A UNIFORM INF SUP CONDITION WITH APPLICATIONS TO PRECONDITIONING

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1 A UNIFORM INF SUP CONDIION WIH APPLICAIONS O PRECONDIIONING KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER Abstract. A uniform inf sup condition related to a parameter dependent Stokes problem is establised. Suc conditions are intimately connected to te construction of uniform preconditioners for te problem, i.e., preconditioners wic beave uniformly well wit respect to variations in te model parameter as well as te discretization parameter. For te present model, similar results ave been derived before, but only by utilizing extra regularity ensured by convexity of te domain. e purpose of tis paper is to remove tis artificial assumption. As a byproduct of our analysis, in te two dimensional case we also construct a new projection operator for te aylor Hood element wic is uniformly bounded in L 2 and commutes wit te divergence operator. is construction is based on a tigt connection between a subspace of te aylor Hood velocity space and te lowest order Nedelec edge element. 1. Introduction e purpose of tis paper is to discuss preconditioners for finite element discretizations of a singular perturbation problem related to te linear Stokes problem. More precisely, let Ω R n be a bounded Lipscitz domain and ɛ (0, 1] a real parameter. We will consider singular perturbation problems of te form (1.1) (I ɛ 2 )u grad p = f in Ω, div u = g in Ω, u = 0 on Ω, were te unknowns u and p are a vector field and a scalar field, respectively. For eac fixed positive value of te perturbation parameter ɛ te problem beaves like te Stokes system, but formally te system approaces a so called mixed formulation of a scalar Laplace equation as tis parameter tends to zero. In pysical terms tis means tat we are studying fluid flow in regimes ranging from linear Stokes flow to porous medium flow. Anoter motivation for studying preconditioners of tese systems is tat tey frequently arises as subsystems in Date: October 31, Matematics Subject Classification. 65N22, 65N30. Key words and prases. parameter dependent Stokes problem, uniform preconditioners. e first autor was supported by te Researc Council of Norway troug Grant and a Centre of Excellence grant to te Centre for Biomedical Computing at Simula Researc Laboratory. e tird autor was supported by te Researc Council of Norway troug a Centre of Excellence grant to te Centre of Matematics for Applications. 1

2 2 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER time stepping scemes for time dependent Stokes and Navier Stokes systems, cf. for example [3, 6, 16, 18, 19]. e prase uniform preconditioners for parameter dependent problems like te ones we discuss ere, refers to te ambition to construct preconditioners suc tat te preconditioned systems ave condition numbers wic are bounded uniformly wit respect to te perturbation parameter ɛ and te discretization. Suc results ave been obtained for te system(1.1) in several of te studies mentioned above, but a necessary assumption in all te studies so far as been a convexity assumption on te domain, cf. [17]. However, below in Section 3 we will present a numerical example wic clearly indicates tat tis assumption sould not be necessary. ereafter, we will give a teoretical justification for tis claim. e basic tool for acieving tis is to introduce te Bogovskiĭ operator, cf. [8], as te proper rigt inverse of te divergence operator in te continuous case. e construction of uniform preconditioners for discretizations of systems of te form (1.1), is intimately connection to te well posedness properties of te continuous system, and te stability of te discretization. In fact, if we obtain appropriate ɛ independent bounds on te solution operator, ten te basic structure of a uniform preconditioner for te continuous system is an immediate consequence. Furtermore, under te assumption of proper stability properties of te discretizations, te basic structure of uniform preconditioners for te discrete system also follows. We refer to [15] and references given tere for a discussion of tese issues. e main tool for analyzing te well posedness properties of saddle point problems of te form (1.1) is te Brezzi conditions, cf. [4, 5]. In particular, te desired uniform bounds on te solution operator is closely tied to a uniform inf sup condition of te form (2.4) stated below. Furtermore, te verification of suc uniform conditions are closely tied to te construction of uniformly bounded projection operators wic properly commute wit te divergence operator. In te present case, tese projection operators ave to be bounded bot in L 2 and in H 1. In Section 4 we will construct suc operators in te case of te Mini element and te aylor Hood element, were te latter construction is restricted to quasi uniform meses in two space dimensions. 2. Preliminaries o state te proper uniform inf sup condition for te system (1.1) we will need some notation. If X is a Hilbert space, ten X denotes its norm. We will use H m = H m (Ω) to denote te Sobolev space of functions on Ω wit m derivatives in L 2 = L 2 (Ω). e corresponding spaces for vector fields are denoted H m (Ω; R n ) and L 2 (Ω; R n ). Furtermore,, is used to denote te inner products in bot L 2 (Ω) and L 2 (Ω; R n ), and it will also denote various duality pairings obtained by extending tese inner products. In general, we will use H0 m to denote te closure in H m of te space of smoot functions wit compact support in Ω, and te dual space of H0 m wit respect to te L 2 inner product by H m. Furtermore, L 2 0 will denote te space of L 2 functions wit mean value zero. We will use L(X, Y ) to denote te space of bounded linear operators mapping elements of X to Y, and if Y = X we simply write L(X) instead of L(X, X).

