Gunar Matthies 1, Piotr Skrzypacz 2 and Lutz Tobiska 2. Introduction

Transcription

1 A UNIFIED CONVERGENCE ANALYSIS FOR LOCAL PROJECTION STABILISATIONS APPLIED TO THE OSEEN PROBLE Gunar atthies 1, Piotr Skrzypacz 2 and Lutz Tobiska 2 Abstract. The discretisation of the Oseen problem by finite element methods suffers in general from two reasons. First, the discrete inf-sup Babuška Brezzi condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard twolevel version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. oreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modeling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach athematics Subject Classification. 65N12, 65N30, 76D05. October 6, Introduction The discretisation of the Oseen problem by finite element methods suffers in general from two reasons. First, the discrete inf-sup Babuška Brezzi condition can be violated. Second, spurious oscillations occur in case of higher Reynolds numbers due to the dominating convection. The idea of streamline upwind Petrov Galerkin SUPG stabilisation has been proposed for the advective term in [9] and extended to the Stokes equations in [18] where a stabilised method is considered accommodating low equal-order interpolation to be stable and convergent. This formulation circumvented the need to abide by inf-sup condition for many interpolations. In an attempt to get the stability features of these works, a method is proposed in [13] that is at the same time advective stable and overcomes the inf-sup restrictions of the standard Galerkin method. A detailed error analysis of these SUPG-type stabilisations applied to the incompressible Navier Stokes equations, including both the case of inf-sup stable and equal-order interpolations, can be found in [32]. Despite the progress of the SUPG method in theory and application, an essential drawback of this method is in particular for higher order interpolations that various terms need to be added to the weak formulation to guarantee the consistency of the method in a strong way Galerkin orthogonality holds for smooth solutions. Residual-based stabilisation methods which use inf-sup stable pairs of elements reduce the number of terms which have to be added to the Keywords and phrases: Stabilised finite elements, Navier Stokes equations, equal-order interpolation Partially supported by the German Research Foundation DFG through grants To143 and FOR Fakultät für athematik, Ruhr-Universität Bochum, Universitätsstraße 150, D Bochum, Germany; gunar.matthies@ruhr-uni-bochum.de 2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität agdeburg, Postfach 4120, D agdeburg, Germany; piotr.skrzypacz@mathematik.uni-magdeburg.de & tobiska@mathematik.uni-magdeburg.de

2 2 Galerkin formulation [14, 25]. However, an additional coupling term between velocity and pressure makes their analysis difficult. Over the last years, several approaches have been developed to relax the strong coupling of velocity and pressure in SUPG-type stabilisations and to introduce symmetric versions of the stabilising terms, for an overview see [6, 23]. The local projection method is designed for equal-order interpolation and allows a separation of velocity and pressure in the stabilisation terms. It has been introduced for the Stokes problem in [3], extended to the transport equation in [4], and analysed for low order discretisations of the Oseen equations in [5]. Some variants and applications are discussed in [7, 8]. In the local projection method, the stabilisation term is based on a projection π h : Y h D h of the finite element space Y h approximating velocity and pressure into a discontinuous space D h. Stabilisation of the standard Galerkin method is achieved by adding terms which give a weighted L 2 -control over the fluctuations id π h of the gradients of the quantity of interest. The key idea in the error analysis of the local projection scheme is the construction of an interpolant into Y h which exhibits an additional orthogonality property with respect to the discontinuous space D h. In [5], the case of low order Q r -elements r = 1, 2 on quadrilaterals d = 2 and hexahedra d = 3 has been considered. There, π h has been chosen to be the L 2 -projection onto the space of discontinuous Q r 1 -elements on a coarser mesh. Unfortunately, this two-level approach leads to a stencil being less compact than for the SUPG-type stabilisation. The main objective of this paper is to give a general convergence theory of local projection schemes leading to the same optimal a-priori error estimates which are known for the SUPG method. To this end, we study under which conditions an interpolant into Y h with additional orthogonality properties with respect to D h can be constructed. We show that an inf-sup condition for the spaces Y h and D h is sufficient for the existence of such an interpolant. Our general theory allows us to consider large classes of spaces Y h and D h, including the two-level approach on simplices, quadrilaterals, and hexahedra for arbitrary but fixed polynomial degree r 1. oreover, we can derive local projection schemes not only as a two-level approach but also for pairs of spaces Y h /D h which are defined on the same mesh family T h. This opens the way to circumvent the disadvantage of the classical form of the local projection scheme which produces a larger stencil. As we will show, this new approach of enriched approximation spaces works also on simplices, quadrilaterals, and hexahedra for arbitrary polynomial degree r 1. It is well known that stabilised methods can also be derived from a variational multiscale formulation [17, 19, 20, 31]. Based on a scale separation of the underlying finite element spaces, it has been shown that it is sufficient to stabilise only the fine scale fluctuations. This results into a stabilising term which gives a weighted L 2 -control over the gradient of fluctuations instead of the fluctuations of gradients [12, 16]. We will discuss the relation between this subgrid modeling approach and the local projection scheme in detail. In particular, we show that for linear elements on simplices both approaches lead to the same discrete problem. The plan of the paper is as follows. In Section 1, a weak formulation of the Oseen equations and its standard Galerkin discretisation is given. We formulate the local projection method in an abstract setting. Section 2 is devoted to the convergence analysis of the local projection method in this abstract setting. The basic tool is the construction of a special interpolant based on the fulfilment of a local inf-sup condition. Proving the stability independent of the Reynolds number and the approximated Galerkin orthogonality, we conclude optimal a-priori estimates. The application of the theory in the framework of two-level methods is studied in Section 3 where the focus is on defining pairs of finite element spaces satisfying the local inf-sup condition given in Section 2. We extent in Section 4 the analysis to spaces which are defined on the same mesh. Starting from the space D h, the space Y h is obtained by enriching standard finite element spaces. Additionally, we study the relation of local projection schemes to the subgrid modeling approach in Section 5. We summarise our results in Section 6. Notation. Throughout the paper C will denote a generic positive constant which is independent of the Reynolds number and the mesh. Subscripted constants such as C 1 are also independent of the Reynolds number and the mesh, but have a fixed value. We will write shortly α β, if there are positive constants C and C such that Cβ α Cβ

