Stenberg s sufficiency criteria for the LBB condition for Axisymmetric Stokes Flow

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1 Stenberg s sufficiency criteria for the LBB condition for Axisymmetric Stokes Flow V.J. Ervin E.W. Jenkins Department of Mathematical Sciences Clemson University Clemson, SC, USA. Abstract In this article we investigate the LBB condition for axisymmetric flow problems. Specifically, the sufficiency condition for approximating pairs to satisfy the LBB condition established by Stenberg in the Cartesian coordinate setting is presented for the cylindrical coordinate setting. For the cylindrical coordinate setting, the aylor-hood (k = 2) and conforming Crouzeix-Raviart elements are shown to be LBB stable. A priori error bounds for approximations to the axisymmetric Stokes flow problem using aylor-hood and Crouzeix-Raviart elements are given. he computed numerical convergence rates for the error for an axisymmetric Stokes flow problem support the theoretical results. Key words. axisymmetric flow; LBB condition; Stenberg sufficiency condition AMS Mathematics subject classifications. 65N30 Introduction Accurate numerical simulations of 3-D fluid flow problems is a computationally challenging problem, involving the approximate solution of large (sparse) systems of linear equations. However, in the case the domain of the problem is a volume of revolution about a central axis, and the fluid flow is also invariant with respect to rotation about the central axis, a change of variable from a Cartesian to a cylindrical coordinate system significantly reduces the computational complexity. Specifically, the 3-D fluid flow problem decouples into a 2-D fluid flow problem and a scalar flow equation. However, this transformation from the 3-D problem to a 2-D problem results in differential operators with singularities on the the central axis, requiring the analysis to be done in suitably weighted Sobolev spaces. In the approximation of a fluid flow problem based on a weak formulation of the modeling equations, specifically those modeling Navier-Stokes and Stokes, an important component in the approximation vjervin@clemson.edu, lea@clemson.edu.

2 algorithm is ensuring that the velocity and pressure approximation spaces, X h X and Q h Q, respectively, satisfy the LBB condition, i.e. inf q Q h for some β IR +, where in Cartesian coordinates b(q, v) = b(q, v) sup β, (.) v X h q Q v X q v dx. (.2) Compatible pairs of velocity and pressure approximation spaces for fluid flow problems in Cartesian coordinates are well documented in the literature, see for example [6, 4. Commonly used elements include the mini-element, P isop 2 P, and aylor-hood pairs. here are a number of ways of establishing that (.) is satisfied for given approximation spaces X h and Q h [6. Of particular interest in this article is the general sufficient condition derived by Stenberg in [9. Briefly stated, in [9 Stenberg showed that if the partition of the domain can be classified into a finite number of macroelements such that for each macroelement, M, the dimension of { } N M = q Q h : q v dx = 0, v {w X h : w M = 0} (.3) M is equal to one, then (.) is satisfied. In the case of axisymmetric flow in cylindrical coordinates one requires a 2-D LBB condition (.) be satisfied. Here however b(q, v) = b a (q, v) := q a v r dx + q v r dx, (.4) where a = [ / r, / z, v = [v r, v z, dx = drdz, and the function spaces (and norms) for X and Q differ significantly from the Cartesian case. Ruas in [8 showed that (.)(.4) was satisfied for rectangular based Q 2 discp elements, and for P 2 + bubble discp on a restricted triangulation of. In [2 Belhachmi, Bernardi, Deprais showed that (.)(.4) was satisfied on a regular triangulation of for P isop 2 P elements (which also implied (.)(.4) for aylor Hood P 2 P elements). In this paper we establish that the sufficient condition of Stenberg also applies to (.)(.4). Using this setting we then show that the LBB condition is satisfied by aylor Hood P 2 P elements and the conforming Crouzeix-Raviart P 2 + bubble discp elements on a general triangulation of the domain. For applications where mass conservation is of particular importance using P 2 + bubble discp elements is attractive, as the computed approximations are mass conservative over each triangle in the partition of. he paper is organized as follows. In the following section we present the axisymmetric Stokes flow problem, introduce the appropriate function space setting, give the corresponding weak formulation, and describe the setting for the finite element approximation. Section 3 contains a discussion of Stenberg s sufficiency condition for the LBB condition and shows how it extends to the axisymmetric setting. In Section 4 we use the Stenberg sufficiency condition to show that the aylor-hood (k = 2) and the conforming Crouzeix-Raviart elements are LBB stable. Combining the approximation 2

3 properties derived by Belhachmi, Bernardi, Deprais in [2 with the LBB stability, in Section 5 we give a priori error bounds for the approximation to the axisymmetric Stokes flow problem computed using aylor-hood and Crouzeix-Raviart elements. A numerical example is given for which the experimental rates of convergence for the approximation error agree with the theoretically predicted rates. 2 Mathematical Preliminaries In this section we give the mathematical framework for the investigation of the LBB condition (.)(.4). We follow the setting used in [2 for the axisymmetric Stokes problem. 2. Problem Description Let IR 3 denote a domain symmetric with respect to the z-axis. With respect to cylindrical coordinates, (r, θ, z), we let denote the half section of, := {(r, 0, z) : r > 0, z IR}. For the description of the boundary we let Γ :=, and Γ 0 the intersection of and the z-axis, Γ 0 := {(0, 0, z) : z IR}. Note that = Γ Γ 0. In addition, we assume that is a simply connected domain with a polygonal boundary. (See Figure 2..) Γ 0 Γ Figure 2.: Illustration of axisymmetric flow domain. Consider Stokes equation (in Cartesian coordinates) in, subject to homogeneous boundary conditions on : where ŭ = u x u y u z directions, respectively. η ŭ + p = f in, (2.) ŭ = 0 in, (2.2) ŭ = 0 on, (2.3) = u x e x + u y e y + u z e z, for e x, e y, e z denoting unit vectors in the x, y and z 3