3 A UNIFORM INF SUP CONDIION 3 If X and Y are Hilbert spaces, bot continuously contained in some larger Hilbert space, ten te intersection X Y and te sum X + Y are bot Hilbert spaces wit norms given by x 2 X Y = x 2 X + x 2 Y and z 2 X+Y = inf x X,y Y z=x+y ( x 2 X + y 2 Y ). Furtermore, if X Y are dense in bot te Hilbert spaces X and Y ten (X Y ) = X +Y and (X + Y ) = X Y, cf. [2]. e system (1.1) admits te following weak formulation: Find (u, p) H 1 0(Ω; R n ) L 2 0(Ω) suc tat (2.1) u, v + ɛ 2 Du, Dv + p, div v = f, v, v H 1 0(Ω; R n ), div u, q = g, q, q L 2 0(Ω) for given data f and g. Here Dv denotes te gradient of te vector field v. More compactly, we can write tis system in te form (2.2) A ɛ u = f, were A ɛ = I ɛ2 grad. p g div 0 For eac fixed positive ɛ te coefficient operator A ɛ is an isomorpism mapping X = H0(Ω; 1 R n ) L 2 0(Ω) onto X = H 1 (Ω; R n ) L 2 0(Ω). However, te operator norm A 1 ɛ L(X,X) will blow up as ɛ tends to zero. o obtain a proper uniform bound on te operator norm for te solution operator A 1 ɛ are forced to introduce ɛ dependent spaces and norms. We define te spaces X ɛ and Xɛ X ɛ = (L 2 ɛ H0)(Ω; 1 R n ) ((H 1 L 2 0) + ɛ 1 L 2 0)(Ω) and Xɛ = (L 2 + ɛ 1 H 1 )(Ω; R n ) (H0 1 ɛ L 2 0)(Ω). Here H0 1 L 2 0 corresponds to te dual space of H 1 L 2 0. Note tat te space X ɛ is equal to X as a set, but te norm approaces te L 2 norm as ɛ tends to zero. Our strategy is to use te Brezzi conditions [4, 5] to claim tat te operator norms (2.3) A ɛ L(Xɛ,X ɛ ) and A 1 ɛ L(X ɛ,x ɛ) are bounded independently of ɛ. In fact, te only nontrivial condition for obtaining tis is tat we need to verify te uniform inf sup condition div v, q (2.4) sup α q H v H0 1(Ω;Rn ) v 1 +ɛ 1 L 2, q L2 0(Ω), L 2 ɛ H 1 were te positive constant α is independent of ɛ (0, 1]. Of course, if ɛ > 0 is fixed, and α is allowed to depend on ɛ, ten tis is just equivalent to te standard inf sup condition for te stationary Stokes problem. As explained, for example in [15], te mapping property (2.3) implies tat te Riesz operator B ɛ, mapping Xɛ isometrically to X ɛ, is a uniform preconditioner for te operator we by

4 4 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER A ɛ. More precisely, up to equivalence of norms te operator B ɛ can be identified as te block diagonal and positive definite operator B ɛ : Xɛ X ɛ, given by (2.5) B ɛ = (I ɛ2 ) 1. 0 ( ) 1 + ɛ 2 I is means tat te preconditioned coefficient operator B ɛ A ɛ is a uniformly bounded family of operators on te spaces X ɛ, wit uniformly bounded inverses. erefore, te preconditioned system B ɛ A ɛ u = B ɛ f p g can, in teory, be solved by a standard iterative metod like a Krylov space metod, wit a uniformly bounded convergence rate. We refer to [15] for more details. Of course, for practical computations we are really interested in te corresponding discrete problems. is will be furter discussed in Section 4 below. 3. e uniform inf sup condition e rest of tis paper is devoted to verification of te uniform inf sup condition (2.4), and its proper discrete analogs. We start tis discussion by considering te standard stationary Stokes problem given by: Find (u, p) H 1 0(Ω; R n ) L 2 0(Ω) suc tat (3.1) Du, Dv + p, div v = f, v, v H 1 0(Ω; R n ), div u, q = g, q, q L 2 0(Ω), were (f, g) H 1 (Ω; R n ) L 2 0(Ω). e unique solution of tis problem satisfies te estimate (3.2) u H 1 + p L 2 c ( f H 1 + g L 2), cf. [13]. Furtermore, if te domain Ω is convex, f L 2 (Ω; R n ), and g = 0 ten u H 2 H 1 0(Ω; R n ), p H 1 L 2 0(Ω), and an improved estimate of te form (3.3) u H 2 + p H 1 c f L 2, olds ([9]). We define R L(H 1 (Ω, R n ), L 2 0(Ω)) to be te solution of operator of te system (3.1), wit g = 0, given by f p = Rf. Hence, if te domain Ω is convex, tis operator will also be a bounded map of L 2 (Ω; R n ) into H 1 L 2 0(Ω). Furtermore, let S L(L 2 0(Ω), H 1 0(Ω; R n ) denote te corresponding solution operator, defined by (3.1) wit f = 0, given by g u = Sg. en S is a rigt inverse of te divergence operator, and te operator S is te adjoint of R, since f, Sg = Rf, div Sg + Du, DSg = Rf, g + p, div u = Rf, g.

5 A UNIFORM INF SUP CONDIION 5 Here u and p are components of te solutions of (3.1) wit data (f, 0) and (0, g), respectively. As a consequence of te improved estimate (3.3), we can terefore conclude tat if te domain Ω is convex ten S can be extended to an operator in L(H0 1 (Ω), L 2 (Ω; R n )). In oter words, in te convex case we ave (3.4) S L(L 2 0, H 1 0) L(H 1 0, L 2 ), and div Sg = g. However, te existence of suc a rigt inverse of te divergence operator implies tat te uniform inf sup condition olds, since for any q L 2 0(Ω) we ave q H 1 +ɛ 1 L 2 = sup g H 1 g, q div Sg, q c sup c sup 0 ɛ g L2 H 1 0 ɛ L2 g H 1 0 ɛ Sg L2 L 2 ɛ H 1 v L 2 ɛ H 1 0 div v, q v L 2 ɛ H 1. On te oter and, if te domain Ω is not convex, ten te estimate (3.3) is not valid, and as a consequence, te operator S cannot be extended to an operator in L(H0 1 (Ω), L 2 (Ω; R n )). erefore, te proof of te uniform inf sup condition (2.4) outlined above breaks down in te nonconvex case General Lipscitz domains. e main purpose of tis paper is to sow tat te problems encountered above for nonconvex domains are just tecnical problems wic can be overcome. As a consequence, preconditioners of te form B ɛ given by (2.5) will still beave as a uniform preconditioner in te nonconvex case. o convince te reader tat tis is indeed a reasonable ypotesis we will first present a numerical experiment. We consider te problem (1.1) on tree two dimensional domains, referred to as Ω 1, Ω 2 and Ω 3. Here Ω 1 is te unit square, Ω 2 is te L saped domain obtained by cutting out te an upper rigt subsquare from Ω 1, wile Ω 3 is te slit domain were a slit of lengt a alf is removed from from Ω 1, cf. Figure 1. Hence, only Ω 1 is a convex domain. e corresponding problems Figure 1. e domains Ω 1, Ω 2, and Ω 3. (1.1) were discretized by te standard aylor Hood element on a uniform triangular grid to obtain a discrete anolog of tis system (2.2) on te form A ɛ, u = f. p Here te parameter indicates te mes size. We ave computed te condition numbers of te operator B ɛ, A ɛ, for different values of ɛ and for te tree domains. e operator B ɛ, is given as te corresponding discrete version of (2.5), i.e., exact inverses of te discrete elliptic operators appearing in (2.5) are used. Hence, in te notation of [15] a canonical preconditioner is applied. e results are given in able 1 below. g

