and nothing is known of the boundary conditions on the remaining partΓ (cid:48) D := ∂ Ω \ Γ D . This lack of boundary information makes the problem ill-posed and we assumethat some other data is known such as: • The normal stress in some part of the boundary Γ N ⊂ ∂ Ω and Γ N ∩ Γ D (cid:54) = ∅ ,( − n · ∇ u + pn ) · n = ψ. (2.4)We will refer to this problem as the Cauchy problem below. • The measured value of ( u, p ) in some subdomain ω ⊂ Ω. We will refer to this problem asthe data assimilation problem below.In the first case it is known that if a solution exists, then g D = ψ = 0 implies u = 0, p = 0 in Ωby unique continuation [13], however, no quantitiative estimates appear to exist in the literaturefor the pure Cauchy problem; see [5] for results using additional measurements on the boundary.In the second case stability may be proven in the form of a three balls inequality and associated2ocal stability estimates, see [19, 5]. For completeness of the analysis we focus on the secondcase for the error estimates below. In particular we consider the case where no data are knownon the boundary, i.e. Γ D = Γ N = ∅ . In the data assimilation case the following Theorem from[19] provides us with a conditional stability estimate. Assuming an optimal conditional stabilityestimate for the Cauchy problem in the spirit of [3], it is straightforward to extend the anaysis tothis case following [7]. Theorem 2.1. (Conditional stability for the Stokes’ problem) There exists a positive number ˜ R < such that if < R < R < R ≤ R and R /R < R /R < ˜ R , then if B R ( x ) ⊂ Ω (cid:90) B R ( x ) | u | d x ≤ C (cid:32)(cid:90) B R ( x ) | u | d x (cid:33) τ (cid:32)(cid:90) B R ( x ) | u | d x (cid:33) − τ for ( u, p ) ∈ [ H ( B R ( x ))] d +1 , satisfying (2.1) with f = g = 0 in B R ( x ) , where the constant C depends on R /R and < τ < depends on R /R , R /R and d . For fixed R and R , theexponent τ behaves like / ( − log( R )) when R is sufficiently small.Proof. For the proof we refer to [19].In the data assimilation problem corresponding to Theorem 2.1 measured data u M : ω (cid:55)→ R d are available in ω such that u M satisfies (2.1) in ω and there exists u defined on Ω satisfying (2.1)such that u | ω = u M . Our objective is to design a method for the reconstruction of u , given ˜ u M := u M + δu , where δu ∈ [ L ( ω )] d is a perturbation of the exact data resulting from measurement erroror interpolation of pointwise measurements inside ω . Observe that the considered configuration isalso closely related to a pure boundary control problem, where we look for data on the boundarysuch that u = u M in the subset ω .We will first cast the problem (2.1), with the notation f = f and with g = 0, on weak form.For the derivation of the weak formulation we introduce the spaces V := { v ∈ [ H (Ω)] d } and W := { v ∈ [ H (Ω)] d } for velocities and Q := L (Ω) and Q := L (Ω), where the zero–subscriptin the second case as usual indicates that the functions have zero integral over Ω.We may the multiply the first equation of (2.1) by w ∈ W and first integrate over Ω and thenapply Green’s formula to obtain (cid:90) Ω ∇ u : ∇ w d x − (cid:90) Ω p ∇ · w d x = (cid:90) Ω f w d x, ∀ w ∈ W similarly we may multiply the second equation by q ∈ L (Ω) and integrate over Ω to get (cid:90) Ω q ∇ · u d x = 0 . Introducing the forms a ( u, w ) := (cid:90) Ω ∇ u : ∇ w d x,b ( p, w ) = − (cid:90) Ω p ∇ · w d x and l ( w ) := (cid:90) Ω f w d x we may formally write the problem as: find ( u, p ) ∈ V × Q such that u | ω = u M and a ( u, w ) + b ( p, w ) = l ( w ) , ∀ w ∈ W (2.5) b ( y, u ) = 0 , ∀ y ∈ Q. (2.6)3bserve that this problem is ill-posed. In particular observe that we are not allowed to test with w = u because of the homogeneous Dirichlet conditions set on the functions in W . To regularizethe problem we cast it on the form of a minimization problem, first writing A [( u, p ) , ( w, y )] := a ( u, w ) + b ( p, w ) − b ( y, u )and then introducing the Lagrangian L [( u, p ) , ( z, x )] := 12 m ( u − ˜ u M , u − ˜ u M ) + A [( u, p ) , ( z, x )] − l ( z ) , where m ( · , · ) is a bilinear form that depends on what data we wish to integrate. For the dataassimilation problem that is our main concern we simply have m ( u, v ) := γ M (cid:90) ω uv d x, where γ M > u, v ) ω := (cid:90) ω uv d x. The optimality system of the associated constrained minimization problem takes the form A [( u, p ) , ( w, y )] = l ( w ) (2.7) A [( v, q ) , ( z, x )] + m ( u, v ) = m (˜ u M , v ) . (2.8)This problem is ill-posed in general, but in the data assimilation case we know that if a solutionexists and l ( w ) = 0 then this solution must satisfy the conditional stability of Theorem 2.1. Aconsequence of this is that if the system admits a solution ( u, p ) ∈ V × L (Ω) for the exact data u M , then this solution is unique. To show this assume that there are two solutions u ∈ V and u ∈ V that solve (2.7)–(2.8), then v = u − u ∈ V solves the homogenous Stokes’ equation andhas v | ω = 0 and the uniqueness is a consequence of unique continuation based on Theorem 2.1.Below we will assume that there exists a unique solution ( u, p ) ∈ [ H (Ω)] d × H (Ω) that satisfies(2.1) in Ω with u = u M in ω . Let {T h } h denote a family of shape regular and quasi uniform tesselations of Ω into nonoverlappingsimplices, such that for any two different simplices κ , κ (cid:48) ∈ T h , κ ∩ κ (cid:48) consists of either the empty set,a common face or a common vertex. The outward pointing normal of a simplex κ will be denoted n κ . We denote the set of element faces in T h by F and let F i denote the set of interior faces F in F . To each face F we associate a unit normal vector, n F . For interior faces its orientation isarbitrary, but fixed. On the boundary ∂ Ω we identify n F with the outward pointing normal of Ω.We define the jump over interior faces F ∈ F i by [ v ] | F := lim (cid:15) → + ( v ( x | F − (cid:15)n F ) − v ( x | F + (cid:15)n F ))and for faces on the boundary, F ∈ ∂ Ω, we let [ v ] | F := v | F . Similarly we define the average of afunction over an interior face F by { v }| F := lim (cid:15) → + ( v ( x | F − (cid:15)n F ) + v ( x | F + (cid:15)n F )) and for F onthe boundary we define { v }| F := v | F . The classical nonconforming space of piecewise affine finiteelement functions (see [11]) then reads X h := { v h ∈ L (Ω) : (cid:90) F [ v h ] d s = 0 , ∀ F ∈ F i and v h | κ ∈ P ( κ ) , ∀ κ ∈ T h } where P ( κ ) denotes the set of polynomials of degree less than or equal to one restricted to theelement κ , and with homogeoneous Dirichlet boundary conditions X h := { v h ∈ L (Ω) : (cid:90) F [ v h ] d s = 0 , ∀ F ∈ F and v h | κ ∈ P ( κ ) , ∀ κ ∈ T h } .