4 Multiplying (2.) through by a suitable smooth function v, v = 0, integrating over, and multiplying (2.2) through by a suitable smooth function q and integrating over we obtain η ŭ : v dv p v dv = f v dv (2.4) q ŭ dv = 0. (2.5) u r Expressing ŭ in cylindrical coordinates, ŭ = u θ = u r e r + u θ e θ + u z e z, and assuming that u z the flow is axisymmetric, i.e. ŭ(r, θ, z) = u(r, z), f(r, θ, z) = f(r, z), p(r, θ, z) = p(r, z), u r (0, z) = 0, u θ (0, z) = 0, equations (2.4)(2.5) transform into [ [ [ ur vr η a : u a r dx + η u z v r v r z r dx vr p a r dx p v v r dx z [ [ fr vr = r dx, (2.6) f z v z η a u θ a v θ r dx + η u θ v θ r dx = f θ v θ r dx, (2.7) [ ur q a r dx + q u r dx = 0, (2.8) u z where a := [ / r / z and dx := dr dz. Note that the angular flow equation for u θ is decoupled from the flow equations for u r and [ u z. For ur simplicity of our discussion of the LBB condition we will assume u θ = 0, and let u =, u [ z vr v =, etc. v z 2.2 Function Spaces and Weak Formulation Let Θ denote a domain in IR 2. For any real α and p <, the space α L p (Θ) is defined as the set of measurable functions w such that ( /p w αl p (Θ) = w p r dx) α <, Θ where r = r(x) is the radial coordinate of x, i.e. the distance of a point x in Θ from the symmetry axis. he subspace L 2 0 (Θ) of L 2 (Θ) denotes the functions q with weighted integral equal to zero: q r dx = 0. Θ 4

5 We define the weighted Sobolev space W l,p (Θ) as the space of functions in L p (Θ) such that their partial derivatives of order less that or equal to l belong to L p (Θ). Associated with W l,p (Θ) is the semi-norm W l,p (Θ) and norm W l,p (Θ) defined by w W l,p (Θ) = ( l k=0 k r l k z w p L p (Θ)) /p, w W l,p (Θ) = ( l /p w p W (Θ)). k,p k=0 When p = 2, we denote W l,2 (Θ) as H l (Θ). Also used in the analysis is the space V (Θ), a subset of H l (Θ), given by V (Θ) = { w H (Θ) : w L 2 (Θ) }, ( /2 with norm w V (Θ) = w 2 H (Θ) + w 2 L (Θ)). 2 It can be proven that all functions in V () have a null trace on Γ 0, [2, 7. In order to incorporate the homogeneous boundary condition for the velocity on Γ, let H () = { w H () : w = 0 on Γ }, and V () = { w V () : w = 0 on Γ }. For convenience of notation, let X := V () H () and for v = [v r, v z, v X(Θ) = ( v r 2 V (Θ) + v z 2 H (Θ)) /2, and Q := L 2 0 () with Q = L 2 (). When Θ =, we write v X := v X(Θ). With X we associate the innerproduct ( v, w X = a v : a w + v r w ) r r dx. (2.9) r r Using as the pivot space ( L 2 () ) 2 with innerproduct f, g := f g r dx, let X denote the dual space of X, i.e. X is the completion of ( L 2 () ) 2 with respect to the norm f X = sup g X f, g g X. For Θ a domain in IR n, n = 2, 3, we use the standard definitions for L 2 (Θ), L 2 0 (Θ), Hk (Θ), and H0 k (Θ) (see [). he weak axisymmetric formulation for the Stokes equations can be stated as: Given f X, determine (u, p) (X Q) satisfying a(u, v) b a (p, v) = f, v X,X v X, (2.0) b a (q, u) = 0, q Q, (2.) where a(u, v) := b a (q, v) := η a u : a v r dx + η u r v r dx, (2.2) r q a v r dx + q v r dx, (2.3) 5

6 and, X,X denotes the duality pairing between X and X. For the discussion of existence and uniqueness of (2.0)(2.) see [3, 2. In particular we note that there exists β > 0 such that inf sup b a (q, v) β. (2.4) q Q v X q Q v X 2.3 Finite Element Approximation Setting In this section we describe, as in [2, the setting for the finite element approximation to (2.0)(2.). We assume that is a convex polygonal domain and ( h ) h denotes a family of uniformly regular triangulations of satisfying: (i) he domain is the union of the triangles of h. (ii) k j is a side, a node, or empty for all triangles k, j, k j, in h. (iii) here exists a constant σ, independent of h, such that for all h its diameter h is smaller that h and contains a circle of radius σ h. Additionally we assume that each triangle in h has at least one vertex inside (i.e. not on Γ Γ 0 ). he properties that is convex and the triangulations uniformly regular are used in the proof of Lemma 2. Let P k ( ) denote the set of restriction to of polynomials of degree less than or equal to k. For the velocity approximation space we consider X h = {w (C 0 ()) 2 : w Γ = 0, w r Γ0 = 0, w (P k ( )) 2, h } X. (2.5) For the pressure space, Q h = {q C 0 () : q r dx = 0, q P k ( ), h } Q. (2.6) he approximation pair (X h, Q h ) given by (2.5)(2.6) with k = 2 represent the aylor-hood P 2 P pair. For h, let (λ (x, y), λ 2 (x, y), λ 3 (x, y)) denote the normalized (i.e. λ +λ 2 +λ 3 = ) barycentric coordinates of (x, y). Introduce the bubble function on, he approximation pair b (x, y) := 27λ (x, y)λ 2 (x, y)λ 3 (x, y), and B := span{b }. (2.7) X h = {w (C 0 ()) 2 : w Γ = 0, w r Γ0 = 0, w (P 2 ( ) B ) 2, h } X, (2.8) Q h = {q : q r dx = 0, q P ( ), h } Q, (2.9) 6