3 holds. Our Oseen problem will be considered in the domain Ω R d, d = 2, 3, which is assumed to be a polygonal or polyhedral domain with boundary Ω. For a measurable subset G of Ω, the usual Sobolev spaces W m,p G with norm m,p,g and semi-norm m,p,g are used. In the case p = 2, we have H m G = W m,2 G and the index p will be omitted. The L 2 inner product on G is denoted by, G. Note that the index G will be omitted for G = Ω. This notation of norms, semi-norms, and inner products is also used for the vector-valued and tensor-valued case. 1. Local projection stabilisation in an abstract setting 1.1. Weak formulation of the Oseen problem Let us consider the Oseen problem ν u + b u + σu + p = f in Ω, u = 0 in Ω, u = 0 on Ω, 1.1 as a linearisation of the steady σ = 0 and the non-steady σ > 0 time-discretised Navier Stokes equations. Here, we assume b W 1, Ω with b = 0. Let V = H 1 0Ω, Q = L 2 0Ω = {q L 2 Ω : q, 1 = 0}. We introduce on the product space V Q the bilinear form A given by A u, p; v, q := ν u, v + b u, v + σu, v p, v + q, u. 1.2 A weak formulation of the Oseen problem 1.1 reads: The property Find u, p V Q such that for all v, q V Q : A u, p; v, q = f, v b v, v = b v v, 1 = b, v v = 0 v V allows to apply the Lax ilgram Lemma in the subspace of divergence-free functions and to establish a unique velocity field u. A unique pressure p Q such that u, p solves 1.3 follows from the Babuška Brezzi condition for the pair V, Q [15] Galerkin discretisation For the finite element discretisation of the Oseen problem 1.3, we introduce a shape regular decomposition T h of Ω into d-dimensional simplices, quadrilaterals or hexahedra. The diameter of a cell K will be denoted by h K and the mesh parameter h represents the maximum diameter of the cells K T h. Let Y h H 1 Ω be a finite element space of continuous, piecewise polynomial functions defined over T h. Assumption A1: There is an interpolation operator i h : H 1 Ω Y h such that i h : H 1 0 Ω Y h H 1 0 Ω and w i h w 0,K + h K w i h w 1,K C h l K w l,ωk w H l ωk, K T h, 1 l r + 1, 1.5 where ωk denotes a certain local neighbourhood of K which appears in the definition of these interpolation operators for non-smooth functions, see [11, 29] for more details. We will also apply this type of interpolation operator for vector-valued functions in a component-wise manner. We indicate this by using boldface notations, for example i h : V Y d h V. Remark 1.1. The existence theory of interpolation operators for non-smooth functions satisfying assumption A1 is well established in the literature, see [1, 11, 29]. 3

4 4 For simplicity of presentation, we consider in this paper the case of equal-order interpolation, thus assuming V h := Y d h V and Q h := Y h Q. Now, the standard Galerkin discretisation of 1.3 reads: Find u h, p h V h Q h such that for all v h, q h V h Q h : A u h, p h ; v h, q h = f, v h 1.6 In general, this formulation suffers from two reasons: the violation of the discrete Babuška Brezzi condition β 0 > 0 : inf q h Q h q h, v h sup β 0, 1.7 v h V h q h 0 v h 1 and the dominating advection in case of ν 1. Both instability phenomena can be handled by the local projection technique which will be the topic of the next subsection Stabilisation by local projection In order to explain the stabilisation method, we start with some additional notations. By a macro element we denote the union of one or more neighbouring cells K T h. The diameter of a macro element is denoted by h. We assume that the decomposition of Ω into macro elements h is non-overlapping and also shape regular, in particular h K h, K, h. One can think of having first the decomposition h into macro elements from which the decomposition T h is generated by certain refinement rules. Note that we also allow the case h = T h. Let D h denote a discontinuous finite element space defined on the macro decomposition h and D h := {q h : q h D h }. Further, let π : L 2 D h be a local projection which defines the projection π h : L 2 Ω D h by π h w := π w. Associated with the projection π h is the fluctuation operator κ h : L 2 Ω L 2 Ω defined by κ h := id π h, where id : L 2 Ω L 2 Ω is the identity. As in the previous subsection, we apply these operators for vector-valued functions in a component-wise manner and indicate this by using boldface notations, e.g. π h : L 2 Ω D h and κ h : L 2 Ω L 2 Ω. Assumption A2: Let the fluctuation operator κ h satisfy the following approximation property: κ h q 0, C h l q l, q H l, h, 0 l r. 1.8 Remark 1.2. We shortly discuss the case in which π h is the L 2 -projection in D h and the space D h contains the space P r 1 of polynomials of degree less than or equal to r 1, r 1. This means that π h w w, w h = 0 w h D h, w L 2 Ω, 1.9 and P r 1 D h Since D h is discontinuous over the macro element faces, 1.9 can be localised and π : L 2 D h is locally defined by π w w, w h = 0 w h D h, w L In this case, the L 2 -projection π : L 2 D h becomes the identity on the subspace P r 1 H l. Now, the Bramble Hilbert Lemma gives the approximation properties for κ h = id π h stated in assumption A2.

5 5 We will modify the discrete problem 1.6 by adding the stabilisation term S h uh, p h ; v h, q h := τ κ h b u h, κ h b v h + µ κ h u h, κ h v h + α κ h p h, κ h q h, 1.12 where τ, µ, and α are user-chosen constants. The optimal mesh-dependent choice will follow from the error analysis of the method. Now, our stabilised scheme reads: Find u h, p h V h Q h such that for all v h, q h V h Q h : A u h, p h ; v h, q h + S h uh, p; v h, q h = f, v h Existence, uniqueness, and convergence properties of the solutions u h, p h V h Q h will be studied in the next section Special interpolant 2. Convergence analysis The key ingredient of the error analysis of the local projection method is the construction of an interpolant j h : H 1 Ω Y h such that the error w j h w is L 2 -orthogonal to D h without loosing the standard approximation properties. Let Y h := {w h : w h Y h, w h = 0 on Ω\}. Assumption A3: Let the local inf-sup condition be satisfied. β 1 > 0, h > 0 h : inf sup q h D h v h Y h v h, q h v h 0, q h 0, β 1 > Remark 2.1. It is clear that Y h compared to D h has to be rich enough for satisfying A3. In particular, a necessary requirement is dim Y h dim D h. 2.2 On the other hand D h has to be large enough to guarantee A2. In Section 3, we follow this idea for a given space Y h by choosing D h as a discontinuous finite element space on a coarser mesh; its dimension small enough to satisfy A3 but big enough to fulfil A2. A different strategy is used in Section 4 where both spaces are defined on the same mesh, D h such that A2 holds and Y h is enriched by additional functions to fulfil A3. Theorem 2.2. Let assumptions A1, A3 be satisfied. Then, there are interpolation operators j h : H 1 Ω Y h and j h : V V h satisfying the following orthogonality and approximation properties: w j h w, q h = 0 q h D h, w H 1 Ω, 2.3 w j h w 0 + h w j h w 1 C h l w l w H l Ω, 1 l r + 1, 2.4 w j h w, q h = 0 q h D h, w V, 2.5 w j h w 0 + h w j h w 1 C h l w l w V H l Ω, 1 l r