4e may then define the spaces V h := [ X h ] d and W h := [ X h ] d . For the pressure spaces we define Q h := { q h ∈ L (Ω) : q | κ ∈ R , ∀ κ ∈ T h } and Q h := Q h ∩ L (Ω) . To make the notation more compact we introduce the composite spaces V h := V h × Q h and W h := W h × Q h . By writing the equations (2.7)–(2.8) with arguments in the discrete spaces, the formulation maynow naively be written: find ( u h , p h ) × ( z h , x h ) ∈ V h × W h such that, A h [( u h , p h ) , ( w h , y h )] = l ( w ) (3.1) A h [( v h , q h ) , ( z h , x h )] + m ( u h , v h ) = m (˜ u M , v h ) . (3.2)for all ( v h , q h ) × ( w h , y h ) ∈ V h × W h . The discrete bilinear form is defined by A h [( u h , p h ) , ( w h , y h )] := a h ( u h , w h ) + b h ( p h , w h ) − b h ( y h , u h ) (3.3)where the forms are defined by a h ( u h , w h ) = (cid:88) κ ∈T h (cid:90) κ ∇ u h : ∇ w h d x,b h ( p h , w h ) = − (cid:88) κ ∈T h (cid:90) κ p h ∇ · w h d x. To obtain a stable formulation we need to add stabilizing terms. This can be done in severaldifferent ways, resulting in different methods with different stability, accuracy and conservationproperties. Our choice herein has been guided by the principle that stabilization is added only if itis necessary for accuracy and has minimal influence on the conservation properties of the scheme.We will also comment on some variants. For the primal velocities we suggest to use the standardjump stabilization that has been shown to stabilize the Crouzeix-Raviart element in a number ofapplications [15, 16, 10], s j,t ( u h , v h ) := (cid:88) F ∈F i (cid:90) F h tF [ u h ][ v h ] d s. (3.4)For the pressure on the other hand we propose to use the following weak penalty term s p,t ( p h , q h ) := (cid:90) Ω h t p h q h d x. (3.5)We also propose the compact form: find ( U h , Z h ) ∈ V h × W h , where U h := ( u h , p h ) ∈ V h × Q h and Z h := ( z h , x h ) ∈ W h × Q h , such that, A h [( U h , Z h ) , ( X h , Y h )] + S h [( U h , Z h ) , ( X h , Y h )] + m ( u h , v h ) = l ( w h ) + m (˜ u, v h ) (3.6)for all ( X h , Y h ) ∈ V h × W h , X h := ( v h , q h ) and Y h := ( w h , y h ). The bilinear forms are then givenby A h [( U h , Z h ) , ( X h , Y h )] := A h [( u h , p h ) , ( w h , y h )] + A h [( v h , q h ) , ( z h , x h )] (3.7)and S h [( U h , Z h ) , ( X h , Y h )] := S p [( u h , p h ) , ( v h , q h )] − S a [( z h , x h ) , ( w h , y h )] , (3.8)where S a and S p are positive semi-definite, symmetric bilinear forms. In the following, we shallalso make use of the following bilinear form G [( U h , Z h ) , ( X h , Y h )] := A h [( U h , Z h ) , ( X h , Y h )] (3.9)+ S h [( U h , Z h ) , ( X h , Y h )] + m ( u h , v h ) . S p [( u h , p h ) , ( v h , q h )] := γ u s j, − ( u h , v h ) + γ p s p, ( p h , q h ) , γ u > , γ p ≥ S a [( z h , x h ) , ( w h , y h )] := γ x s p, ( x h , y h ) , γ x ≥ . (3.11)Observe that the minimal stabilization that allows for optimal error estimates is γ u > γ p = γ x = 0. In the analysis below we will focus on this case, noting that the case with added pressurestabilization follows in a similar way, but is slightly more elementary. From the theoretical pointof view the choice γ p > γ x > Consider the dual mass conservation equation in the formulation (3.6) with the stabilization givenby (3.10) and (3.11) and γ p > b ( q h , z h ) + s p, ( p h , q h ) = 0 , ∀ q h ∈ Q h . Observing that γ − p h − ∇ · w h ∈ Q h we may eliminate the physical pressure from the formulation,since b ( p h , w h ) = − ( p h , ∇ · w h ) h = − s p, ( p h , γ − p h − ∇ · w h ) = b ( γ − p h − ∇ · w h , z h )= ( γ − p h − ∇ · w h , ∇ · z h ) h . Similarly, for γ x > x h may be eliminated. Starting from the mass conservationequation − b ( y h , u h ) − s x, ( x h , y h ) = 0we use that y h = ∇ · v h is a valid test function to deduce − b ( x h , v h ) = − ( x h , ∇ · v h ) h = s x, ( x h , ∇ · v h ) = − b ( ∇ · v h , u h ) = ( ∇ · v h , ∇ · u h ) h . The resulting formulation is an equal order interpolation formulation for the Stokes’ system usingthe nonconforming Crouzeix-Raviart element for both the forward and the dual system. Find( u h , z h ) ∈ V h × W h such that a h ( u h , w h ) − ( γ − p h − ∇ · w h , ∇ · z h ) h = l ( w h ) (3.12) a h ( z h , v h ) + ( ∇ · u h , ∇ · v h ) h + s j, − ( u h , v h ) + m ( u h , v h ) = m (˜ u M , v h )for all ( v h , w h ) ∈ V h × W h . We identify this scheme as a discretization of the continuous