7 correspond to the conforming Crouzeix-Raviart mixed finite element pair. In Section 4 we show that the pairs (2.5)(2.6),for k = 2, and (2.8)(2.9) are both LBB stable. Below, all constants C, C, C 2,... used are independent of h. However their values may change from line to line. 3 Mathematical Preliminaries In [9 Stenberg established a sufficient condition on the family of partitions ( h ) h and the approximation spaces X h, and Q h, for the LBB condition (.)(.2) to be satisfied. For the axisymmetric flow formulation we have a different operator b(, ) and different velocity and pressure spaces. he proof of the Stenberg sufficiency condition in [9 follows easily from two lemmas, generalized as Lemma and Lemma 2 below. he proof of Lemma follows as in [9. However, because of the singular operators and different norms arising in the axisymmetric formulation, the proof of Lemma 2 is considerably more complicated. As in [9, the proof of the sufficiency condition follows from Lemmas and 2. We discuss the case for a triangulation of the domain. he results can be extended to a partition of the domain into regular quadrilateral elements. 3. Stenberg sufficient condition A macroelement M is said to be equivalent to a reference macroelement M if there is a mapping F M : M M satisfying the conditions: (i) F M is continuous and one-to-one. (ii) F M ( M) = M. (iii) If M = m j= j, where j, j =, 2,..., m, are the triangles in M, then j = F M ( j ), j =, 2,..., m, are the triangles in M. (iv) F M j = F j F, j =, 2,..., m, where F j j and F j are the affine mappings from the reference triangle with vertices (0, 0), (, 0) and (0, ) onto j and j, respectively. he family of macroelements equivalent with M is denoted E M. For a macroelement M define the spaces X h,m, Q h,m and N h,m as X h,m = {w (C 0 ()) 2 : w Γ = 0, w r Γ0 = 0, w \M = 0, w (P k ( )) 2, M} X (3.), Q h,m = {q : q r dx = 0, q P l ( ), M} L 2 0(M), (3.2) N h,m = {q Q h,m : b a (q, w) = 0, w X h,m }. (3.3) heorem [9 [Stenberg Sufficiency Condition If 7

8 (i) there exists a finite set of classes E Mi, i =,..., n, n, such that for each M E Mi, i =,..., n, the space N M is one dimensional consisting of functions which are constant on M, (ii) for each h ( h ) h, the triangles can be grouped together to form macroelements M j, j =,..., m, such that the so obtained macroelement partitioning of, M h satisfies that M j belongs to some E Mi, for all M j M h, then (.)(.2) is satisfied. In the case linear elements are used for the velocity approximation there is one additional constraint on h. (iii) If γ is the common part of two macroelements in (ii) then γ is connected and contains at least two edges of triangles in h. Remark: he stated theorem trivially extends to the case where the velocity approximating space is enriched with bubble functions, i.e. w (P k ( ) B k ( )) 2, where B k ( ) = {v (P k+ ( )) 2 : v = λ λ 2 λ 3 w, w (P k 2 ( )) 2 }. he following two lemmas are analogues of the key lemmas used by Stenberg in [9. Let Π h denote the projection, with respect to the innerproduct q, p := q p r dx, from Q h onto the space Q C h := {q Q : q M is constant M M h }. (3.4) Lemma [See [9, Lemma 3.2 Under the conditions of heorem, there is a constant C > 0 such that for all q h Q h there is a v h X h satisfying b a (q h, v h ) = q h a v h r dx + q h v h r dx = (I Π h )q h a v h dx + (I Π h )q h v h r dx C (I Π h )q h 2 Q, (3.5) and v h X (I Π h )q h Q. (3.6) Proof: he proof of this lemma follows as that of Lemma 3.2 in [9. Lemma 2 [[9, Lemma 3.3 Under the conditions of heorem, there is a constant C > 0 such that for all q h Q h there is a v h X h satisfying b a (q h, v h ) = q h a v h r dx + q h v h r dx = Π h q h 2 Q, (3.7) and v h X C Π h q h Q. (3.8) he proof of Lemma 2 involves three steps. First, for q h Q h given, the identification of v X satisfying b a (q h, v) = Π h q h, and v X C Π h q h Q. Step 2 is the construction of an approximation v h X h of v such that b a (q h, v h ) = b a (q h, v). he third step involves establishing that 8

9 v h X C v X. Because of the norms involved, it is this step that differs significantly from [9. o do step 3 we follow the approach from Ruas in [8. Steps and 2 in proof of Lemma 2 Let q 0 h Q h be given. As Π h q 0 h Q h from (2.4) we have that there exists a v 0 X satisfying a v 0 + r v0 r = Π h q 0 h, and v0 X β Π hq 0 h Q. (3.9) Let P h : X X h denote the orthogonal projection defined by, X. Let a i, i =,..., n e denote a labeling of the triangle edges in the triangulation h, with n i and τ i a unit normal and tangent vector to a i, respectively. Assume that we have a Lagrangian basis for X h, and that along each edge, a i, the nodes are located at the endpoints and the Gaussian quadrature points. (For k = 3 modified Gaussian quadrature points are used. See (3.25).) Let M i denote an interior nodal point on a i, with φ Mi the associated local basis function, such that φ Mi r ds 0. (3.0) a i Denote the other nodal points as S,..., S nb. Introduce R h : X X h an approximation operator defined by For v 0 defined in (3.9), let R h v(s j ) = P h v(s j ), j =,..., n b, (3.) R h v(m i ) τ i = P h v(m i ) τ i, i = i,..., n e, (3.2) R h v n i r ds = v n i r ds, i = i,..., n e. (3.3) a i a i v 0 h = R hv 0, e 0 h = v0 h P hv 0, and e 0 = v 0 P h v 0, By construction Also, Π h q 0 h 2 Q = b a (q 0 h, v0 ) = b a (q 0 h, v0 h ). v 0 h X( ) e 0 h X( ) + P h v 0 X( ) e 0 h X( ) + v 0 X( ), o complete the proof it suffices to show that e 0 h X C e 0 X. o establish this inequality requires us to look closely at the triangulation of and the interpolation. We introduce the additional notation. For h, (see Figure 3.) let [ [ r (r2 r F (ξ) = J ξ +, J = ) (r 3 r ), (3.4) (z 2 z ) (z 3 z ) z a i, i =, 2, 3, denote the edges of, with M i denoting the associated edge point used in (3.2), n i the unit normal used in (3.3), l i its length, and I e, 2, 3 an index set such that i I e implies that a i Γ 0, i.e. a i does not lie on the z-axis. 9