6 6 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER domain ɛ \ Ω Ω Ω able 1. Condition numbers for te operators B ɛ, A ɛ, ese results indicate clearly tat te condition numbers of te operators B ɛ, A ɛ, are not dramatically effected by lack of convexity of te domains. Actually, we will sow below tat tese condition numbers are indeed uniformly bounded bot wit respect to te perturbation parameter ɛ and te discretization parameter. We will now return to a verification of te inf sup condition (2.4) for general Lipscitz domains. e problem we encountered above in te nonconvex case is caused by te lack of regularity of te solution operator for Stokes problem on general Lipscitz domains. However, to establis (2.4) we are not restricted to suc solution operators. It sould be clear from te discussion above tat if we can find any operator S satisfying condition (3.4), ten (2.4) will old. A proper operator wic satisfy tese conditions is te Bogovskiĭ operator, see for example [11, section III.3], or [8, 12]. On a domain Ω, wic is star saped wit respect to an open ball B, tis operator is explicitly given as an integral operator on te form Sg(x) = Ω g(y)k(x y, y) dy, were K(z, y) = z z n z θ(y + r z z )rn 1 dr. Here θ C0 (R n ) wit supp θ B, and θ(x) dx = 1. R n is operator is a rigt inverse of te divergence operator, and it as exactly te desired mapping properties given by (3.4), cf. [8, 12]. Furtermore, te definition of te rigt inverse S can also be extended to general bounded Lipscitz domains, by using te fact tat suc domains can be written as a finite union of star saped domains. e constructed operator will again satisfy te properties given by (3.4). We refer to [11, section III.3], [12, section 2], and [8, section 4.3] for more details. We can terefore conclude our discussion so far wit te following teorem.

7 A UNIFORM INF SUP CONDIION 7 eorem 3.1. Assume tat Ω is a bounded Lipsitz domain. en te uniform inf sup condition (2.4) olds. 4. Preconditioning te discrete coefficient operator e purpose of tis final section is to sow discrete variants of eorem 3.1 for various finite element discretizations of te problem (1.1). More precisely, we will consider finite element discretizations of te system (1.1) of te form: Find (u, p ) V Q suc tat (4.1) u, v + ɛ 2 Du, Dv + p, div v = f, v, v V, div u, q = g, q, q Q. Here V and Q are finite element spaces suc tat V Q H0(Ω; 1 R n ) L 2 0(Ω), and is te discretization parameter. Alternatively, tese problems can be written on te form A ɛ, u = f, p were te coefficient operator A ɛ, is acting on elements of V Q. In te examples below we will, for simplicity, only consider discretizations were te finite element space V Q X ɛ for all ɛ in te closed interval [0, 1]. is implies tat also te pressure space Q is a subspace of H 1. e proper discrete uniform inf sup conditions we sall establis will be of te form (4.2) sup v V div v, q v L 2 ɛ H 1 α q H 1 +ɛ 1 L 2,, q Q, were te positive constant α is independent of bot ɛ and. H 1 +ɛ 1 L 2, is defined as q 2 H 1 +ɛ 1 L 2, = g inf q 1,q 2 Q q=q 1 +q 2 ( q 1 2 H 1 + ɛ 2 q 2 2 L 2), q Q. Here te discrete norm e tecnique we will use to establis te discrete inf sup condition (4.2) is in principle rater standard. We will just rely on te corresponding continuous condition (2.4) and a bounded projection operator into te velocity space V. e key property is tat te projection operator Π commutes properly wit te divergence operator, cf. (4.4) below, and tat it is uniformly bounded in te proper operator norm. Suc projection operators are frequently referred to as Fortin operators. We will restrict te discussion below to two key examples, te Mini element and te aylor Hood element. For bot tese examples we will construct interpolation operators Π : L 2 (Ω; R n ) V wic are uniformly bounded, wit respect to, in bot L 2 and H 1 0. erefore, tese operators will be uniformly bounded operators in L((L 2 ɛ H 1 0)(Ω; R n )), i.e., we ave (4.3) Π L((L 2 ɛ H 1 0 ) is bounded independently of ɛ and.