regu-larization of the Stokes’ Cauchy problem proposed in [6]. It follows that the analysis below alsocovers that method in the special case that the discretization uses the nonconforming space X h . We will end this section by proving some elementary Lemmas that will be useful in the analysisbelow. We will use (cid:107) · (cid:107) X to denote the L –norm over X , subset of R d or R d − .We recall the interpolation operator r h : [ H (Ω)] d → [ X h ] d defined by the (component wise)relation { r h v }| F := | F | − (cid:90) F { r h v } d s = | F | − (cid:90) F v d s F ∈ F and with | F | denoting the ( d − F . It is conventient to introducethe broken scalar product ( x, y ) h := (cid:88) κ ∈T h (cid:90) κ xy d x, with the associated norms (cid:107) x (cid:107) h := ( x, x ) h and (cid:107) x (cid:107) ,h := (cid:107) x (cid:107) h + a h ( x, x ) . The following inverse and trace inequalities are well known (cid:107) v (cid:107) ∂κ ≤ C t ( h − (cid:107) v (cid:107) κ + h (cid:107)∇ v (cid:107) κ ) , ∀ v ∈ H ( κ ) ,h κ (cid:107)∇ v h (cid:107) κ + h κ (cid:107) v h (cid:107) ∂κ ≤ C i (cid:107) v h (cid:107) κ , ∀ v h ∈ X h . (3.13)Using the inequalities of (3.13) and standard approximation results from [11] it is straightforwardto show the following approximation results of the interpolant r h (cid:107) u − r h u (cid:107) Ω + h (cid:107)∇ ( u − r h u ) (cid:107) h ≤ Ch t | u | H t (Ω) (cid:107) h − ( u − r h u ) (cid:107) F + (cid:107) h ∇ ( u − r h u ) · n F (cid:107) F ≤ Ch t − | u | H t (Ω) (3.14)where t ∈ { , } . It will also be useful to bound the L -norm of the interpolant r h by its valueson the element faces. To this end we prove a technical lemma. Lemma 3.1.
For any function v h ∈ X h there holds (cid:107) v h (cid:107) Ω ≤ c T (cid:32) (cid:88) F ∈F h F (cid:107){ v h }(cid:107) F (cid:33) Proof.
It follows by norm equivalence of discrete spaces on the reference element and a scalingargument (under the assumption of shape regularity) that for all κ ∈ T h (cid:107) v h (cid:107) κ ≤ C (cid:88) F ∈ ∂κ h F (cid:107) v h (cid:107) F . (3.15)The claim follows by summing over the elements of T h and recalling that (cid:107) v h (cid:107) F = (cid:107){ v h }(cid:107) F .For the analysis below we also need a quasi-interpolation operator that maps piecewise linear,nonconforming functions into the space of piecewise linear conforming functions. Let I cf [ Q h ] d ∪ V h (cid:55)→ V ∩ V h denote the quasi interpolation of u h into V h ∩ V , [17, 2, 18] such that (cid:107) I cf u h − u h (cid:107) Ω + h (cid:107)∇ ( I cf u h − u h ) (cid:107) h (cid:46) (cid:107) h [ u h ] (cid:107) F i and for v h ∈ X h (cid:107) I cf ∇ v h − ∇ v h (cid:107) h (cid:46) (cid:107) h [ ∇ v h ] (cid:107) F i . (3.16)Below, the global conservation properties of this operator will be important and we thereforepropose the following perturbed variant that satisfies a global conservation property. We definethe modified interpolant by ˜ u h := I cf u h + d h (3.17)where the perturbation d h ∈ V h ∩ V is the solution to the following constrained problem, ¯ p ∈ R ( d h , w h ) h + ( ∇ d h , ∇ w h ) h + (¯ p, ∇ · w h ) h = 0( ∇ · d h , ¯ q ) h = ( −∇ · I cf u h , ¯ q ) h , (3.18)7or all ( w h , ¯ q ) ∈ ( V h ∩ V ) × R . This implies that (cid:90) ∂ Ω ˜ u h · n d s = (cid:90) Ω ∇ · ˜ u h d x = | Ω |∇ · ( I cf u h + d h ) = 0with | Ω | denoting the d -measure of Ω and ∇ · I cf u h := | Ω | − (cid:90) Ω ∇ · I cf u h d x. Lemma 3.2.
The problem (3.18) is well-posed and the solution satisfies (cid:107) d h (cid:107) H (Ω) ≤ (cid:107)∇ · I cf u h (cid:107) Ω (cid:46) (cid:107) h − [ u h ] (cid:107) F i + (cid:107)∇ · u h (cid:107) h Proof.
Since the linear system corresponding to (3.18) is square, existence and uniqueness is aconsequence of the stability estimate. Take w h = d h + α ¯ p x , with α >
0, ¯ q = ¯ p in (3.18) andobserve that for α small enough, there exists c ( α ) > (cid:107) d h (cid:107) H (Ω) + c ( α ) (cid:107) ¯ p (cid:107) (cid:46) (cid:107)∇ · I cf u h (cid:107) (cid:46) (cid:107)∇ · ( I cf u h − u h ) (cid:107) h + (cid:107)∇ · u h (cid:107) h (cid:46) (cid:107) h − [ u h ] (cid:107) F i + (cid:107)∇ · u h (cid:107) h . (3.19)An immediate consequence of Lemma 3.2 is that ˜ u h satisfies similar approximation estimatesas I cf , but with improved global conservation. We collect these results, the proof of which is animmediate consequence of the discussion above, in a corollary. Corollary 3.3.