10 η (0, ) a 2 (0, 0) S S 3 a 3 a S F S 2 (, 0) ξ z (r, z) (r 3, z3 ) a 2 S 3 a 3 a (r, z) S r Figure 3.: Mapping of the reference triangle to triangle. By assumption of a regular triangulation, there exists constants c J, C J > 0 such that c J h 2 det(j ) = J C J h 2. For Θ, let r max (Θ) := max{r : (r, z) Θ}, and r min (Θ) := min{r : (r, z) Θ}. As h is a regular triangulation of we have that there exists c > 0 such that r max ( ) c h, for all h. (3.5) It is useful to categorization the triangles of h into three types. For constants c 2, c 3, c 4 > 0 the following inequalities hold. ype : Γ 0 is empty. For these triangles we have that r min ( ) c 2 h. (3.6) ype 2: Γ 0 is a side. For these triangles, without loss of generality (WLOG), we assume that the local counter-clockwise labeling of is such that the vertices S and S 3 (equivalently a 2 ) lie on Γ 0. hen, under the transformation F, r = r 2 ξ = r max ( ) ξ, and r max ( ) c 3 h. (3.7) ype 3: Γ 0 is a point. For these triangles, WLOG, we assume that the local counter-clockwise labeling of is such that the vertex S lies on Γ 0. Under the transformation F, r = r 2 ξ + r 3 η min{r 2, r 3 }(ξ + η) c 4 h (ξ + η). (3.8) Step 3 in proof of Lemma 2 For each h we now estimate e 0 h X( ). As h is considered fixed we omit the superscript in the notation of a i, M i, etc. 0

11 We have e 0 h = 3 e 0 h (M i) n i φ i n i, where φ i is the canonical local basis function of X h associated with M i. hus, i= e 0 h 2 X( ) = e 0 h r 2 V ( ) + e0 h z 2 H ( ) 3 2 = e 0 h (M i) n i a φ i n i r r dx + i= e 0 h (M i) n i a φ i n i z r dx i= i= i= 3 2 e 0 h (M i) n i φ i n i r r 2 r dx 3 ( ) e ( ) h (M i ) n i φ i 2 H ( ) + φ i 2 L 2 ( ). (3.9) i= Mapping from to the reference triangle we have φ i 2 H ( ) = For a ype triangle, a φ i 2 r dx C ( ˆφ i ξ 2 + ˆφ i η 2 ) h 2 ˆr h2 dξ C r max ( ). (3.20) φ i 2 L 2 ( ) = φ i 2 r 2 r dx = ˆφ i 2 ˆr h2 dξ C r min ( ) h2 C h. (3.2) For a ype 2 triangle, for i I e, ˆφ i vanishes along ξ = 0 thus ˆφ i = ξ ˆψ i, with ˆψ i P k ( ). φ i 2 L 2 ( ) = φ i 2 r 2 r dx = ˆφ i 2 r max ( ) ξ h2 dξ C ξ ˆψ i 2 h dξ C h. (3.22) For a ype 3 triangle, φ i 2 L 2 ( ) = φ i 2 r 2 r dx = C ˆφ i 2 r 2 ξ + r 3 η h2 dξ ˆφ i 2 ξ + η h dξ C h. (3.23) hus, combining (3.20) (3.23), we have for i I e φ i 2 H ( ) + φ i 2 L 2 ( ) C r max ( ). (3.24)

12 Next we need to construct an estimate for e 0 h (M i) n i. By construction of v 0 h, e 0 h (M i) n i φ i r ds = a i i.e. e 0 h (M i) n i φ i r ds a i e 0 n i r ds, a i e 0 n i r ds. a i Note that φ i r is a polynomial of degree k + in s along a i which vanishes at the endpoints. For k = 2 and k 4 the k Gaussian quadrature formula exactly evaluates a i φ i r ds. For k = 3 the modified Gaussian quadrature formula, exactly evaluates the integral. f(t) dt 5 6 f( / 5) f(/ 5), (3.25) With r Mi the r coordinate of M i, applying the quadrature formula we have that there exists c > 0 such that c r Mi l i e 0 h (M i) n i e 0 n i r ds. a i For a i Γ 0, e 0 h (M i) n i = 0. Otherwise, for a i Γ 0, there exists a constant C > 0 such that C r max ( ), and l i 2σh. hus r Mi e 0 h (M i) n i C (r max ( ) h ) a i e 0 n i r ds. (3.26) Next we use the obvious bound a i e 0 n i r ds a i e 0 r r ds + a i e 0 z r ds. (3.27) For a i e 0 r r ds, again using the mapping of the triangle to the reference triangle e 0 r r ds h re 0 r dŝ h re 0 r L 2 ( a i â ). i Applying the race heorem to, and using J c J h 2, we then have e 0 r r ds C h re 0 r H ( a ) i ( C h re 0 r 2 J dx + C ( e 0 r 2 r r dx + h 2 ) /2 a re 0 r 2 h 2 J dx a e 0 r 2 r r dx + h 2 e 0 r r 2 r r dx) /2 (3.28) C (r max ( )) /2 ( e 0 r 2 L 2 ( ) + h2 e 0 r 2 V ( )) /2. (3.29) 2