6 6 Proof. We use a result in [15, Lemma I.4.1]: The linear continuous operator B h : Y h D h defined by B h v h, q h Dh := v h, q h v h Y h, q h D h is an isomorphism from W h onto D h with β 1 v h 0, B h v h Dh v h W h if and only if A3 holds. Here, D h denotes the dual space of D h, W h := {v h Y h : v h, q h = 0 q h D h }, and W h the L 2 -orthogonal complement of W h in Y h. Consequently, for each w H 1 Ω we have a unique z h w W h such that B h z h w, q h Dh = z h w, q h = w i h w, q h q h D h, 2.7 z h w 0, 1 β 1 w i h w 0,. 2.8 We take j h w := i h w + z h w for all h. According to Y h Y h, a global interpolation j h : H 1 Ω Y h is defined which satisfies w j h w β1 w i h w 0 C h l w l w H l Ω, 1 l r The orthogonality property 2.3 follows from 2.7 and the definition of j h. It remains to show the approximation property for the H 1 -seminorm. To this end, we apply an inverse inequality, use the approximation property 1.5, and sum up over all macro cells h to get z h w 1, Ch 1 z hw 0, Ch 1 w i hw 0,, /2 z h w 1 z h w 2 1, C h l 1 w l Using the triangle inequality w j h w 1 w i h w 1 + z h w 1, we get 2.4. Now, 2.5 and 2.6 are extensions to the vector-valued case. Taking into consideration the definition j h w = i h w + z h w, the mapping property i h : V V h, and that z h w vanishes on the boundary Ω, we get j h : V V h, 2.5, and 2.6. Remark 2.3. We see from 2.9 that the constant β 1 should be independent of the mesh size h to guarantee the approximation properties 2.4 and 2.6 of the interpolant j h. Note that 2.2 does not guarantee that the constant β 1 is independent of the mesh size h. Remark 2.4. Note that by setting q h = 1 in 2.3 we get j h w, 1 = w, 1 for all w H 1 Ω. This implies in particular j h : H 1 Ω Q Q h. Remark 2.5. Following the ideas in [30] and assuming a family of macro elements which is equivalent to a reference macro element, one can show that 2.1 reduces to show that holds true. N := {q h D h : q h, v h = 0 v h V h } = {0}

7 2.2. Stability Let us introduce the mesh-dependent norm on the product space V Q by v h, q h := ν v h σ v h ν + σ q h S h vh, q h ; v h, q h 1/ We show that the bilinear form A + S h satisfies an inf-sup condition on V h Q h. Lemma 2.6. Assume A1, A3, and maxν, σ, τ, µ, h 2 /α C. Then, there is a positive constant β 2 independent of ν and h such that holds true. A + S h vh, q h ; w h, r h inf sup β 2 > v h,q h V h Q h w h,r h V h Q h v h, q h w h, r h Proof. Let us consider an arbitrary v h, q h V h Q h. Choosing w h, r h = v h, q h, we have A + S h v h, q h ; v h, q h = ν v h σ v h S h vh, q h ; v h, q h 2.14 due to property 1.4. Now we consider another choice to generate an L 2 -norm control over the pressure. For any q h Q h, the continuous Babuška Brezzi condition guarantees the existence of a function v qh V such that v qh, q h = q h, q h, v qh 1 C q h We choose w h, r h = j h v qh, 0 where j h is the interpolant of Theorem 2.2 satisfying 2.5 and 2.6. Thus, we obtain A v h, q h ; j h v qh, 0 = q h 2 0 q h, j h v qh v qh + b v h, j h v qh +ν v h, j h v qh + σv h, j h v qh We estimate the last four terms on the right hand side. Starting with an integration by parts of the first of them, we get q h, j h v qh v qh = q h, j h v qh v qh = κ h q h, j h v qh v qh, q h, j h v qh v qh α κ h q h 2 0, 1/2 1 α j h v qh v qh 2 0, 1/2 C S h vh, q h ; v h, q h 1/2 vqh 1 C S h vh, q h ; v h, q h 1/2 qh 0 q h C S h vh, q h ; v h, q h Integrating by parts the third term in 2.16, using the H 1 stability of j h which follows from Theorem 2.2 and 2.15, we obtain 7 b v h, j h v qh = v h, b j h v qh C v h 0 j h v qh 1 q h C v h

8 8 For estimating the remaining terms in 2.16, we use maxν, σ C to get ν v h, j h v qh + σv h, j h v qh ν v h 1 + σ v h 0 jh v qh 1 C ν 1/2 v h 1 + σ 1/2 v h 0 qh 0 q h The Cauchy Schwarz inequality and the L 2 -stability of κ h give + C ν v h σ v h Sh vh, q h ; j h v qh, 0 C Sh vh, 0; v h, 0 1/2 jh v qh 1 C S h vh, q h ; v h, q h 1/2 qh 0 q h C S h vh, q h ; v h, q h Let X := ν v h σ v h S h vh, q h ; v h, q h 1/2 denote the part of the triple norm without L 2 -control over the pressure. Using , we get from 2.16 A + S h v h, q h ; j h v qh, 0 q h Now, we multiply 2.21 by 2ν + σ and use the Poincare inequality to estimate Hence, we obtain 2ν + σ v h 2 0 C ν v h σ v h 2 0. C X 2 C v h A + S h v h, q h ; 2ν + σj h v qh, 0 ν + σ q h 2 0 C 1 X with a suitable constant C 1. We define for an arbitrary v h, q h V h Q h Then, we have and A + S h v h, q h ; w h, r h w h, r h := v h, q h + 2ν + σ 1 + C 1 j h v qh, 0 V h Q h. ν + σ q h C 1 X 2 1 = v h, q h C C C 1 2ν + σ 2ν + σ w h, r h v h, q h + j h v qh, 0 v h, q h + j h v qh C C 1 v h, q h + Cν + σ q h 0 C 2 v h, q h From 2.23 and 2.24 we conclude 2.13 with β 2 = 1/C C 1. Remark 2.7. Note that for σ > 0 we have control over the L 2 -norm of the pressure and the velocity uniformly with respect to ν > 0. However, in the case σ = 0 we loose this control for ν 0. Remark 2.8. The unique solvability of the stabilised discrete problem 1.13 follows directly from Lemma 2.6.