8 8 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER It is easy to see tat tis is equivalent to te requirement tat Π is uniformly bounded wit respect to in L(L 2 ) and L(H 1 0). Furtermore, te operators Π will satisfy a commuting relation of te form (4.4) div Π v, q = div v, q v H 1 0(Ω; R n ), q Q. As a consequence, te discrete uniform inf sup condition (4.2) will follow from te corresponding condition (2.4) in te continuous case. We recall from eorem 3.1 tat (2.4) olds for any bounded Lipscitz domain, and witout any convexity assumption, and as a consequence of te analysis below te discrete condition (4.2) will also old witout any convexity assumptions. e key ingredient in te analysis below is te construction of a uniformly bounded interpolation operator Π. In te case of te Mini element te construction we will present is rater standard, and resembles te presentation already done in [1], were tis element was originally proposed (cf. also [5, Capter VI]). However, for te aylor Hood element te direct construction of a bounded, commuting interpolation operator is not obvious. In fact, most of te stability proofs found in te literature for tis discretization typically uses an alternative approac, cf. for example [5, Section VI.6] and te discussion given in te introduction of [10]. An exception is [10], were a projection satisfying (4.4) is constructed. However, tis operator is not bounded in L 2. Below we propose a new construction of projection operators satisfying (4.4) by utilizing a tecnique for te aylor Hood metod wic is similar to te construction for Mini element presented below. e new projection operator will be bounded in bot L 2 and H 1, and ence it satisfies (4.3). is analysis is restricted to quasi uniform meses in two space dimensions. In te present case, te discrete inf sup condition (4.2) will imply uniform stability of te discretization in te proper norms introduced above. As a consequence, we are able to derive preconditioners B ɛ,, suc tat te condition numbers of te corresponding operators B ɛ, A ɛ, are bounded uniformly wit respect to te perturbation parameter ɛ and te discretization parameter. e operator B ɛ, can be taken as a block diagonal operator of te form (2.5), but were te elliptic operators are replaced by te corresponding discrete analogs. In fact, to obtain an efficient preconditioner te inverses of te elliptic operators wic appear sould be replaced by corresponding elliptic preconditioners, constructed for example by a standard multigrid procedure. We refer to [15], see in particular Section 5 of tat paper, for a discussion on te relation between stability estimates and te construction of uniform preconditioners. In particular, te results for te aylor Hood metod presented below explains te uniform beavior of te preconditioner B ɛ, observed in te numerical experiment reported in able 1 above e discrete inf sup condition. e rest of te paper is devoted to te construction of proper interpolation operators Π for te Mini element and te aylor Hood element, i.e, we will construct interpolation operators Π : L 2 (Ω; R n ) V suc tat (4.3) and (4.4) olds. We will assume tat te domain Ω is a polyedral domain wic is triangulated by a family of sape regular, simplicial meses { } indexed by decreasing values of te mes parameter = max. Here is te diameter of te simplex. We recall tat te mes is sape regular if te tere exist a positive constant γ 0 suc tat for all values of te

9 A UNIFORM INF SUP CONDIION 9 mes parameter Here denotes te volume of. n γ 0, e Mini element. We recall tat for tis element te velocity space, V, consists of linear combinations of continuous piecewise linear vector fields and local bubbles. More precisely, v V if and only if v = v 1 + c b, were v 1 is a continuous piecewise linear vector field, c R n, and b P n+1 ( ) is te bubble function wit respect to, i.e. te unique polynomial of degree n + 1 wic vanis on and wit b dx = 1. e pressure space Q is te standard space of continuous piecewise linear scalar fields. In order to define te operator Π we will utilize te fact tat te space V can be decomposed into two subspaces, V b, consisting of all functions wic are identical to zero on all element boundaries, i.e. V b 1 is te span of te bubble functions, and V consisting of continuous piecewise linear vector fields. Let Π b : L2 (Ω; R n ) V b be defined by, Πv, b z = v, z z Z, were Z denotes te space of piecewise constants vector fields. Clearly tis uniquely determines Π b. Furtermore, a scaling argument, utilizing equivalence of norms, sows tat te local operators Π b are uniformly bounded, wit respect to, in L2 (Ω; R n ). e operator Π b will satisfy property (4.4) since for all v H1 0(Ω; R n ) and q Q, we ave (4.5) div Π b v, q = Π b v, grad q = v, grad q = div v, q, were we ave used tat grad Q Z. e desired operator Π will be of te form were R : L 2 (Ω; R n ) V 1 Π = Π b (I R ) + R, will be specified below. Note tat I Π = (I Π b )(I R ), and terefore div(i Π )v, q = div(i Π)(I b R )v, q ) = 0 for all q Q. Hence, te operator Π satisfies (4.4). We will take R to be te Clement interpolant onto piecewise linear vector fields, cf. [7]. Hence, in particular, te operator R is local, it preserves constants, and it is stable in L 2 and H 1 0. More precisely, we ave for any tat (4.6) (I R )v H j ( ) c k j Ω v H k (Ω ), 0 j k 1, were te constant c is independent of and v. Here Ω denote te macroelement consisting of and all elements suc tat, and Ω = max, Ω. It

10 10 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER domain ɛ \ Ω Ω Ω able 2. Condition numbers for te operators B ɛ, A ɛ, discretized wit te Mini element. also follows from te sape regularity of te family { } tat te covering {Ω } as a bounded overlap. erefore, it follows from (4.6) and te L 2 boundedness of Π b tat Π is uniformly bounded in L(L 2 (Ω; R n )). Furtermore, by combining (4.6) wit a standard inverse estimate for polynomials we ave for any tat Π v H 1 ( ) Π b (I R )v H 1 ( ) + R v H 1 ( ) c( 1 Πb (I R )v L 2 ( ) + v H 1 ( )) c( 1 (I R )v L 2 ( ) + v H 1 ( )) c( 1 Ω + 1) v H 1 (Ω ) c v H 1 (Ω ), were we ave used tat 1 Ω is uniformly bounded by sape regularity. is implies tat Π is uniformly bounded in L(H 1 0(Ω; R n )). We ave terefore verified (4.3). ogeter wit (4.4) tis implies (4.2). In able 2 below we present results for te Mini element wic are completely parallel to results for te aylor Hood element presented in able 1 above. As we can see, by comparing te results of te two tables, te effect of te different discretizations seems to minor, as long as te mes is te same e aylor Hood element. Next we will consider te classical aylor Hood element. We will restrict te discussion to two space dimensions, and we will assume tat te family of meses { } is quasi uniform. More precisely, we assume tat tere is a mes independent constant γ 1 > 0 suc tat (4.7) γ 1,, were we recall tat = max. For te aylor Hood element te velocity space, V, consists of continuous piecewise quadratic vector fields, and as for te Mini element above