13 In order to bound a i e 0 z r ds we consider the three types for h. In each case we establish ( /2 e 0 z r ds C (r max ( )) /2 e 0 z 2 L 2 ( ) + h2 e 0 z 2 H ( )). (3.30) a i ype. Γ 0 is empty. Estimate (3.30) is established by mapping to and applying the race heorem. (See [5 for details.) ype 2. Γ 0 is a side. Estimate (3.30) is established by mapping to, revolving around the η-axis to generate a reference cone Ê, and then applying the race heorem to Ê. (See [5 for details.) ype 3. Γ 0 is a point. In this case, after mapping to, the integral is split into two pieces. One piece is handled as in case ype 2, by forming a reference cone by rotating around the η-axis. he other piece is handled similarly, by forming a reference cone by rotating around the ξ-axis. (See [5 for details.) Combining (3.27),(3.29) and (3.30) we obtain ( /2 e 0 n i r ds C (r max ( )) /2 e 0 r 2 L 2 ( ) + h2 e 0 r 2 V ( ) + e0 z L 2 ( ) + h 2 e 0 z 2 H ( )). a i From (3.26) and (3.3) e 0 h (M i ) n i 2 C (rmax ( )) h 2 Combining (3.9),(3.24), and (3.32) yields e 0 h 2 X( ) C h 2 Summing over the triangles we obtain C (3.3) ( ) e 0 r 2 L 2 ( ) + h2 e 0 r 2 V ( ) + e0 z L 2 ( ) + h 2 e 0 z 2 H ( ). (3.32) ( e 0 r 2 L 2 ( ) + h2 e 0 r 2 V ( ) + e0 z L 2 ( ) + h 2 e 0 z 2 H ( ) e 0 h 2 X( ) hus, what remains to show is that h h 2 ). (3.33) e0 2 L 2 ( ) + e0 2 X. (3.34) h h 2 e0 2 L 2 ( ) C e 0 2 X. (3.35) Using the fact that the mesh is a uniformly regular triangulation implies that there exists a constant c > 0 such that c h h. Hence (3.35) may be replaced by showing that e 0 L 2 () C h e 0 X. (3.36) o establish (3.36) we use the following Propositions relating function spaces defined on to axisymmetric functions, with zero angular component, defined on. 3

14 ( 3, Proposition [3 he axisymmetric reduction of functions in L ( )) 2 with zero angular component, to functions in ( L 2 () ) 2 is an isometry (up to a factor of 2π). Proposition 2 he axisymmetric reduction of functions in ( H 2 ( )) 3, with zero angular component, to functions in ( H 2 () ) 2 is a bounded mapping satisfying w (H 2 ( )) 3 w ( H 2 ()) 2. Proof: he proof follows by direct calculation. Let ĕ 0 denote the axisymmetric extension of e 0 to. From Proposition we have that ( 3 ĕ 0 L ( )) 2 and ĕ 0 L 2 ( ) = ( ( )) 3 2π e 0 L 2 (). Let w H0 be given by w = ĕ 0, in. (3.37) As ĕ 0 is axisymmetric, then w is also. Additionally, as ( 3, is convex, w H ( )) 2 and w H 2 ( ) C ĕ 0 L 2 ( ) C e0 L 2 (). Let w be the reduction of w to. From Proposition 2 we have that w ( H 2 () ) 2, and w ( H 2 ()) 2 w (H 2 ( )) 3 C e0 L 2 (). hen 2π e 0 2 L 2 () = ĕ 0 2 L 2 ( ) = (ĕ 0, ĕ 0 ) L2 ( ) hus we have that = ( w, ĕ 0 ) L 2 ( ) = 2π w, e 0 X = 2π w χ, e 0 X, for χ X h, 2π e 0 X inf χ X h w χ X 2π e 0 X C h w ( H 2 2 ()) (from [2, heorem 5) C h e 0 X e 0 L 2 (). (3.38) e 0 L 2 () C h e 0 X. (3.39) Proof of heorem : In view of Lemmas and 2, the proof of heorem now follows as in [9. 4 he LBB condition for aylor-hood and Crouzeix-Raviart elements In this section we show that the Stenberg sufficiency criteria for satisfying the LBB condition (.)(.3) is satisfied for aylor-hood P 2 P and the conforming Crouzeix-Raviart approximating elements on triangles. 4