The conforming approximation ˜ u h satisfies the discrete estimate, (cid:107) ˜ u h − u h (cid:107) ,h (cid:46) (cid:107) h − [ u h ] (cid:107) F i + (cid:107)∇ · u h (cid:107) h (3.20) and has the global conservation property (cid:90) ∂ Ω ˜ u h · n d s = 0 . Using the regularity assumptions on the data in l ( w ) it is straightforward to show that theformulation satisfies the following weak consistency Lemma 3.4. (Weak consistency) Let ( u, p ) be the solution of (2.1) , with f ∈ L (Ω) , and let ( u h , p h ) ∈ V be the solution of (3.6) . Then, for all w h ∈ W h , there holds, | a h ( u h − u, w h ) + b h ( p h − p, w h ) |≤ inf ( ν h ,η h ) ∈V (cid:88) F ∈F (cid:90) F | n F · ( σ ( u, p ) − { σ ( ν h , η h ) } ) || [ w h ] | d s. (3.21) where σ ( u, p ) := ∇ u − I p , with I the identity matrix.Proof. Multiplying (2.1) with w h ∈ W h and integrating by parts we have (cid:90) Ω f w h d x = − (cid:90) Ω ∇ · ( ∇ u − I p ) · w h d x = − (cid:88) κ ∈T h (cid:88) F ∈ ∂κ (cid:90) F σ ( u, p ) · n κ · w h d s + a h ( u, w h ) + b h ( p, w h ) (3.22)8r by rearranging terms a h ( u, w h ) + b h ( p, w h ) = l ( w h ) + (cid:88) κ ∈T h (cid:88) F ∈ ∂κ (cid:90) F σ ( u, p ) · n κ · w h d s. Using (3.6) we obtain a h ( u h − u, w h ) + b h ( p h − p, w h ) = − (cid:88) κ ∈T h (cid:88) F ∈ ∂κ (cid:90) F σ ( u, p ) · n κ · w h d s. Since every internal face appears twice with different orientation of n κ we have for all ν h ∈ V h , (cid:88) F ∈ ∂κ (cid:90) F σ ( u, p ) · n κ · w h d s = (cid:88) F ∈ ∂κ (cid:90) F n κ · ( σ ( u, p ) − { σ ( ν h , η h ) } ) · w h d s. We now observe that by replacing w h with the jump [ w h ] we may write the sum over the faces ofthe mesh, replacing n κ by n F . The conclusion follows by taking absolute values on both sides andmoving the absolute values under the integral sign resulting in the desired inequality. Lemma 3.5.
Let U := ( u, p ) ∈ V × Q denote the solution to (2.7) - (2.8) with δu = 0 . Then thereholds |A [( U − U h , Z h ) , ( X h , Y h )] − S h [( U h , Z h ) , ( X h , Y h )] + m (˜ u M − u h , v h ) |≤ inf ( ν h ,η h ) ∈V (cid:88) F ∈F (cid:90) F | n F · ( σ ( u, p ) − { σ ( ν h , η h ) } ) || [ w h ] | d s for all ( X h , Y h ) := ([ w , y h ] , [ v h , q h ]) ∈ V × W .Proof. Subtract (3.6) from (2.7)-(2.8) and apply Lemma 3.4 to the equation for the primal variable U . Lemma 3.6.
For any v ∈ [ H (Ω)] d , y ∈ L (Ω) and for all w h ∈ W h , q h ∈ Q h there holds a h ( v − r h v, w h ) = 0 , b h ( q h , v − r h v ) = 0 and b h ( y − π y, w h ) = 0 Proof.
By integration by parts we have, using the orthogonality property on the faces of r h , a h ( v − r h v, w h ) = (cid:88) κ ∈T h (cid:88) F ∈ ∂κ (cid:90) F ( v − r h v ) · ( n κ · ∇ w h ) d s = 0 ,b h ( q h , v − r h v ) = (cid:88) κ ∈T h (cid:88) F ∈ ∂κ (cid:90) F ( v − r h v ) · n κ q h d s = 0 , and by definition b h ( p − π p, w h ) = ( p − π p, ∇ · w h ) h = 0 . Lemma 3.7.
Let ( u h , p h ) ∈ V then there holds (cid:107) h n F · [ ∇ u h ] (cid:107) F i + (cid:107) h [ p h ] (cid:107) F i (cid:46) (cid:107) h n F · [ ∇ u h − I p h ] (cid:107) F i + (cid:107)∇ · u h (cid:107) Ω + (cid:107) h − [ u h ] (cid:107) F i . Proof.
Let u i , i = 1 , . . . , d denote the components of u h and define the tangential projection ofthe gradient matrix on the face F by T ∇ u h := ( I − n F ⊗ n F ) ∇ u h where ⊗ denotes outer product.Considering one face F ∈ F i we have (cid:107) h n F · [ ∇ u h − I p h ] (cid:107) F = (cid:107) h n F · [ ∇ u h ] (cid:107) F + (cid:107) h [ p h ] (cid:107) F − (cid:90) F h F n F · [ ∇ u h ] · ( n F · [ I p h ])d s. n F · [ ∇ u h ] · ( n F · [ I p h ]) = [ p h ] d (cid:88) i =1 d (cid:88) j =1 n F,i n F,j [ ∂ x j u i ] . By applying the following identity d (cid:88) i =1 d (cid:88) j =1 n F,i n F,j ∂ x j u i = ∇ · u h − tr( T ∇ u h ) , where tr( T ∇ u h ) denotes the trace of T ∇ u h , we may write[ p h ] d (cid:88) i =1 d (cid:88) j =1 n F,i n F,j [ ∂ x j u i ] = [ p h ] ([ ∇ · u h ] − [tr( T ∇ u h )]) . Observe that since the tangential component of the gradient of the conforming approximation I cf u h does not jump we have [tr( T ∇ u h )] = [tr( T ( ∇ u h − ∇ I cf u h )] . Collecting these identities we obtain (cid:90) F h F n F · [ ∇ u h ] · n F · [ I p h ]d s = (cid:90) F h F [ p h ] (cid:16) [ ∇ · u h ] − [tr( T ∇ ( u h − I cf u h ))] (cid:17) d s ≤ (cid:107) h [ p h ] (cid:107) F C i ( (cid:107)∇ ( u h − I cf u h ) (cid:107) ∆ F + (cid:107)∇ · u h (cid:107) ∆ F ) , where ∆ F denotes the union of the elements that have F as common face. Consequently2 (cid:90) F h F n F · [ ∇ u h ] · n F · [ I p h ]d s ≤ (cid:107) h [ p h ] (cid:107) F + C (cid:107) h − [ u h ] (cid:107) F ∆ F + C (cid:107)∇ · u h (cid:107) F . Summing over F ∈ F i we see that (cid:107) h n F · [ ∇ u h ] (cid:107) F i + 12 (cid:107) h [ p h ] (cid:107) F i (cid:46) (cid:107) h n F · [ ∇ u h − I p h ] (cid:107) F i + (cid:107)∇ · u h (cid:107) + C (cid:107) h − [ u h ] (cid:107) F i which proves the claim. Lemma 3.8. (Discrete Poincar´e inequality) For all ( u h , p h ) ∈ V h × Q h there holds (cid:107) hu h (cid:107) ,h (cid:46) (cid:107) h n F · [ ∇ u h ] (cid:107) F i + (cid:107) h − [ u h ] (cid:107) F i + (cid:107) u h (cid:107) ω and (cid:107) hp h (cid:107) Ω (cid:46) (cid:107) h [ p h ] (cid:107) F i . Proof.