9 2.3. Approximated Galerkin orthogonality In contrast to residual-based stabilisation schemes [6], we do not have the Galerkin orthogonality. Therefore, we investigate in this subsection the consistency error. Lemma 2.9. Let u, p V Q be the solution of 1.3 and u h, p h V h Q h be the solution of 1.13, respectively. Then, Au u h, p p h ; v h, q h = S h u h, p h ; v h, q h v h, q h V h Q h Proof. We get 2.25 simply by subtracting 1.13 from 1.3. Lemma Let the fluctuation operator κ h satisfy A2. Further, assume maxτ, µ, α Ch and b W r, h with max h b r,, C. Then, for u, p H r+1 Ω H r+1 Ω we have Sh u, p; vh, q h C h r+1/2 u r+1 + p r+1 vh, q h v h, q h V h Q h Proof. From the definition of the stabilising term we get S h u, p; v h, q h 1/2 1/2 1/2 vh S h u, p; u, p S h v h, q h ; v h, q h S h u, p; u, p, q h. Using the approximation properties of κ h and the asymptotic behaviour of the parameters τ, µ, and α, we see that S h u, p; u, p C h 2r+1 b u 2 r, + u 2 r, + p 2 r, and 2.26 follows A-priori error estimate From stability and consistency we get an a-priori error estimate in the usual way. The important aspect is that the constant in the error bound will be independent of the viscosity ν and h. Theorem Assume A1 A3 and τ, µ, α h. Let u, p H 1 0Ω H r+1 Ω L 2 0Ω H r+1 Ω be the solution of 1.3 and u h, p h V h Q h be the solution of the local projection method Then, there is a positive constant C independent of ν and h such that holds true. u u h, p p h Cν 1/2 + h 1/2 h r u r+1 + p r+1 Proof. Starting with Theorem 2.2, we get an estimate for the error to the interpolants: j h u u h, j h p p h 1 A + S h jh u u h, j h p p h ; w h, r h sup β 2 w h,r h V h Q h w h, r h 1 A + S h u uh, p p h ; w h, r h sup β 2 w h,r h V h Q h w h, r h + 1 A + S h jh u u, j h p p; w h, r h sup. β 2 w h,r h V h Q h w h, r h 2.27

10 10 Using Lemmata 2.9 and 2.10, we estimate the first term by A + S h u u h, p p h ; w h, r h = S h u, p; wh, r h C h r+1/2 u r+1 + p r+1 wh, r h. For the estimation of the second term, we consider each individual addend in A+S h j h u u, j h p p; w h, r h separately. The estimation of ν j h u u, w h + σj h u u, w h C ν 1/2 + σ 1/2 hh r u r+1 w h, r h is standard. When estimating the next three terms, we use the interpolant constructed in Theorem 2.2. Integrating by parts, we get for τ, µ, α h, b j h u u, w h = jh u u, b w h = jh u u, κ h b w h Finally, we obtain C h r+1/2 u r+1 S h wh, 0; w h, 0 1/2, 2.28 p jh p, w h = p jh p, κ h w h C h r+1/2 p r+1 S h wh, 0; w h, 0 1/2, 2.29 rh, j h u u = rh, j h u u = κh r h, j h u u C h r+1/2 u r+1 S h 0, rh ; 0, r h 1/ S h jh u u, j h p p; w h, r h S h jh u u, j h p p; j h u u, j h p p 1/2S h wh, r h ; w h, r h 1/2 Collecting all estimates above, we have shown By using the triangle inequality and the approximation property the statement of the Theorem follows. C h r+1/2 u r+1 + p r+1 wh, r h. j h u u h, j h p p h C ν 1/2 + h 1/2 h r u r+1 + p r u u h, p p h u j h u, p j h p + j h u u h, j h p p h u j h u, p j h p C ν 1/2 + h 1/2 h r u r+1 + p r+1, Remark In comparison with the SUPG method, we obtain the same rate of convergence with respect to the norm 1/2 v, q ν v σ 1/2 v ν + σ q 2 0 for equal-order interpolation [28, 32]. oreover, the SUPG method gives additional control over τk b v + q 2 0,K + µ K v 2 1/2 0,K. K T h

11 Recently, it has been shown [24] that by choosing the parameter τ K appropriately, the SUPG also allows a separate control over the terms b v 0,K and q 0,K. This behaviour of the SUPG method is related to the behaviour of the local projection method where an additional control is only guaranteed over fluctuations of these quantities, i.e. with respect to a slightly weaker norm. On the other hand, it is well-known that too much stabilisation can lead to smearing out boundary layers. Thus, the right amount of stabilisation seems to be important. Later we will see that the amount of stabilisation in the local projection method can be controlled by choosing the spaces Y h and D h, respectively. This flexibility makes the local projection method very attractive. Finally, we discuss two slightly modified approaches resulting in the same error estimates as those given in Theorem The first modification consists in replacing the stabilising term S h from 1.12 by S 1 hu h, p h ; v h, q h := τ κ h u h, κ h v h + α κ h p h, κ h q h 2.32 which gives control over the fluctuations of the gradients of the velocities instead of separate control over the fluctuations of the derivatives in the streamline direction and the divergence, respectively. In the second modification, we replace the stabilising term S h from 1.12 by a term Sh 2 which is spectral equivalent, i.e., there are positive constants C 3, C 4, independent of ν and h, such that C 3 S h vh, q h ; v h, q h S 2 h vh, q; v h, q h C 4 S h vh, q h ; v h, q h v h, q h V h Q h Note that the choice of the parameters τ, µ, and α defining S h influences the selection of possible stabilising terms Sh 2 satisfying When replacing S h by Sh i in 2.12, i = 1, 2, two new mesh-dependent norms appear which will be denoted by, i, i = 1, 2. Corollary Assume A1 A3 and τ, µ, α h. Let u, p H 1 0Ω H r+1 Ω L 2 0Ω H r+1 Ω be the weak solution of 1.3 and u h, p h V h Q h be the solution of the local projection method 1.13 with S h replaced by Sh 1. Then, for σ > 0 there is a positive constant C σ independent of ν such that holds true. u u h, p p h 1 C σ ν 1/2 + h 1/2 h r u r+1 + p r Proof. A careful check shows that Lemma 2.6 with S h and, replaced by S 1 h and, 1, respectively, is valid. Further, the additional smoothness assumption concerning b in Lemma 2.10 can be omitted since now the asymptotic behaviour of the parameters τ, α and the approximation properties of the fluctuation already give Sh 1 u, p; u, p C h 2r+1 u 2 r, + p 2 r,. However, the estimates 2.28 and 2.29 in the proof of Theorem 2.11 have to be modified. Consider first 2.29: p jh p, w h = p jh p, κ h w h C C τ 1/2 h r+1 p r+1, κ h w h 0, hr+1 p r+1, τ 1/2 κ h w h 0, C h r+1/2 p r+1 S 1 hw h, 0; w h, 0 1/2. 11