15 4. aylor-hood P 2 P approximation pair We begin by identifying an appropriate macroelement, M, and then show that the corresponding vector space N h,m has dimension one. Let M be given by the collection of three triangles in Figure 4.. z (r 5, z5) R 5 R R 3 R 4 (r 4, z4) (r 3, z3) z R R 6 R 7 R R 3 R, ) R 6 R 2 (r z (r 2, z2) R 8 R 2 ( ) R 4 r r Figure 4.: P 2 P. Macroelement for aylor-hood Figure 4.2: P 2 P. Macroelement for aylor-hood Let X h,m, Q h,m, N h,m, be given by (3.)-(3.3) with k = 2 and l =, and X 0 h,m = {w (C0 ()) 2 : w = 0, w \M = 0, w (P 2 ( )) 2, M} X h,m, (4.) N 0 h,m = { q Q h,m : b a (q, w) = 0, w X 0 h,m} Nh,M. (4.2) As the function q = constant is contained in N h,m and N 0 h,m, we have dim(n h,m) dim(n 0 h,m ). Hence it suffices to show that dim(n 0 h,m ) =. Note that Xh,M 0 differs from X h,m in that for M such that M Γ 0, w Xh,M 0 satisfies w Γ 0 = 0, whereas for w X h,m, w r Γ0 = 0. Xh,M 0 is introduced for convenience so that we do not need to separately consider those macroelements which have a nontrivial intersection with the symmetry boundary, Γ 0. For notational convenience we suppress the h subscript and 0 superscript, i.e. N M N 0 h,m and X M X 0 h,m. We have that X M = span { [ v = q 6 (r, z) 0 [ 0, v 2 = q 6 (r, z) [, v 3 = q 7 (r, z) 0 [ 0, v 4 = q 7 (r, z) where q 6, q 7 represent the (continuous) Lagrangian quadratic basis function which has value at node R 6, R 7, respectively, and vanish at all other nodes. Q M = span{l (r, z), l 2 (r, z), l 3 (r, z), l 4 (r, z), l 5 (r, z)}, where l i, i =,..., 5, represents the (continuous) Lagrangian linear basis function which has value at node R i and vanishes at nodes R j, j =, 2,..., 5, j i. Note that the defining equation for N M generates four equations for the five unknown constants p, p 2, p 3, p 4, p 5, where p(r, z) = p l (r, z) + p 2 l 2 (r, z) + p 3 l 3 (r, z) + p 4 l 4 (r, z) + p 5 l 5 (r, z). }, 5

16 Using Green s theorem, p a v r dx + M M p v r dx = M v a p r dx = 3 j= j v a p r dx. (4.3) Also, j v a p r dx = where J j denotes the absolute value of the determinant of J j, J t j of J j, and ĝ(ξ, η) := g(f j (r, z)), (see (3.4)). ˆ ˆv J t j ξ,η ˆp ˆr J j dξ dη, (4.4) the transpose of the inverse For v X M, p Q M, ˆv is a (vector) quadratic function, ξ,η ˆp is a constant vector, and J j is a constant matrix. Hence the integrand in (4.4) is a polynomial of degree 3. Introduce the following Lagrangian quadratic and linear basis functions on. ˆq (ξ, η) = ( ξ η)( 2ξ 2η), ˆq 2 (ξ, η) = ξ (2ξ ), ˆq 3 (ξ, η) = η (2η ), ˆq 4 (ξ, η) = 4 ξ η, ˆq 5 (ξ, η) = 4 η ( ξ η), ˆq 6 (ξ, η) = 4 ξ ( ξ η), and ˆl (ξ, η) = ( ξ η), ˆl2 (ξ, η) = ξ, ˆl3 (ξ, η) = η. (4.5) Also note that the quadrature formula 8 ( ) ˆf(ξ, η) dξ dη ˆf(/2, 0) + ˆf(/2, /2) + ˆf(0, /2) 20 is exact for polynomials of degree ( ) ˆf(0, 0) + ˆf(, 0) + ˆf(0, ) ˆf(/3, /3), (4.6) 4.. Computation of 2 v a p r dx In terms of the mapping of 2 to the reference triangle, relative to (3.4), associate S R, S 2 R 3, and S 3 R 5. We have that v (r, z) 2 v 3 (r, z) 2 [ = ˆq 6 (F 2 (r, z)) 0 [ = ˆq 4 (F 2 (r, z)) 0 [, v 2 (r, z) 2 = ˆq 6 (F 0 2 (r, z)) [, v 4 (r, z) 2 = ˆq 4 (F 0 2 (r, z)),, and J 2 = p(r, z) = p ˆl (F 2 (r, z)) + p 3 ˆl3 (F 2 (r, z)) + p 5 ˆl5 (F 2 (r, z)), [ (r3 r ) (r 5 r ). (z 3 z ) (z 5 z ) 6

17 Using (4.4) and (4.6) we have that and v a p r dx = 2 30 (2r + 2r 3 + r 5 ) ((z 3 z 5 )p + (z 5 z )p 3 + (z z 3 )p 5 ), (4.7) v 2 a p r dx = 2 30 (2r + 2r 3 + r 5 ) ((r 3 r 5 )p + (r 5 r )p 3 + (r r 3 )p 5 ), (4.8) v 3 a p r dx = 2 30 (r + 2r 3 + 2r 5 ) ((z 3 z 5 )p + (z 5 z )p 3 + (z z 3 )p 5 ), (4.9) 2 v 4 a p r dx = 30 (r + 2r 3 + 2r 5 ) ((r 3 r 5 )p + (r 5 r )p 3 + (r r 3 )p 5 ). (4.0) Similar equations to (4.7)-(4.0) are obtained from considering v a p r dx, and 3 v a p r dx. (See [5 for details.) 4..2 Dimension of N M Let α := 2r + r 2 + 2r 3, α 2 := 2r + 2r 3 + r 5, α 3 := r + 2r 3 + 2r 5, α 4 := 2r 3 + r 4 + 2r 5. Note, as r i 0, i =, 2,..., 5, and the geometry of the triangles, that α, α 2, α 3 and α 4 > 0. Corresponding to M p a v r dx + M p v r dx = M v ap r dx = 0, v X M, we obtain (after minor simplification) the following linear system of equations, Ap = 0, where A is given by α (z 2 z 3) + α 2(z 3 z 5) α ( z + z 3) α (z z 2) + α 2( z + z 5) α 2(z z 3) α (r 2 r 3) + α 2(r 3 r 5) α ( r + r 3) α (r r 2) + α 2( r + r 5) α 2(r r 3) α 3(z 3 z 5) α 3( z + z 5) + α 4(z 4 z 5) α 4( z 3 + z 5) α 3(z z 3) + α 4(z 3 z 4). α 3(r 3 r 5) α 3( r + r 5) + α 4(r 4 r 5) α 4( r 3 + r 5) α 3(r r 3) + α 4(r 3 r 4) (4.) Note that p = p 2 = p 3 = p 4 = p 5 is a solution to (4.), i.e. N M contains the constant functions. o show that dim(n M ) = it suffices to show that the matrix A has full rank, i.e. the rows of A are linearly independent. Lemma 3 he rows of the matrix A given in (4.) are linearly independent. Proof: Let p = p + (α 3 /α 4 )p 4, p 5 = p 5 + (α 2 /α )p 2, p = [ p p 2 p 3 p 4 p 5, and consider in place of Ap = 0 the corresponding linear system à p = 0. o see that the rows of à are linearly independent, consider: C Ã, + C 2 à 2, + C 3 à 3, + C 4 à 4, = 0. (4.2) 7