For the first inequality use the Poincar´e inequality for nonconforming finite elements anda triangle inequality (cid:107) hu h (cid:107) ,h (cid:46) (cid:107) h ( ∇ u h − I cf ∇ u h ) (cid:107) h + (cid:107) hI cf ∇ u h (cid:107) h . Then observe that for I cf ∇ u h constant, (cid:107) u h (cid:107) ω = 0 implies that I cf ∇ u h = 0 and therefore [12,Lemma B.63] (cid:107) hI cf ∇ u h (cid:107) h ≤ (cid:107) h ∇ ( I cf ∇ u h − ∇ u h ) (cid:107) h + (cid:107) u h (cid:107) ω . Using (3.16) componentwise twice we then have (cid:107) hu h (cid:107) ,h (cid:46) (cid:107) h [ ∇ u h ] (cid:107) F i + (cid:107) u h (cid:107) ω . ∇ u h is decomposed on the normal and tangential component on eachface F and we observe that using an elementwise trace inequality, (cid:107) h ( I − n F ⊗ n F )[ ∇ u h ] (cid:107) F i = (cid:107) h ( I − n F ⊗ n F )[ ∇ ( u h − I cf u h )] (cid:107) F i (cid:46) (cid:107)∇ ( u h − I cf u h ) (cid:107) h (cid:46) (cid:107) h − [ u h ] (cid:107) F i . Similarly for the proof of the second inequality observe that since (redefining I cf to act on a scalarvariable, and once again by [12, Lemma B.63]) (cid:107) hI cf p h (cid:107) Ω (cid:46) (cid:107) h ∇ I cf p h (cid:107) Ω + (cid:82) Ω hI cf p h d x there holds (cid:107) hp h (cid:107) h (cid:46) (cid:107) h ( p h − I cf p h ) (cid:107) h + (cid:107) h ∇ ( I cf p h − p h ) (cid:107) h + (cid:90) Ω h ( I cf p h − p h ) d x. It then follows using an inverse inequality that (cid:107) hp h (cid:107) h (cid:46) (cid:107) p h − I cf p h (cid:107) Ω (cid:46) (cid:107) h [ p h ] (cid:107) F i and the proof is complete. We will now focus on the formulation (3.6) with γ p = γ x = 0. An immediate consequence of thischoice is that any solution to the system must satisfy ∇ · u h | κ = ∇ · z h | κ = 0 , ∀ κ ∈ T h . (4.1)The issue of stability of the discrete formulation is crucial since we have no coercivity or inf–sup stability of the continuous formulation (2.7)–(2.8) to rely on. Indeed here the regularizationplays an important part, since it defines a semi-norm on the discrete space. We introduce amesh-dependent norm for the primal variable X h := ( v h , q h ) ∈ V||| X h ||| V,Q := (cid:107) h n F · [ ∇ v h ] (cid:107) F i + (cid:107) h [ q h ] (cid:107) F i + γ M (cid:107) v h (cid:107) ω + (cid:107) h − [ v h ] (cid:107) F i , (4.2)We will also use the following triple norm with control of both the dual pressure variabel x h and the dual velocities z h . ||| ( U h , Z h ) ||| := ||| U h ||| V,Q + (cid:107) x h (cid:107) Ω + (cid:107)∇ z h (cid:107) h . Since Dirichlet boundary conditions are set weakly on W h , ||| ( U h , Z h ) ||| can be shown to be anorm on V h × Q h using Lemmata 3.7–3.8. We now prove a fundamental stability result for thediscretization (3.6). Theorem 4.1.
Let γ u , γ M > , γ p = γ x = 0 in (3.6) – (3.8) . There exists a positive constant c s , that is independent of h , but not of γ u , γ M or the local mesh geometry, such that for all ( U h , Z h ) ∈ V × W there holds c s ||| ( U h , Z h ) ||| ≤ sup ( X h ,Y h ) ∈V×W G [( U h , Z h ) , ( X h , Y h )] ||| ( X h , Y h ) ||| Proof.