11 A UNIFORM INF SUP CONDIION 11 Q is te standard space of continuous piecewise linear scalar fields. Note tat if we ave establised te discrete inf sup condition for a pair of spaces (V, Q ), were V is a subspace of V, ten tis condition will also old for te pair (V, Q ). is observation will be utilized ere. For tecnical reasons we will assume in te rest of tis section tat any as at most one edge in Ω. Suc an assumption is frequently made for convenience wen te aylor Hood element is analyzed, cf. for example [5, Proposition 6.1], since most approaces requires a special construction near te boundary. On te oter and, tis assumption will not old for many simple triangulations. erefore, in Section below we will refine our analysis, and, as a consequence, tis assumption will be relaxed. We let V V H0(Ω; 1 R 2 ) be te space of piecewise quadratic vector fields wic as te property tat on eac edge of te mes te normal components of elements in V are linear. Eac function v in te space V can be determined from its values at eac interior vertex of te mes, and of te mean value of te tangential component along eac interior edge. In fact, in analogy wit te discussion of te Mini element above, te space V can be decomposed as V = V 1 V b 1. As above te space V is te space of continuous piecewise linear vector fields, wile te space V b in tis case is spanned by quadratic edge bubbles. o define tis space of bubbles we let 1 ( ) = i 1( ) 1( ) be te set of te edges of te mes, were i 1( ) are te interior edges and 1( ) are te edges on te boundary of Ω. Furtermore, if ten 1 ( ) are te set of edges of, and i 1( ) = 1 ( ) i 1( ). For eac e 1 ( ) we let Ω e be te associated macroelement consisting of te union of all wit e 1 ( ). e scalar function b e is te unique continuous and piecewise quadratic function on Ω e wic vanis on te boundary of Ω e, and wit b e e ds = e, were e denotes te lengt of e. e space V b is defined as V b = span{b e t e e i 1( ) }, were t e is a tangent vector along e wit lengt e. Alternatively, if x i and x j are te vertices corresponding to te endpoints of e ten te vector field ψ e = b e t e is determined up to a sign as ψ e = 6λ i λ j (x j x i ), were {λ i } are te piecewise linear functions corresponding to te barycentric coordinates, i.e., λ i (x k ) = δ i,k for all vertices x k. In particular, ψ e (x j x i ) ds = 6 λ i λ j ds e 2 = e 3. e As above te desired interpolation operator Π will be of te form Π = Π b (I R ) + R, were R : L 2 (Ω; R 3 ) V 1 is te same Clement operator as above, and were Πb : L 2 (Ω; R n ) V b needs to be specified. In fact, to perform a construction similar to te one we did for te Mini element it will be sufficient to construct Π b suc tat it is L2 stable, and satisfies te commuting relation (4.5). e

12 12 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER We will need to separate te triangles wic ave an edge on te boundary of Ω from te interior triangles. Wit tis purpose we define = { Ω 1( ) } and i = \. In order to define te operator Π b we introduce Z as te lowest order Nedelec space wit respect to te mes. Hence, if z Z ten on any, z is a linear vector field suc tat z(x) x is also linear. Furtermore, for eac e 1 ( ) te tangential component of z is continuous. As a consequence, Z H(curl; Ω) were te operator curl denotes te two dimensional analog of te curl operator given by curl z = curl(z 1, z 2 ) = x2 z 1 x1 z 2. It is well known tat te proper degrees of freedom for te space Z is te mean value of te tangential components of v, v t, wit respect to eac edge in 1 ( ). Furtermore, we let Z 0 = {z Z z t ds = 0, }. Alternatively, te elements of Z 0 are tose vector fields in Z wit te property tat curl z = 0 if, i.e., z is a constant vector field on for. It is a key observation tat te mes assumption given above, tat any intersects Ω in at most one edge, implies tat te spaces V b and Z0 ave te same dimension. Furtermore, we note tat grad q Z 0 for any q Q. For eac e 1 ( ) let φ e Z be te basis function corresponding to te Witney form, i.e., φ e satisfies (φ e t e ) ds = e, and (φ e t e ) ds = 0, e e, e e were, as above, t e is a tangent vector of lengt e. Hence, if e = (x i, x j ) ten te vector field φ e can be expressed in barycentric coordinates as Any z Z 0 φ e = λ i grad λ j λ j grad λ i. can be written uniquely on te form z = a e φ e + e i 1 ( ) e 1 ( ) c e φ e, were te coefficients a e corresponding to interior edges can be cosen arbitrarily, but were te coefficients c e for eac boundary edge sould be cosen suc tat curl z = 0 on te associated triangle in. We note tat tere is a natural mapping Φ between te spaces V b and Z0 given by Φ (ψ e ) = φ e for all interior edges, or alternatively, (4.8) Φ (v) t e ds = e 2 v t e ds, e i 1( ). e Below, we will use V b( ) and Z0 b ( ) to denote te restriction of te spaces V and Z0 to a single element, and Φ will denote te corresponding restriction of te map Φ. We will define te operator Π b : L2 (Ω; R n ) V b by, (4.9) Π b u, z = u, z z Z 0. e