18 Corresponding to columns and 2 in (4.2) we have [ α (z 2 z 3 ) + α 2 (z 3 z 5 ) α (r 2 r 3 ) + α 2 (r 3 r 5 ) α ( z + z 3 ) α ( r + r 3 ) [ C C 2 = [ 0 0. Note that if r = r 3, as R R 3, then the second equation implies C = 0. he first equation then gives C 2 = 0. herefore, assume r r 3. A non-trivial solution for C, C 2 requires the determinant of the 2 2 matrix to be zero. his implies (z 2 z 3 ) + α 2 α (z 3 z 5 ) (r 2 r 3 ) + α 2 α (r 3 r 5 ) = z z 3. (4.3) r r 3 Consider now the quadrilateral formed by R, R 2, R 3, R P, where R P denotes the point on the halfline passing through R 5 and terminating at R 3 given by R P := (r 3 α 2 α (r 3 r 5 ), z 3 α 2 α (z 3 z 5 ). Equation (4.3) implies that the vector R P R 2 has the same slope as the vector R R 3, which is impossible as they form opposite diagonals of the quadrilateral. Hence C = C 2 = 0. An analogous argument using columns 4 and 5 in (4.2) leads to C 3 = C 4 = 0. Hence, the rows of à are linearly independent, i.e. rank(ã) = 4 = rank(a). By modifying the matrix in (4.), it is straight forward to show that the three triangles depicted in Figure 4.2 also form a macroelement for aylor-hood P 2 P approximation pair. hus, we could conclude that for any triangulation of the domain of, which can be partitioned into groups of three adjacent triangles, the aylor-hood P 2 P approximation pair is LBB stable. However often the number of triangles in a triangulation is not exactly divisible by three. Next we demonstrate that there are many choices of macroelements for the aylor-hood P 2 P approximation pair. Lemma 4 Suppose M is a macroelement with N M =, consisting of functions which are constant on M. Let M be formed from M by adding an adjacent triangle (i.e. sharing an edge with M). hen M is also a macroelement with the desired property that N M =, consisting of functions which are constant on M. Proof: We consider separately the two cases corresponding to M being formed by adding a triangle to M that: (i) shares two edges with M, and (ii) shares one edge with M. Case (i): he added triangle shares two edges with M. (For example, see Figures 4., 4.3.) Let A and à be the matrices associated with N M and N M, respectively. For n M,Q the dimension of Q M, we have that rank(a) = n M,Q, and that A is a m n M,Q matrix with m = n M,Q. à is therefore a m n M,Q matrix with m n M,Q +, as à must have at least one more interior edge that M. Note that every row in à comes from M v i p da = 0, for v i X M. As, v i X M, M v i p da = 0 is satisfied for p a constant function, then p = p 2 =... = p nm,q satisfies Ãp = 0, and rank(ã) n M,Q. Since the rank(a) = n M,Q, this implies that A has n M,Q 8

19 z R R 4 R 5 R R R 3 R R 8 6 R 2 z R R 5 R R 6 R 7 R 0 R 8 R 2 R r Figure 4.3: New macroelement for aylor- Hood P 2 P. r Figure 4.4: New macroelement for aylor- Hood P 2 P. A linearly independent rows. he fact that for v X M, v M\M = 0; we have à =, which B implies that à has at least n M,Q linearly independent rows. Hence rank(ã) = n M,Q and the dimension of N M =. Case (ii): he added triangle shares one edges with M. Let R 2 and R 3 denote the endpoints of the shared triangle edge. (For example, see Figures 4., 4.4.) In this case, along with two new triangle edges, an additional triangle vertex is added to M in forming M. herefore, the dimension of Q M = dim(q M ) +, with the increase in dimension corresponding to the new added vertex. Again, as v i X M, M v i p da = 0 is satisfied for p a constant function, then p = p 2 =... = p nm,q + satisfies Ãp = 0, which implies that A 0 rank(ã) n M,Q. Also, as for v X M, v M\M = 0; we have à =, where the number B b of rows in the matrix B is two, corresponds to the velocity basis functions ṽ, ṽ 2, associated with the shared triangle edge, added to Q M to form Q M. As the added triangle lies in the support of ṽ, and ṽ 2, then from (4.4) (and corresponding minor simplifications) b = [α( z 2 +z 3 ) α( r 2 +r 3 ), for α > 0. As R 2 R 3, the number of independent rows in à must be greater than the number of independent rows in A = n M,Q. Hence rank(ã) = n M,Q and the dimension of N M =. Corollary he aylor-hood P 2 P approximation pair is LBB stable on a regular triangulation of. 4.2 Crouzeix-Raviart approximation pair Again, we begin by identifying a macroelement M for the conforming Crouzeix-Raviart elements. In this case we simply take M to be an arbitrary triangle in h, see Figure 3.. 9