12 12 The estimation of 2.28 needs more care. We start as in the proof of Theorem 2.11 b j h u u, w h = jh u u, b w h = jh u u, κ h b w h C κh b w 0, h. h r+1 u r+1, Let b be the L 2 -projection of b in the space of piecewise constant functions with respect to the macro decomposition h. Using the L 2 -stability of κ h, an inverse inequality, and κ h b w h = b κ h w h, we get Since σ > 0, we end up with κ h b w h 0, κ h b b w h 0, + κ h b w h 0, b j h u u, w h C C h b 1,, w h 0, + C κh w h 0, C w h 0, + C κ h w h 0,. h r+1 u r+1, wh 0, + κ h w h 0, C σ h r+1/2 u r+1 σ w h S 1 hw h, 0; w h, 0 1/2. The remaining arguments are identical to those given in the proof of Theorem 2.11 and will be omitted. We come back to the second modification which replaces S h by a spectrally equivalent stabilisation term S 2 h. We assume 2.33, the consistency estimate S 2 h u, p; vh, q h C h r+1/2 u r+1 + p r+1 vh, q h v h, q h V h Q h 2.35 and the approximation property S 2 h jh u u, j h p p; v h, q h C h r+1/2 u r+1 + p r+1 vh, q h v h, q h V h Q h 2.36 to be satisfied. Corollary Assume A1, A3, and τ, µ, α h. Let u, p H 1 0Ω H r+1 Ω L 2 0Ω H r+1 Ω be the solution of 1.3 and u h, p h V h Q h be the solution of the local projection method 1.13 with S h replaced by Sh 2 satisfying 2.33, 2.35, and Then, there is a positive constant C independent of ν and h such that holds true. u u h, p p h 2 Cν 1/2 + h 1/2 h r u r+1 + p r Proof. A careful check shows that Lemma 2.6 with S h and, replaced by Sh 2 and, 2, respectively, is valid. Lemma 2.10 is replaced by Now following the lines of proof of Theorem 2.11 and bounding S h by C3 1 S2 h, we get j h u u h, j h p p h 2 Cν 1/2 + h 1/2 h r u r+1 + p r+1. The statement follows from the triangle inequality.

13 13 â 3 a 3 F K 2 K 2 K 1 a 0 K 1 â 0 K3 F 1 a 1 K 3 â 1 â 2 a 2 Figure 1. Reference macro triangle left and macro triangle h right. 3. Schemes based on local projection onto coarser meshes The triangulation T h consists of generic cells K whereas the macro mesh h consists of macro cells. The partition T h is formed by a suitable refinement of the macro mesh h which will be indicated by the notation h = T 2h Simplices Let be the unit d-simplex with the vertices â i, i = 1,..., d + 1. The barycentre of is denoted by â 0. The refinement of is done in the following way. Each child K i is given by the vertices â 0 and â j, j i, see left picture in Figure 1 for the 2d case. Let F : denote the affine mapping from the reference macro onto the macro cell h. This mapping defines cells K T h by setting K = F K i, i = 1,..., d + 1, h, see right picture in Figure 1 for the 2d case. For a function v : R, we define ˆv := v F : R. Furthermore, we consider the affine mapping F K : K K from a reference d-simplex K onto an arbitrary cell K T h. We choose for the approximation of velocity and pressure the finite element space of continuous, piecewise polynomials of degree r N. Let the projection space consist of discontinuous, piecewise polynomials of degree r 1 on T 2h, i.e. shortly where Y h /D h = P r,h /P disc r 1,2h P r,h : = {v H 1 Ω : v K F K P r K K T h }, P disc r 1,2h : = {v L 2 Ω : v F P r 1 T 2h }. We define the auxiliary space Ŷ as a counterpart of Y h by Ŷ := {ˆv : ˆv F 1 Y h},

14 14 i.e., on the reference d-simplex we have Ŷ = {w H 1 0 : w bki P r K i, i = 1,..., d + 1} where K i are children of resulting from the decomposition of. The existence of the interpolation operators satisfying the assumption A1 is obvious. Further, we can choose the local projection π in such a way that π = id on the subspace P r 1. This guarantees assumption A2 too. Lemma 3.1. Let the local projection scheme be defined for the pair Y h /D h = P r,h /Pr 1,2h disc with an arbitrary but fixed polynomial degree r N. Then, on shape regular simplicial meshes the local inf-sup condition A3 holds with the constant β 1 independent of h. Proof. We make use of the reference transformation F : and observe that the local inf-sup condition A3 can be in virtue of the constant det DF investigated merely on the reference macro inf sup q D h v Y h q, v q 0, v 0, = inf sup ˆq P r 1 c ˆv Y b c ˆq, ˆv c ˆq 0, c ˆv 0, c. 3.1 Let ˆq P r 1 be arbitrarily chosen. By ˆb : R we denote the piecewise linear hat function associated with â 0, i.e., ˆb bki is linear, ˆbâ 0 = 1, ˆbâ i = 0, i = 1,..., d + 1, and we set ˆv := ˆq ˆb. We note that ˆv Ŷ since ˆv is continuous, ˆv c = 0 and ˆv bki P r K i on each child K i, i = 1,..., d + 1. Since ˆb > 0 in, we state that 1/2 ˆq ˆq, ˆq ˆb 0, c is a norm on the space P r 1. Using the fact that all norms on finite dimensional spaces are equivalent, we get ˆq, ˆv c = ˆq, ˆq ˆb c β 1 ˆq 2 0, c 3.2 where the constant β 1 is independent of h. Due to ˆbˆx 1 ˆx, we have on the other hand ˆv 0, c ˆq 0, c. 3.3 From we conclude A Quadrilaterals and Hexahedra Let = 1, 1 d be the reference hyper-cube with the vertices â i, i = 1,..., 2 d. is refined into 2 d congruent cubes K i, i = 1,..., 2 d. Let F : be the multilinear reference mapping. The refinement of induces a refinement of into 2 d cells, see Figure 2 for the 2d case. The union of all these cells forms the principal mesh T h = {F K i : i = 1,..., 2 d }. Furthermore, we consider the multilinear mapping F K : K K from the reference hyper-cube K = 1, 1 d onto an arbitrary cell K T h. One can define the projection space D h in two ways, namely as an image of a space on the reference macro or directly on the macro. This leads to different finite element spaces, so we distinguish between both variants of the projection space D h. The mapped version of D h takes advantage of fixing the projection space locally on the reference macro. However, the interpolation property A2 turns out to be missing on arbitrary families of meshes [2,26]. To keep the notation clear, we use the extra superscript m for mapped finite element spaces D h.