13 A UNIFORM INF SUP CONDIION 13 o sow tat tis operator is well defined te following general formula for integration of products of barycentric coordinates over a triangle will be useful (cf. for example [14, Section 2.13]) (4.10) λ α 1 1 λ α 2 2 λ α 3 3 dx = 2α!, (2 + α )! were α! = α 1!α 2!α 3!, α = i α i and is te area of. Lemma 4.1. Let i wit edges e 1, e 2, e 3. For any v V b ( ) we ave v Φ (v) dx 1 5 a 2, were Φ = Φ, v = i a iψ ei and a 2 = i a2 i. Proof. A direct computation gives 3 3 v Φ (v) dx = a i a j i=1 j=1 were te 3 3 matrix M is given by M = {M i,j } 3 i,j=1 = { ψ ei φ ej dx = 3 i=1 3 a i a j j=1 ψ i φ ej dx} 3 i,j=1{ b ei φ ej t ei dx} 3 i,j=1, e desired result will follow from te diagonal dominance of tis matrix. b ei φ ej t ei dx = a Ma, Let x j be te vertex opposite e j, and let λ j be te corresponding barycentric coordinate on. en te first diagonal element M 1,1 of te matrix M is given by M 1,1 = 6 λ 2 λ 3 (λ 2 grad λ 3 λ 3 grad λ 2 )(x 3 x 2 ) dx = 6 (λ 2 2λ 3 + λ 2 λ 2 3) dx = 2 /5, were we ave used formula (4.10) in te final step. Actually, from tis formula we derive tat all te diagonal elements are given by M i,i = 2 /5, and similar calculations for te off diagonal elements gives M i,j = /10. In addition, te matrix M is symmetric. e matrix M is terefore strictly diagonally dominant, and by te Gersgorin circle teorem all eigenvalues are bounded below by /5. By combining tis wit te fact tat M is symmetric we conclude tat a Ma a 2 /5, and tis is te desired bound. e next lemma is a variant of te result above for. Lemma 4.2. Let wit interior edges e 1, e 2, and were e 3 is te edge on te boundary. For any v V b ( ) we ave were v = a 1 ψ e1 + a 2 ψ e2 and a 2 = a a 2 2. v Φ (v) dx 1 2 a 2,

14 14 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER Proof. Let v = a 1 ψ e1 +a 2 ψ e2 = 6a 1 λ 2 λ 3 (x 3 x 2 )+6a 2 λ 1 λ 3 (x 3 x 1 ). It is a key observation tat in tis case Φ (v) is simply given as Φ(v) = a 1 grad λ 2 a 2 grad λ 1. Since grad λ i t ei 0 we terefore obtain from (4.10) tat v Φ (v) dx = 6a 2 1 λ 2 λ 3 grad λ 2 (x 2 x 3 ) dx + 6a 2 2 λ 1 λ 3 grad λ 1 (x 1 x 3 ) dx = 6a 2 1 λ 2 λ 3 dx + 6a 2 2 λ 1 λ 3 dx = 1 2 a 2. is completes te proof. Lemma 4.3. ere is a positive constant c 0, independent of, suc tat for eac v V b sup z Z 0 v, z z L 2 (Ω) c 0 v L 2 (Ω). Proof. Let v V b be given, i.e., v = e i 1 ( ) a eψ e. We simply coose te corresponding z = Φ (v) = e i 1 ( ) a eφ e + e 1 ( ) c eφ e Z 0. It follows from scaling and sape regularity tat te two norms of z, given by (4.11) z L 2 (Ω) and ( a 2 e) 1/2 e i 1 ( ) are equivalent uniformly in. Correspondingly, te two norms (4.12) v L 2 (Ω) and ( 2 e i 1 ( ) a 2 e) 1/2 are uniformly equivalent. As a consequence of tese properties, combined wit Lemmas 4.1 and 4.2, we obtain v, Φ (v) = Φ (v) v dx 1 a 2 e c 0 Φ (v) L 5 2 (Ω) v L 2 (Ω). e i 1 ( ) were c 0 > 0 is independent of. is completes te proof. It is a direct consequence of Lemma 4.3 tat te operators Π b are uniformly bounded in L(L 2 (Ω; R 2 )). In fact, te associated operator norm is bounded by c 1 0. Note tat in contrast to te situation for te Mini element, te operator Π b is not local in tis case. However, if te mes is quasi uniform we obtain from (4.7) tat (4.13) Πu b H 1 (Ω) c ( ) 2 Πb u 2 1/2 L 2 ( ) cγ Πu b L 2 (Ω). As a furter consequence we now obtain. eorem 4.4. e operator Π satisfies properties (4.3) and (4.4).

15 A UNIFORM INF SUP CONDIION 15 Proof. o sow tat te operator Π fulfills te condition (4.4), it is enoug to sow tat te operator Π b satisfies te corresponding condition (4.5). However, since grad Q Z 0 tis follows exactly as before, since div Π b v, q = Π b v, grad q = v, grad q = div v, q. Furtermore, to sow (4.3) it is enoug to sow tat Π is uniformly bounded wit respect to in bot L(L 2 (Ω; R 2 )) and L(H0(Ω; 1 R 2 )). However, te L 2 result follows from te corresponding bounds for te operators Π b and R. Finally, by combining (4.6), (4.13) and te boundedness of Π b in L2 we obtain Π v H 1 (Ω) Π b (I R )v H 1 (Ω) + R v H 1 (Ω) c( 1 Π b (I R )v L 2 (Ω) + v H 1 (Ω)) c(c (I R )v L 2 (Ω) + v H 1 (Ω)) c v H 1 (Ω), and tis is te desired uniform bound in L(H 1 0(Ω; R 2 )) More general triangulations. e analysis of te aylor Hood metod given above leans avily on te assumption tat tere are no triangles in wit more tan one edge on te boundary of Ω. is assumption simplifies te analysis, but it is not necessary. e purpose of tis section is to relax tis assumption. We let,1 and,2 denote te subset of triangles in wit one or two edges in Ω, respectively. We let =,1,2 be te set of all boundary triangles, and as before i = \,2. We note tat ten tere is a unique associated triangle suc tat i 1( ). We will denote tis interior edge associated any,2 by e, and we will use to denote te macroelement defined by te two triangles and. e set of all interior edges of te form e,,2, will be denoted i,2 1 ( ), wile i,1 1 ( ) = i 1( ) \ i,2 1 ( ). rougout tis section we will assume tat all te triangles of te form are interior triangles, i.e., i for all,2. e interpolation operators Π and Π b will be defined as above, wit te only exception tat te definition of te bubble space V b is canged sligtly in te neigborood of te edges in i,2 1 ( ). We note tat te result of Lemma 4.2 still olds if,1. o establis te result of Lemma 4.3, and as a consequence eorem 4.4, in te present case we basically need an anolog of Lemma 4.2 for triangles,2. More precisely, we need to modify te definition of te map Φ, used in proof of Lemma 4.3, to te present case. Actually, te map Φ will not be defined locally on te triangles,2, but rater on te corresponding macroelement. As in te previous section te space Z 0 is taken to be te subspace of te Nedelec space Z suc tat z Z 0 is constant on te triangles in. o be able to define te interpolation operator Π b b, mapping into te bubble space, by (4.9), te two spaces V and Z 0 must be balanced. In particular, tey sould ave te same dimension. However, in te present case te dimension of te space Z 0 is not te same as te number of interior edges. o see tis just consider te restriction Z 0( ) of Z 0 to a macroelement macroelement,