20 With ˆl i (ξ, η), i =, 2, 3, defined in (4.5), let ˆb(ξ, η) = 27 ˆl (ξ, η) ˆl 2 (ξ, η) ˆl 3 (ξ, η). (4.4) ˆb(ξ, η) is the cubic bubble function which vanishes on the boundary of, and is equal to at (ξ, η) = (/3, /3). With b (x, y) as defined in (2.7), let { [ [ } Xh,M 0 0 = span b, b 0 X h,m, (4.5) N 0 h,m = { q Q h,m : b a (q, w) = 0, w X 0 h,m} Nh,M. (4.6) We note, as commented at the beginning of Section 4., that q = constant is contained in N h,m and N 0 h,m, we have dim(n h,m) dim(n 0 h,m ). Hence it suffices to show that dim(n 0 h,m ) =. Again, for notational convenience we suppress the h subscript and 0 superscript, i.e. N M N 0 h,m and X M X 0 h,m. he defining equation for N M generates two equations for the three unknown constants p, p 2, p 3, where p(r, z) = p l (r, z) + p 2 l 2 (r, z) + p 3 l 3 (r, z). From (4.3)(4.4) we have p a v r dx + M Let val = J M p v r dx = ˆb(ξ, η) ˆr(ξ, η) dξ dη = J ˆv J t j ξ,η ˆp ˆr J j dξ dη. ˆb(ξ, η) (r + (r 2 r )ξ + (r 3 r )η) dξ dη. Note that, as ˆb(ξ, η) and ˆr(ξ, η) are greater than 0 for (ξ, η) \, val > 0. [ For v = b, we obtain 0 M p a v r dx + and for v = b [ 0 M, p a v r dx + M M p v r dx = ((z 3 z )(p 2 p ) (z 2 z )(p 3 p )) val, (4.7) p v r dx = ((r 3 r )(p 2 p ) (r 2 z )(p 3 p )) val. (4.8) From (4.7)(4.8), N M is given by the solutions to the linear system of equations [ z2 z 3 z + z 3 z z 2 p = 0, (4.9) r 2 r 3 r + r 3 r r 2 where p = [p p 2 p 3. he two rows of the coefficient matrix in (4.9) are linearly independent unless the points (r, z ), (r 2, z 2 ), (r 3, z 3 ) all lie along a line. hat, however, would contradict the facts that the points form the vertices of a non-degenerate triangle. Hence dim(n M ) =. In summary, we have the following. 20

21 Corollary 2 he conforming Crouzeix-Raviart (P 2 + bubble discp ) approximation pair is LBB stable on a regular triangulation of. 5 Numerical Experiment From the continuity and positivity of a(, ), the continuity of b a (, ), and the inf-sup condition (.)(.4) we have that approximations (u h, p h ) to (2.0)(2.) satisfy { } u u h X + p p h Q C inf u v X + inf p q Q. v X h q Q h From [2, for X h, Q h given by (2.5),(2.6), respectively, and k = 2 inf u v X C h 2 u v X H 3 () and inf p q Q C h 2 p h q Q H 2 (). h Hence, for u H 3 (), p H 2 (), we have that u u h X + p p h Q C h 2. (5.) We investigate this a priori error estimate in the following example. Let = (0, /2) ( /2, /2), Γ 0 = {0} [ /2, /2, Γ = \Γ 0. We consider a modified aylor-green vortex flow problem [ r cos(ωπr) sin(ωπz) u(r, z) =, cos(ωπr) cos(ωπz) + r sin(ωπr) cos(ωπz) 2 ωπ p(r, z) = sin(ωπz)( cos(ωπr) + 2ωπr sin(ωπr)). he computations are performed for ω =. A plot of the velocity field u, and the pressure p, is given in Figures 5. and 5.2, respectively. For the aylor-hood (k = 2) and Crouzeix-Raviart approximation pairs the errors for the velocity, pressure, and divergence ( div axi (u) = a u + u r /r), along with their experimental convergence rates are given in ables 5. and 5.2. (he Crouzeix-Raviart approximation is mass conservative over each triangle in the triangulation h, i.e. div axi(u h ) r dx = 0.) he experimental convergence rates are consistent with those predicted in (5.). References [ R.A. Adams and J.J.F. Fournier. Sobolev spaces, volume 40 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, [2 Z. Belhachmi, C. Bernardi, and S. Deparis. Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math., 05(2):27 247,

22 Figure 5.: Plot of the velocity flow field u. Figure 5.2: Plot of the pressure function p. h u u h X Cvg. rate p p h Q Cvg. rate div axi (u h ) L 2 () Cvg. rate / E-.585E-.224E- /3 2.53E E E /4.40E E E / E E E /6 6.26E E E / E E E / E E E /9 2.75E E E / E E E able 5.: Experimental convergence rates for the aylor-hood (k = 2) approximation pair. [3 C. Bernardi, M. Dauge, and Y. Maday. Spectral methods for axisymmetric domains, volume 3 of Series in Applied Mathematics (Paris). Gauthier-Villars, 999. Numerical algorithms and tests due to Mejdi Azaïez. [4 F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 99. [5 V.J. Ervin and E.W. Jenkins. Stenberg s sufficiency condition for axisymmetric Stokes flow. echnical report, Dept. Math. Sci., Clemson University,

23 h u u h X Cvg. rate p p h Q Cvg. rate div axi (u h ) L 2 () Cvg. rate /2.755E E-.082E- / E E E /4.56E E E / E E E /6 5.88E E E / E E E / E E E / E E E /0.884E E E able 5.2: Experimental convergence rates for the Crouzeix-Raviart approximation pair. [6 M.D. Gunzburger. Finite element methods for viscous incompressible flows. Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, 989. [7 B. Mercier and G. Raugel. Résolution d un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et séries de Fourier en θ. RAIRO Anal. Numér., 6(4):405 46, 982. [8 V. Ruas. Mixed finite element methods with discontinuous pressures for the axisymmetric Stokes problem. ZAMM Z. Angew. Math. Mech., 83(4): , [9 R. Stenberg. Analysis of mixed finite elements methods for the Stokes problem: A unified approach. Math. Comp., 42:9 23,