15 15 a 3 â 4 â 3 K 4 K 3 F a 4 K 4 K 3 a 0 K 1 â 0 K 2 K 1 K 2 F 1 a 2 â 1 â 2 a 1 Figure 2. Reference macro quadrilateral left and macro quadrilateral h right Projection spaces based on mapped finite elements First, we consider the following finite element pair where Y h /D h = Q r,h /Q disc,m r 1,2h Q r,h : = {v H 1 Ω : v K F K Q r K K T h }, Q disc,m r 1,2h : = {v L2 Ω : v F Q r 1 T 2h }. The auxiliary space Ŷ is defined in analogy to the simplicial macros, i.e., on the reference hyper-cube we have Ŷ = {w H 1 0 : w bki Q r K i, i = 1,..., 2 d } where K i are children of resulting from the decomposition of. The construction of interpolation operators satisfying the assumption A1 is standard. oreover, since P r 1 Q disc,m r 1 holds true, the Bramble Hilbert lemma provides the optimal interpolation properties A2 on families of shape regular meshes. Following the lines of proof of Lemma 3.1 for establishing assumption A3, we find that in general det D F is no longer constant, and consequently 3.1 holds only in special cases parallelograms/parallelepipeds. Before we continue our investigation of the assumption A3, we recall some general properties of the multilinear reference mapping F T : T T from T = 1, 1 d onto an arbitrary quadrilaterals/hexahedra T. F T can be expanded as follows F T ˆx = m T + B T ˆx + G T ˆx 3.4 where m T := F T 0, B T := DF T 0, and G T ˆx = F T ˆx F T 0 DF T 0ˆx. Furthermore, we define by means of the Euclidian norm the distortion parameter γ T := sup ˆx b T B 1 T DG T ˆx and assume γ T γ < 1. Then, the mapping F T is one-to-one and from [27, Lemma 2] the estimate C 1 d!1 γ T d h d T det DF T ˆx Cd!1 + γ T d h d T 3.5

16 16 follows. oreover, on a family of uniformly refined meshes we have lim γ T = 0. h 0 For a parallelogram/parallelepiped T, the reference mapping F T is affine and γ T = 0. Lemma 3.2. Let the local projection scheme be defined for the pair Y h /D h = Q r,h /Q disc,m r 1,2h with an arbitrary but fixed polynomial degree r N. Then, the local inf-sup condition A3 holds with the constant β 1 = β 1 d, γ independent of h. Proof. From 3.5 we get q 2 0, Cd!1 + γ d h d ˆq 2 0, c q D h. 3.6 Let ˆb : R be the multilinear hat function associated with â0, i.e., ˆb bki Q 1 K i, i = 1,..., 2 d, ˆbâ 0 = 1, ˆbâi = 0, i = 1,..., 2 d. For an arbitrary q D h we choose vx := ˆq ˆb F 1 x where ˆq Q r 1. Since ˆq ˆb is continuous on, ˆq ˆb bki Q r K i, i = 1,..., 2 d, and ˆb c = 0, we have ˆvˆx := ˆqˆxˆbˆx Ŷ. Then, it follows from the estimate 3.5 q, v = qxvx dx = ˆqˆxˆvˆx det DF ˆx dˆx = ˆqˆxˆqˆxˆbˆx det DF ˆx dˆx Cd!1 γ d h d ˆqˆx 2 ˆbˆx dˆx. c The equivalence of norms on the finite dimensional space Q r 1 implies ˆq ˆb 0, c C ˆq 0, c ˆq Q r 1 c c and hence Using ˆbˆx 1 ˆx, we get v 2 0, q, v K Cd!1 γ d h d ˆq 2 0, c. 3.7 ˆqˆx 2 det DF ˆx dˆx Cd!1 + γ d h d ˆq 2 0, c. 3.8 From it follows immediately c q, v q 0, v 0, C d 1 γ C 1 + γ d 1 γ =: β 1 d, γ 1 + γ and thus, the local inf-sup condition A3 is proven. Alternatively, one can choose a smaller projection spaces D h as follows P disc,m r 1,2h := {v L2 Ω : v F P r 1 T 2h }. This results in more stabilisation in the sense that the stabilising term vanishes on the smaller subset P disc,m Q disc,m r 1,2h r 1,2h. The assumption A1 holds without any change. However to guarantee assumption A2 we have to restrict ourselves to suitably refined families of meshes, see [2] for quadrilaterals and [26] for hexahedra.

17 Corollary 3.3. Let the local projection scheme be defined for the pair Y h /D h = Q r,h /P disc,m r 1,2h with an arbitrary but fixed polynomial degree r N. Then, the local inf-sup condition A3 holds. Remark 3.4. To discuss other choices of the projection space D h, we mention that in the case Y h /D h = Q r,h /Q disc,m r 1,2h we have dim Y h = 2r 1 d r d = dim D h, in particular for r = 1 the dimensions of both spaces coincide. In the case r 2, one could think of choosing a larger projection space in order to minimise the stabilisation. A possible candidate would be D h = Q disc,m r,2h since now dim Y h = 2r 1 d r + 1 d = dim D h, r 2. The assumption A1 holds without any change, A2 would be satisfied with a higher rate than required. However, whether the inf-sup condition A3 holds, is an open problem Projection spaces based on unmapped finite elements We choose for the projection space D h the space of discontinuous, piecewise polynomials of degree r 1 posed directly on the generic cells, i.e. Y h /D h = Q r,h /P disc r 1,2h where Q r,h : = {v H 1 0 Ω : v K F K Q r K K T h }, P disc r 1,2h : = {v L 2 Ω : v P r 1 T 2h }. Again, the assumption A1 holds without any change but in contrast to the mapped version the approximation property A2 is fulfilled also. The local inf-sup condition follows from Lemma 3.5. Let the local projection scheme be defined for the pair Y h /D h = Q r,h /Pr 1,2h disc with an arbitrary but fixed polynomial degree r N. Then, on quadrilateral/hexahedral meshes the local inf-sup condition A3 holds with the constant β 1 = β 1 d, γ. Proof. Due to the fact that Pr 1,2h disc is contained in Qdisc,m r 1,2h the proof is a straightforward consequence of Lemma Schemes based on enrichment of approximation spaces This class of schemes takes advantage of constructing the finite element spaces Y h and D h on the same mesh. There are some indications of using only one principal mesh given recently in [6]. However, their choice of polynomial spaces of order at least r 2 and the construction of a nodal fluctuation operator is not optimal with respect to the order of convergence. We propose a novel method based on the enrichment of approximation spaces in order to satisfy the local inf-sup condition A3. The main benefit of this enrichment is the reduced stencil of the stiffness matrix and the avoidance of the difficult handling of data structures in the local projection methods which use coarser meshes. We present the method for two types of mesh geometries Simplices Let d+1 ˆbˆx := d + 1 d+1 ˆλ i ˆx 4.1 be the bubble function which takes the value 1 in the barycentre of the reference simplex K. Thereby ˆλ i, i = 1,..., d + 1, are barycentric coordinates on K. Furthermore, we define the enriched space i=1 P bubble r K := P r K + ˆb P r 1 K. 17