16 16 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER x 0 x 1 x 2 x 3 Figure 2. e macroelement. cf. Figure 2 below. e dimension of te space Z 0( ) is four, wile tere are only tree interior edges, namely te tree edges of. o compensate for tis we will extend te space of bubble functions V b, by including also normal bubbles on te edges e,,2. In te present case we define te space V b V by V b = span({ b e t e e i 1( ) } { b e n e e i,2 1 ( ) }). Here t e and n e are tangent and normal vectors to te edge e wit lengt e. Wit tis definition te space V b as te same dimension as Z0. Furtermore, te map Φ : V b Z0 will be defined to satisfy property (4.8) for all edges in i,1 1 ( ). Note tat tis specifies Φ on all triangles in, except for te ones tat belongs to te macroelements,,2. o complete te definition of Φ we need to specify its restriction to eac macroelement. Consider a macroelement of te form given in Figure 2. Here te edge e as endpoints denoted by x 1 and x 2, te tird boundary vertex of is x 0, wile te single interior vertex of is x 3. We will use e i to denote te edge opposite x i, i = 1, 2, of te triangle, wile e i are te corresponding edges of te triangle. We let Φ : V b( ) Z 0( ) be te restriction of Φ to. o be compatible wit te definition of Φ outside te macroelements te map Φ as to satisfy condition (4.8) on te edges e i, i.e., (4.14) were t e i functions e i Φ (v) t e i is a vector tangential to e i. ds = e i 2 e i v t e i ds, i = 1, 2, As a basis for te space Z0 ( ) we will use te φ i = λ i grad λ 3 λ 3 grad λ i, i = 1, 2, wit support only in, combined wit te two functions φ i given by grad λ i on, φ i = ( 1) i φ, on,

17 A UNIFORM INF SUP CONDIION 17 for i = 1, 2, were φ = λ 1 grad λ 2 λ 2 grad λ 1 corresponds to te Witney form associated te edge e = (x 1, x 2 ). e functions φ i, φ i for i = 1, 2 spans te space Z 0 ( ). We will define two basis functions ψ i of V b as a multiple of te scalar bubble function b e, namely, (4.15) ψ i = b e w i = 6λ 1 λ 2 w i i = 1, 2, were te vectors w i will be cosen below. Furtermore, te functions ψ i ψ i = 6λ i λ 3 (x 3 x i ) + β( 1) i λ 1 λ 2 (x 2 x 1 ), i = 1, 2, were β = 6 /(5 + 4 ). We note tat 0 < β < 3/2., are given as e functions ψ i, ψ i for i = 1, 2 span te space V b( ), and we define Φ (ψ i ) = φ i and Φ (ψ i ) = φ i. A map of tis form will satisfy te compatibility condition (4.14) by construction. e motivation for te coice of te constant β is tat we obtain (4.16) ψ i φ j dx = 0, i, j = 1, 2. Lemma 4.5. e ortogonality conditions (4.16) old. Proof. e identities (4.16) can be verified by formula (4.10). For example ψ1 φ 1 dx = β λ 1 λ 2 grad λ 1 (x 2 x 1 ) dx (λ 1 grad λ 2 λ 2 grad λ 1 ) (6λ 1 λ 3 (x 3 x 1 ) βλ 1 λ 2 (x 2 x 1 )) dx = β λ 1 λ 2 dx (6λ 1 λ 2 λ 3 β(λ 2 1λ 2 + λ 1 λ 2 2)) dx = ( 6 + β(5 + 4 )/60 = 0. Furtermore, it is easy to ceck tat ψ1 φ 2 dx = ψ1 φ 1 dx, and as a consequence ψ 1 φ 2 dx = 0. Similar computations can be done for te integrals involving ψ2. We can also verify, again using formula (4.10), tat te 2 2 matrix M = {M i,j } i,j=1,2 = { ψ i φ j } i,j=1,2 is given by (24 β)/60 (6 + β)/60 M =. (6 + β)/60 (24 β)/60 For 0 < β < 3/2 tis symmetric matrix is strictly diagonally dominant wit bot eigenvalues greater tan /4.