First we observe that by testing with X h = U h and Y h = − Z h we have γ u (cid:107) h − [ u h ] (cid:107) F i + γ M (cid:107) u h (cid:107) ω = G [( U h , Z h ) , ( U h , − Z h )] . Then observe that by integrating by parts in the bilinear form a h ( · , · ) and using the zero meanvalue property of the approximation space we have a h ( u h , w h ) + b h ( p h , w h ) = (cid:88) F ∈F i (cid:90) F [ n F · ∇ u h − p h n F ] · { w h } d s. ξ h ∈ W h such that for every face F ∈ F i { ξ h }| F := h F [ n F · ∇ u h − p h n F ] | F . This is possible in the nonconforming finite element space since the degrees of freedom may beidentified with the average value of the finite element function on an element face. Using Lemma3.1 we have (cid:107) ξ h (cid:107) ≤ c T (cid:88) F ∈F i h F (cid:107) h F [ n F · ∇ u h − p h n F ] (cid:107) F . (4.3)Testing with Y h = ( ξ h ,
0) and X h = (0 ,
0) we get (cid:107) h [ n F · ∇ u h − p h n F ] (cid:107) F i = G [( U h , Z h ) , (0 , Y h )] . By testing with X h = ( z h + αr h v x , x h ), where α > v x ∈ [ H (Ω)] d is a function such that ∇ · v x = x h and (cid:107) v x (cid:107) H (Ω) ≤ c x (cid:107) x h (cid:107) Ω , we have (cid:107)∇ z h (cid:107) h + α (cid:107) x h (cid:107) + a h ( z h , αr h v x )+ γ u s j, − ( u h , z h + αr h v x ) + m ( u h , z h + αr h v x ) ω = G [( U h , Z h ) , ( X h , . Observe now that by the Cauchy-Schwarz inequality, the arithmetic-geometric inequality and thestability of r h v x there holds a h ( z h , αr h v x ) ≤ (cid:107)∇ z h (cid:107) h + c x α (cid:107) x h (cid:107) . Then by the trace inequality and Poincar´e’s inequality γ u s j, − ( u h , z h + αr h v x ) + γ M ( u h , z h + αr h v x ) ω (cid:46) ( γ u (cid:107) h − [ u h ] (cid:107) F i + γ M (cid:107) u h (cid:107) ω )( (cid:107)∇ z h (cid:107) h + (cid:107) v x (cid:107) H (Ω) ) ≤ C γ ( γ u (cid:107) h − [ u h ] (cid:107) F i + γ M (cid:107) u h (cid:107) ω ) + 14 ( (cid:107)∇ z h (cid:107) h + α (cid:107) x h (cid:107) ) . The consequence of this is that for α, β > c such that c (cid:16) (cid:107) h − [ u h ] (cid:107) F i + (cid:107)∇ z h (cid:107) h + (cid:107) u h (cid:107) ω + (cid:107) x h (cid:107) + (cid:107) h [ n F · ∇ x h − p h n F ] (cid:107) F (cid:17) ≤ G [( U h , Z h ) , ( X UZ , Y UZ )] , (4.4)where X UZ = U h + ( β ( z h + αr h v x ) , x h ), Y UZ = − Z h + ( ξ h , (cid:107)∇ · u h (cid:107) h = 0 we deduce that C ||| ( U h , Z h ) ||| ≤ G [( U h , Z h ) , ( X UZ , Y UZ )] . (4.5)It remains to prove that ||| ( X UZ , Y UZ ) ||| (cid:46) ||| ( U h , Z h ) ||| . This follows by observing that ||| ( X UZ , Y UZ ) ||| ≤ ||| ( U h , Z h ) ||| + β ( (cid:107) h [ ∇ ( z h + αr h v x )] (cid:107) F i + (cid:107) ( z h + αr h v x ) (cid:107) ω + (cid:107) h − [ z h + αr h v x ] (cid:107) F i ) + (cid:107)∇ ξ h (cid:107) h and (cid:107) h [ ∇ ( z h + αr h v x )] (cid:107) F i + (cid:107) ( z h + αr h v x ) (cid:107) ω + (cid:107) h − [ z h + αr h v x ] (cid:107) F i (cid:46) (cid:107) z h (cid:107) ,h + (cid:107) r h v x (cid:107) ,h (cid:46) (cid:107)∇ z h (cid:107) h + (cid:107) x h (cid:107) Ω . Finally we use an inverse inequality and the bound (4.3) to obtain the bound (cid:107)∇ ξ h (cid:107) h (cid:46) (cid:107) h F [ n F · ∇ u h − p h n F ] (cid:107) F i (cid:46) ||| ( U h , ||| which finishes the proof. 12 orollary 4.2. The formulation (3.6) admits a unique solution ( u h , p h ) ∈ V and ( z h , x h ) ∈ W .Proof. The system matrix corresponding to (3.6) is a square matrix and we only need to showthat there are no zero eigenvalues. Assume that l ( w h ) = 0. It then follows by Theorem 4.1 thatfor any solution ( u h , p h ) there holds c s ||| ( U h , Z h ) ||| ≤ sup ( X h ,Y h ) ∈V×W G [( U h , Z h ) , ( X h , Y h )] ||| ( X h , Y h ) ||| = 0 . Recalling Lemma 3.8 this implies that u h = p h = z h = x h = 0 showing that the solution isunique. Remark 4.3.
Observe that the proof of Theorem 4.1 works also for γ p > and γ x > , the onlymodification in this case is that the contribution (cid:107)∇ · u h (cid:107) h must be added to the norm ||| ( u h , p h ) ||| V and stability must be proven by testing with y h = ∇ · u h . In this section we will use the stability proven in the previous section to derive error estimates.
Proposition 5.1.
Let ( u, p ) ∈ [ H (Ω)] d × H (Ω) be the solution of (2.1) and ( u h , p h ) × Z h thesolution to (3.6) – (3.8) , with γ u , γ M > and γ p = γ x = 0 . Then there holds ||| (( r h u − u h , π p − p h ) , Z h ) ||| (cid:46) h ( (cid:107) u (cid:107) H (Ω) + (cid:107) p (cid:107) H (Ω) ) + γ M (cid:107) δu (cid:107) ω Proof.