18 18 Let Y h /D h := Pr,h bubble /Pr 1,h disc be the pair of finite element spaces defined via the reference mapping P bubble r,h : = {v H 1 Ω : v K F K P bubble r K K T h }, P disc r 1,h : = {v L 2 Ω : v K F K P r 1 K K T h }. Obviously, the assumptions A1 and A2 are fulfilled. At a first glance the enriched space seems to be large, but a more careful look shows P r K + ˆb P r 1 K = P r K ˆb d i=1 P r i K { d } where P r K d = span ˆx αi i, α i = r, ˆx 1,..., ˆx d K is a monomial space. Using the fact that i=1 i=1 the bubble part of the space P r K is ˆb P r d+1 K, we have dim Ŷ K r d d d [ ] r i + d r i + d 1 = + d d d i=1 r 1 r 1 + d r 1 = + = dim P r 1 d d d K. The chosen enrichment is minimal with respect to the required inequality 2.2. Lemma 4.1. Let the local projection scheme be defined for the pair Y h /D h = Pr,h bubble /Pr 1,h disc with an arbitrary but fixed polynomial degree r N. Then, the local inf-sup condition A3 holds. Proof. Since the reference mapping is affine, the proof of the local inf-sup stability can be performed using the identity 3.1 for the reference cell K. For an arbitrary ˆq P r 1 K we choose ˆv := ˆb ˆq with ˆb defined by 4.1. Since ˆb ˆq Pr bubble K and ˆb ˆq K b = 0, we have ˆv Ŷ K, and with the analogous argumentation as in the proof of the Lemma 3.1 we get the required positive lower bound β 1 independent of the mesh size h Quadrilaterals and Hexahedra As for the method based on projection onto coarser meshes, we will distinguish between mapped and unmapped P disc,m r 1,h := {v L2 Ω : v K F K P r 1 K K T h } P disc r 1,h := {v L 2 Ω : v K P r 1 K K T h } finite element spaces for the projection space D h. In order to obtain the optimal order of the interpolation error for the mapped projection space, families of uniformly refined quadrilateral/hexahedral meshes are required, see again [2, 26], whereas for the unmapped spaces the interpolation property A2 holds on general shape regular meshes. We extend the approximation spaces in order to ensure the local inf-sup condition A3.

19 Projection spaces based on mapped finite elements Let d ˆbˆx = 1 ˆx 2 i Q 2 K, ˆx = ˆx 1,..., ˆx d K, d = 2, 3, 4.2 i=1 be the bubble function associated with the reference cell K = 1, 1 d. The enriched finite element space is set to Q bubble,1 r K := Q r K span ˆb ˆx r 1 i, i = 1,..., d. We define a pair of finite element spaces via the reference mapping Q bubble,1 r,h Y h /D h := Q bubble,1 r,h /P disc,m r 1,h : = {v H 1 Ω : v K F K Q bubble,1 r K K T h }. We note that in general the functions of spaces Q bubble,1 r,h, P disc,m r 1,h are not polynomials. Since Q r K Q bubble,1 r K the assumption A1 is satisfied. Assumption A2 holds on uniformly refined meshes, see [2, 26]. Lemma 4.2. Let the local projection scheme be defined for the pair Y h /D h = Q bubble,1 r,h /P disc,m r 1,h with an arbitrary but fixed polynomial degree r N. Then, the local inf-sup condition A3 holds with the constant β 1 = β 1 d, γ. Proof. For an arbitrary q D h K we choose vx := ˆq ˆb F 1 K x where ˆb 0 is the bubble function from 4.2, ˆq P r 1 K. Since ˆq = ˆq 0 + ˆq 1 with ˆq 0 span x r 1 i, i = 1,..., d and ˆq 1 Q r 2, we have ˆvˆx := ˆqˆxˆbˆx Q bubble,1 r K. Then, we proceed as in the proof of Lemma 3.2 and get from 3.5 q, v K q 0,K v 0,K C d 1 γk C 1 + γ K d 1 γ =: β 1 d, γ v Y h K q D h K. 1 + γ 19 This implies the local inf-sup condition A3. Remark 4.3. Note that the space Qr bubble,1 K has for r 2 exactly d base functions more than Q r K, independent of r. Remark 4.4. A comparison of the dimensions of the spaces Y h and D h shows that dim Ŷ K r 1 + d = r 1 d + d = dim P r 1 d K r N d N. In particular, the enrichment is optimal for biquadratic and bicubic elements on quadrilaterals and for triquadratic elements on hexahedra Projection spaces based on unmapped finite elements In order to satisfy A2 on general meshes, we propose an alternative way for setting the finite element pair Y h /D h. We choose the space Q bubble,2 r K := Q r K + ˆb Q r 1 K with the bubble function ˆb from 4.2 and define the enriched space Q bubble,2 r,h := {v H 1 0 Ω : v K F 1 K Qbubble,2 r K K T h }.