First denote the discrete error Θ h = ( r h u − u h , π p − p h ). Then by Theorem 4.1 c s ||| (Θ h , Z h ) ||| ≤ sup ( X h ,Y h ) ∈V×W G [(Θ h , Z h ) , ( X h , Y h )] ||| ( X h , Y h ) ||| . Then applying Lemma 3.5 and 3.6 we have G [(Θ h , Z h ) , ( X h , Y h )] ≤ inf ( ν h ,η h ) ∈ V h × W h (cid:88) F ∈F (cid:90) F | ( σ ( u, p ) − { σ ( ν h , η h ) } ) · n F || [ w h ] | d s − b h ( y h , r h u − u ) + s j, − ( r h u, v h ) + γ M ( r h u − u − δu, v h ) ω . First note thatinf ( ν h ,η h ) ∈V (cid:88) F ∈F (cid:90) F | ( σ ( u, p ) − { σ ( ν h , η h ) } ) · n F || [ w h ] | d s ≤ h ( inf ( ν h ,η h ) ∈V (cid:88) F ∈F (cid:107) σ ( u, p ) − { σ ( ν h , η h ) }(cid:107) F ) (cid:107)∇ w h (cid:107) h (cid:46) h ( (cid:107) u (cid:107) H (Ω) + (cid:107) p (cid:107) H (Ω) ) ||| (0 , Y h ) ||| ,b h ( y h , r h u − u ) = 0 ,s j, − ( r h u, v h ) ≤ Ch (cid:107) u (cid:107) H (Ω) (cid:107) h − [ v h ] (cid:107) F i ≤ Ch (cid:107) u (cid:107) H (Ω) ||| ( X h , ||| . Finally, using a Cauchy-Schwarz inequality and a Poincar´e inequality for η h γ M ( r h u − u, v h ) ω (cid:46) γ M (cid:107) r h u − u (cid:107) ω (cid:107) v h (cid:107) ω (cid:46) h (cid:107) u (cid:107) H (Ω) ||| ( X h , ||| . For the perturbation we have γ M ( δu, w h ) ω ≤ γ M (cid:107) δu (cid:107) ω (cid:107) w h (cid:107) ω . Collecting the above estimates ends the proof. 13 heorem 5.2.
Assume that u ∈ [ H (Ω)] d and p ∈ H (Ω) . Let ˜ u h be defined by (3.17) then (cid:107) ˜ u h (cid:107) H (Ω) + (cid:107) p h (cid:107) Ω (cid:46) (cid:107) u (cid:107) H (Ω) + (cid:107) p (cid:107) H (Ω) + γ M h − (cid:107) δu (cid:107) ω and, if δu = 0 ˜ u h (cid:42) u in [ H (Ω)] d and p h (cid:42) p in L Ω) . Proof.
For the pressure we immediately observe that (cid:107) p h (cid:107) Ω (cid:46) (cid:107) p h − π p (cid:107) Ω + (cid:107) p (cid:107) Ω (cid:46) h − ||| (0 , p h − π p ) ||| V + (cid:107) p (cid:107) Ω Then observe that by a Poincar´e inequality and the H -stability of the interpolation operator r h there holds (cid:107) ˜ u h (cid:107) H (Ω) ≤ (cid:107) ˜ u h − u h (cid:107) ,h + (cid:107) u h (cid:107) ,h ≤ (cid:107) ˜ u h − u h (cid:107) ,h + (cid:107) u h − r h u (cid:107) ,h + (cid:107) r h u (cid:107) ,h (cid:46) (cid:107) h − [ u h − r h u ] (cid:107) F i + (cid:107) u h − r h u (cid:107) ω + (cid:107)∇ ( u h − r h u ) (cid:107) h + (cid:107) u (cid:107) H (Ω) . Therefore (cid:107) ˜ u h (cid:107) H (Ω) (cid:46) h − ||| ( u h − r h u, ||| V + (cid:107) u (cid:107) H (Ω) and the first claim follows by applying Proposition 5.1.It follows that for δu = 0 we may extract a subsequence of pairs (˜ u h , p h ) that converges weaklyin [ H (Ω)] d × L (Ω). By construction the divergence of the H -conforming part satisfies (cid:107)∇ · ˜ u h (cid:107) Ω (cid:46) (cid:107) h − [ u h − r h u ] (cid:107) F i + (cid:107)∇ · u h (cid:107) h (cid:124) (cid:123)(cid:122) (cid:125) =0 + h (cid:107) u (cid:107) H (Ω) (cid:46) h (cid:107) u (cid:107) H (Ω) and hence (cid:107)∇ · ˜ u (cid:107) h → h →
0. It remains to show that the weak limit is a weak solution ofStokes equation. To this end consider, with w ∈ C (Ω), | a (˜ u h , w ) + b ( p h , w ) − l ( w ) | = | a h (˜ u h − u h , w ) + a h ( u h , w − r h w ) + b ( p h , w − r h w ) − l ( w − r h w ) | = | a h (˜ u h − u h , w ) − l ( w − r h w ) | (cid:46) ( (cid:107) h − [ u h ] (cid:107) F i + h (cid:107) f (cid:107) Ω ) (cid:107) w (cid:107) H (Ω) (cid:46) h (cid:107) w (cid:107) H (Ω) We conclude by taking the limit h → Theorem 5.3.
Assume that ( u, p ) ∈ [ H (Ω)] d × H (Ω) is the unique solution of (2.1) with u = u M in ω and the parameters R , R and R satisfy the assumptions of Theorem 2.1. If u h is the solution of (3.6) - (3.8) , with γ u , γ M > , γ p = γ x = 0 and (cid:107) δu (cid:107) Ω ≤ h , h > , then, for h > h , there holds (cid:107) u − u h (cid:107) B R ( x ) (cid:46) h τ where τ is the power from Theorem 2.1 and the hidden constant depends on R /R , the local meshgeometry and (cid:107) u (cid:107) H (Ω) and (cid:107) p (cid:107) H (Ω) .Proof. First let u − u h = u − ˜ u h (cid:124) (cid:123)(cid:122) (cid:125) e u ∈ [ H (Ω)] d + ˜ u h − u h (cid:124) (cid:123)(cid:122) (cid:125) e h ∈ V h , where ˜ u h is defined by (3.17). We recall that (cid:107) e h (cid:107) Ω (cid:46) (cid:107) h − [ u h ] (cid:107) F i ≤ Ch (cid:107) u (cid:107) H (Ω) so we only need to bound (cid:107) e u (cid:107) B R ( x ) . Also introduce e p = p − p h ∈ L (Ω). It follows that ( e u , e p )is a solution to the Stokes’ equation on weak form with a particular right hand side. Indeed wehave for all ( w, q ) ∈ [ H (Ω)] d × Qa ( e u , w ) + b ( e p , w ) = l ( w ) − a (˜ u h , w ) + b ( p h , w ) =: (cid:104) f , w (cid:105) V (cid:48) ,V (5.1)14nd − b ( q, e u ) = b ( q, ˜ u h ) =: ( g , q ) Ω (5.2)where f ∈ V (cid:48) and g ∈ L (Ω). Now consider the problem (2.1) with homogeneous Dirichletboundary conditions on ∂ Ω and the right hand side f and g as defined above. This problem iswell-posed and we call its solution {E u , E p } ∈ [ H (Ω)] d × L (Ω). By the well-posedness of theproblem we know that (cid:107)E u (cid:107) H (Ω) + (cid:107)E p (cid:107) Ω ≤ (cid:107) f (cid:107) H − (Ω) + (cid:107) g (cid:107) Ω We know from equation (3.20), the fact that (cid:107)∇ · u h (cid:107) h = 0 and Proposition 5.1 that (cid:107) g (cid:107) Ω (cid:46) (cid:107) h − [ u h ] (cid:107) F i (cid:46) h and for (cid:107) f (cid:107) V (cid:48) we derive the boundsup w ∈ [ H ] d (cid:107) w (cid:107) =1 (cid:104) f , w (cid:105) V (cid:48) V = l ( w ) − a (˜ u h , w ) − b ( p h , w )= l ( w − r h w ) − a h (˜ u h − u h , w ) (cid:46) h (cid:107) f (cid:107) Ω + s j, − ( u h , u h ) (cid:46) h + (cid:107) δu (cid:107) ω . (5.3)Considering now the functions U := u − ˜ u h − E u and P := p − p h − E p we see that { U, P } is asolution to (2.1) with f = 0 and g = 0. By equation (2.3) we have { U, P } ∈ [ H ( (cid:36) )] d × H ( (cid:36) ) onevery compact (cid:36) ⊂ Ω. We may then apply Theorem 2.1 to U and obtain (cid:90) B R ( x ) | U | d x ≤ C (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) τ (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) − τ . (5.4)These results may now be combined in the following way to prove the theorem. First by thetriangle inequality, writing u − u h = U + E u + ˜ u h − u h , (cid:107) u − u h (cid:107) B R ( x ) ≤ (cid:107)E u (cid:107) B R ( x ) + (cid:107) ˜ u h − u h (cid:107) B R ( x ) + (cid:107) U (cid:107) B R ( x ) = I + II + III.
By (2.2) and (5.3) there holds for the first term I (cid:46) (cid:107)E u (cid:107) H (Ω) (cid:46) h + (cid:107) δu (cid:107) ω and using the discrete interpolation and Proposition 5.1 II = (cid:107) ˜ u h − u h (cid:107) B R ( x ) (cid:46) (cid:107) h − [ u h ] (cid:107) F i (cid:46) h + (cid:107) δu (cid:107) ω . For the last term, using (5.4), we have
III (cid:46) (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) τ/ (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) (1 − τ ) / . By the definition of U and since by assumption B R ( x ) ⊂ ω (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) (cid:46) (cid:107) r h u − u h (cid:107) ω + (cid:107) r h u − u (cid:107) ω + (cid:107) ˜ u h − u h (cid:107) ω + (cid:107)E u (cid:107) B R ( x ) (cid:46) h + (cid:107) δu (cid:107) ω . (5.5)Here we applied Proposition 5.1, (3.14), discrete interpolation (3.20), and (2.2) applied to E u .Finally by the triangle inequality, the a priori assumption u ∈ H (Ω), (2.2) and the first claim ofTheorem 5.2 we have (cid:32)(cid:90) B R ( x ) | U | d x (cid:33) ≤ (cid:107) u (cid:107) H (Ω) + (cid:107) ˜ u h (cid:107) H (Ω) + (cid:107)E u (cid:107) H (Ω) (cid:46) h − (cid:107) δu (cid:107) ω . The claim follows by collecting the bounds on the terms I − III and applying the assumption onthe perturbations in data versus the mesh-size. 15 emark 5.4.
It is straightforward to prove the Proposition 5.1 and the Theorems 5.2 and 5.3also for γ p ≥ and γ x ≥ and thereby extending the analysis to include the method (3.12) . Weleave the details for the reader. Remark 5.5.
One may also introduce perturbations in the right hand side f . Provided theseperturbations are in [ L (Ω)] d the same results holds. Details on the necessary modifications canbe found in [7]. Our numerical example is set in the unit square Ω = (0 , with zero right hand side and data givenin the disc S / := { ( x, y ) ∈ R : (cid:112) ( x − . + ( y − . < . } . The flow is nonsymmetricwith the exact solution given by u ( x, y ) = (20 xy , x − y ) and p ( x, y ) = 60 x y − y − . We consider the formulation (3.6)-(3.8), with l ( w h ) = 0 For the parameters we chose, γ M =800 and γ u = 10 − , γ p = γ z = γ x = 0. First we perform the computation with unperturbeddata. The results are presented in the left graphic of Figure 1. We report the velocity errorboth in the global L -norm (open square markers), the local L -norm in the subdomain where (cid:112) ( x − . + ( y − . < .
375 (filled square markers) and in the residual quantities of (6.1)(circle markers, r filled, r open), r := (cid:32)(cid:90) S / ( u h − u ) d x (cid:33) and r := (cid:107) h − [ u h ] (cid:107) F i . (6.1)The global pressure is plotted with triangle markers. The error plots for this case are given infigure 1. We observe the O ( h ) convergence of the residual quantities (6.1). The global velocityand pressure L -errors appears to have approximately O ( | log ( h ) | − ) convergence. The local errormatches the result of Theorem 5.3. Indeed the dotted line is shows the behavior of the quantity C (cid:107) e (cid:107) . ( r + r ) . + 10 h illustrating the different components of the local error used in the proofof the theorem. We see that this quantity (with a properly chosen constant) gives a good fit withthe local error.The same computation was repeated with a 1% relative random perturbation of data. Theresults for this case is reported in the right plot of Figure 1. As predicted by theory the resultsare stable under perturbation of data as long as the discretization error is larger than the randomperturbation (up to a constant). When the perturbations dominate the errors in all quantitiesappear to stagnate. 16 .01 0.10.0010.010.11 Figure 1: Relative L -error against mesh-size, left unperturbed data, right with 1% relative noise.Reference lines are the same in both plots and of orders, dashed lines ≈ O ( h ) with differentconstants, dash dot ≈ O ( h ) and dotted C (cid:107) e (cid:107) . ( r + r ) . + 10 h References [1] B. A. Abda, I. B. Saad, and M. Hassine. Data completion for the Stokes system.
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