20 20 Now, our alternative choice is Y h /D h = Q bubble,2 r,h /Pr 1,h. disc Then, the properties A1 and A2 are naturally fulfilled. Lemma 4.5. Let the local projection scheme be defined for the pair Y h /D h = Q bubble,2 r,h /Pr 1,h disc with an arbitrary but fixed polynomial degree r N. Then, the local inf-sup condition A3 holds with β 1 = β 1 d, γ. Proof. For an arbitrary q D h, we set v = q b K where b K x := ˆb F 1 K x with the bubble function ˆb defined by 4.2 and the reference mapping F K. Since q P r 1 K, we find ˆq Q r 1 K. Consequently, ˆb ˆq Q bubble,2 r K and ˆb ˆq K b = 0. Then, we have ˆv := ˆb ˆq Ŷ K and v Y h K. In analogy to the proofs of Lemma 3.2 and Lemma 4.1, we obtain from the norm equivalence the positive lower bound Remark 4.6. The space Q bubble,2 r,h ˆq ˆb 0, b K C ˆq 0, b K ˆq Q r 1 K β 1 = β 1 d, γ = C d 1 γ. 1 + γ is more enriched than the space Q bubble,1 r,h. Comparing the dimensions of spaces Y h K and D h K, we can guess that the enriched space could be reduced. However, the question of the local inf-sup condition is still open. 5. Relation to subgrid modeling The idea of subgrid modeling goes back to Guermond and has firstly been applied to a scalar transport equation [16]. It is based on a scale separation of the underlying finite element space Y h = Y H Y H h where Y H stands for the large scale space and Yh H for the small scale space. Associated with the scale separation is a suitable projection operator P H : Y h Y H Y h which is the identity on the subspace Y H. Let κ h := id P H denote the fluctuation operator. We assume that the finite element space Y H is based on a shape regular decomposition of the domain into cells h of diameter h. Then, a stabilising term of the form Su h, v h = h κh u h, κ h v h or Su h, v h = h b κh u h, b κ h v h has been proposed to add to the standard Galerkin approach [12,16]. These stabilisation terms can be interpreted as an artificial diffusion term or an artificial diffusion in the streamline direction for the subscales which are represented by Yh H. This technique has been developed in different directions, for an extension to time-dependent convection-diffusion problems see e.g. [22]. Scale separation plays also an important role in large eddy simulation of turbulent flows, see [21]. The scale separation can be realised in different ways. In the two-level approach, Y h and Y H are standard finite element spaces on different refinement levels we indicate this by writing Y H = Y 2h and Yh H consists just of those hierarchical basis functions which are missing in the coarse space Y 2h to generate Y h. Another view is to consider Y h as a finite element space Y H enriched by Yh H which contains suitable functions, e.g. higher order polynomials. However, both variants differ from the local projection approach since the stabilisation term in the subgrid modeling approach is based on gradients of fluctuations id P H u h whereas the local projection method uses fluctuations of the gradients id π h u h.

21 In applications, the projection P H : Y h Y 2h in the two-level approach has been often chosen as the global Lagrange interpolant I 2h,r into Y 2h [3, 5 8, 23]. This leads to a stabilising term of the form Sh 3 uh, p h ; v h, q h = τ κ h u h, κ h v h + α κ h p h, κ h q h 5.1 instead of S h given by In the following subsections, we study the relations between the stabilising terms S 1 h given by 2.32 and S3 h Two-level approach on piecewise linear elements We consider first the case where T h is generated from a refinement of a shape regular triangulation T 2h by connecting the barycentre of each macro cell with its vertices, see Figure 1 for the 2d case. Let Y h and Y 2h denote the spaces of continuous, piecewise linear finite elements associated with the triangulations T h and T 2h, respectively. Lemma 5.1. Let d 1, π 2h,0 be the L 2 -projection onto the space P0,2h disc of piecewise constant functions, and I 2h,1 : Y h Y 2h be the Lagrange interpolant into the space P 1,2h of continuous, piecewise linear functions. Then, we have π 2h,0 v h = I 2h,1 v h v h P 1,h, T 2h. Thus, the discrete problems of the local projection method and the subgrid modeling approach coincide. Proof. We consider the case of an arbitrary dimension d 1. We restrict ourselves to the scalar case since the assertion for the vector-valued case follows immediately from the scalar one by a component-wise application. Let v h be the restriction of a finite element function v h Y h onto a macro simplex T 2h. Furthermore, let λ i, i = 1,..., d + 1, denote the barycentric coordinates with respect to the simplex. Defining the continuous, piecewise linear function ϕ 0 x = d + 1λ i x x K i, i = 1,..., d + 1, we can represent v h by its nodal functionals N i v = va i, i = 0,..., d + 1, as with Since N i ϕ 0 = 0, i = 1,..., d + 1, we have from which d+1 v h = N i v h λ i + Ñ0v h ϕ i=1 Ñ 0 v = N 0 v 1 d+1 N i v. d + 1 i=1 d+1 I 2h,1 v h = N i v h λ i i=1 d+1 I 2h,1 v h = N i v h λ i i=1 follows. Let h denote the piecewise applied gradient operator. Taking into consideration that h v h is constant on each subdomain K j, j = 1,..., d+1, and that K j = K /d+1, we compute the L 2 -projection onto P 0 21

22 22 to be Next we get from 5.2 for j = 1,..., d + 1 π 2h,0 h v h = 1 d+1 h v Kj h. d + 1 j=1 d+1 v Kj h = N i v h λ i + d + 1Ñ0v h λ j, 1 d+1 v Kj h = d + 1 j=1 i=1 d+1 i=1 N i v h λ i + Ñ0v d+1 h j=1 d+1 λ j = N i v h λ i. Since π 2h,0 v h = I 2h,1 v h holds true, the stabilising terms in both approaches are identical. i=1 Remark 5.2. In general we expect that π 2h,r 1 h v h I 2h,r v h v h P r,h, r 2 holds true where π 2h,r 1 is the L 2 -projection onto the space Pr 1,2h disc of discontinuous, piecewise polynomials of degree r 1 on the coarse mesh T 2h and I 2h,r : Y h P r,2h is the Lagrange interpolant into the space of continuous, piecewise polynomials of degree r on the coarse mesh T 2h. Similarly, we expect that in general on quadrilateral or hexahedral meshes T 2h π 2h,r 1 h v h I 2h,r v h v h Q d r,h, d 2, r 1 holds true where π 2h,r 1 is the L 2 -projection onto the space Q disc r 1,2h of discontinuous, piecewise polynomials of degree r 1 in each variable and I 2h,r : Y h Q r,2h is the Lagrange interpolant into the space of continuous, piecewise polynomials of degree r in each variable. As an example, we consider the case r = 2, d = 1. We get for the reference macro = 1, +1 and the piecewise quadratic function { 4ˆx1 ˆx if 0 ˆx 1, vˆx = 0 if 1 ˆx < 0, the relation ˆπ 2h,1 ˆ ˆv = ˆx 0 = ˆ Î 2h,2ˆv. Thus, in general the two approaches, subgrid modeling and local projection, do not lead to the same stabilisation term. However, we will see later that this does not automatically exclude the possibility of spectral equivalence of the stabilisation terms Enriched piecewise linear elements In the previous section we have seen that both methods, the local projection scheme and the subgrid modeling, results for the two-level approach with Y h = P 1,h and D h = P0,2h disc in the same stabilisation term. Now we will show that the same is also true for enriched piecewise linear elements, i.e., Y h = P1,h bubble and D h = P0,h disc. Lemma 5.3. Let d 1, π h,0 be the L 2 -projection onto the space P0,h disc of piecewise constant functions, and I h,1 : Y h P 1,h be the Lagrange interpolant into the space P 1,h of continuous, piecewise linear functions. Then, we have π h,0 v h = I h,1 v K h K v h P bubble 1,h, K T h. Thus, the discrete problems of the local projection method and the subgrid modeling approach coincide.