18 18 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER x 0 ˆx 0 x 1 x 2 x 3 Figure 3. e macroelement and te associated parallelogram. Finally, we need to investigate te 2 2 matrix M = {M i,j } i,j=1,2 = { ψ i φ j } i,j=1,2. However, first we need to define te functions ψ i = Φ (φ i ) precisely by specifying te vectors w i in (4.15). We let ψ 1 = 6γλ 1 λ 2 (x 3 x 2 ), and ψ 2 = 6γ 1 λ 1 λ 2 (x 3 x 1 ), were te positive constant γ will be cosen below. Assume for a moment tat is a parallelogram. en x 3 x 1 = x 2 x 0 and x 3 x 2 = x 1 x 0, and terefore we would ave easy computable representations of te functions ψ i on bot and. In general, we introduce a new point ˆx 0 R 2, depending on, wit te property tat ˆx 0, x 1, x 2, x 3 corresponds to te corners of a parallelogram, cf. Figure 3. More precisely, ˆx 0 = x 1 (x 3 x 2 ) = x 1 + x 2 x 3. Let {ˆλ i } 2 i=0 be te barycentric coordinates wit respect to te triangle, extended to linear functions on all of R 2. en ˆλ 1 (x 3 ) + ˆλ 2 (x 3 ) > 1 and ˆλ 1 (ˆx 0 ) + ˆλ 2 (ˆx 0 ) = 2 ˆλ 1 (x 3 ) + ˆλ 2 (x 3 ) < 1. In fact, it is a consequence of sape regularity tat tere is a constant α > 0, independent of and te coice of,2, suc tat (4.17) ˆλ1 (ˆx 0 ) + ˆλ 2 (ˆx 0 ) 1 α. If we compute te matrix { ψ i φ j dx} we obtain { ψ i φ j dx} i,j=1,2 = γ(1 ˆλ 1 (ˆx 0 )) γˆλ 2 (ˆx 0 ) 2 γ 1ˆλ1 (ˆx 0 ) γ 1 (1 ˆλ. 2 (ˆx 0 )) o control te full matrix M we also need to consider te contributions from te triangle. A straigtforward computation, using formula (4.10), sows tat te matrix { ψ i φ j } is

19 given by A UNIFORM INF SUP CONDIION 19 { ψ i φ j dx} i,j=1,2 = γ γ. 5 γ 1 γ 1 We will utilize te constant γ to obtain a symmetric matrix M. We define 2 γ = + 5 ˆλ 1 (ˆx 0 ) ˆλ 2 (ˆx 0 ). is coice of γ is motivated by te desired identity γ( 2 ˆλ 2 (ˆx 0 ) + 5 ) = γ 1 ( 2 ˆλ 1 (ˆx 0 ) + 5 ), wic can be seen to old, and terefore te matrix M is symmetric. Furtermore, we note tat γ, γ erefore, it is a consequence of sape regularity tat te positive constant γ is bounded from above and below, independently of and te coice of,2. Lemma 4.6. e matrix M defined above is symmetric and positive definite wit bot eigenvalues bounded below by c 1, were c 1 = α min(γ, γ 1 )/2. Proof. It follows from te calculations above tat γ( M = (1 ˆλ 2 1 (ˆx 0 )) + ) γ( ˆλ (ˆx 0 ) + ) 5 γ 1 ( ˆλ 2 1 (ˆx 0 ) + ) 5 γ 1 ( (1 ˆλ 2 2 (ˆx 0 )) + ). 5 Since ˆλ 1 (ˆx 0 )+ ˆλ 2 (ˆx 0 ) 1 α it follows from Gersgorin circle teorem tat bot eigenvalues of M are bounded below by α 2 min(γ, γ 1 ). We now ave te following result. Lemma 4.7. e conclusion of Lemma 4.3 olds in te present case. Proof. Let v V b be given. We first consider te situation on eac macroelement. If v = i (a+ i ψ i + a i ψ i ) V b( ) ten we write v = v + v +, were v = i a i ψ i. Observe tat Lemma 4.6, togeter wit te ortogonality property (4.16), implies tat v Φ (v + ) dx = v + Φ (v + ) dx c 1 a + 2. Similarly, we ave from te property of te matrix M, te norm equivalences expressed by (4.11) and (4.12), and sape regularity tat v Φ (v ) dx v Φ (v ) dx v + L 2 ( ) Φ (v ) L 2 ( )

20 20 KEN ANDRE MARDAL, JOACHIM SCHÖBERL, AND RAGNAR WINHER 1 4 a 2 c a a a 2 c 2 a + 2, were te constant c 2 is independent of and. By coosing Φ (v) = CΦ(v + ) + Φ(v ), were te constant C is sufficiently large, we can now conclude tat v Φ (v) dx c a 2. We note tat te map Φ will inerit te compatibility condition (4.14) from te map Φ. By combining tis result on eac macroelement, wit te map Φ defined previously on te rest of te triangles in, to a global map Φ mapping V b, we can conclude, as in te proof of Lemma 4.3, tat is completes te proof. sup z Z 0 v, z z L 2 (Ω) v, Φ (v) Φ (v) L 2 (Ω) c 0 v L 2 (Ω). As we ave noted above te result just given implies tat te conclusion of eorem 4.4 olds will old for te more general meses studied in tis section. References [1] F. Brezzi, D.N. Arnold and M. Fortin, A stable finite element metod for Stokes equations, Calcolo 21 (1984), [2] J. Berg and J. Löfström, Interpolation spaces, Springer-Verlag, [3] J.H. Bramble and J.E. Pasciak, Iterative tecniques for time dependent Stokes problems, Comput. Mat. Appl. 33 (1997), no. 1-2, 13 30, Approximation teory and applications. [4] F. Brezzi, On te existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO. Analyse Numérique 8 (1974), [5] F. Brezzi and M. Fortin, Mixed and ybrid finite element metods, Springer-Verlag, [6] J. Caouet and J.-P. Cabard, Some fast 3D finite element solvers for te generalized Stokes problem, Internat. J. Numer. Metods Fluids 8 (1988), no. 8, [7] P. Clement, Approximation by finite element functions using local regularization, RAIRO Anal. Numér. 9 (1975), [8] M. Costabel and A. McIntos, On Bogovskiĭ and regularized Poincaré integral operators for de Ram complexes on Lipscitz domains, ttp://arxiv.org/abs/ v1. [9] M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or tree-dimensional domains wit corners. I. Linearized equations, SIAM Journal on Matematical Analysis 20 (1989), no. 1, [10] R.S Falk, A Fortin operator for two dimensional aylor Hood elements, Matematical Modelling and Numerical Analysis (M2AN) 42 (2008), [11] Giovanni P. Galdi, An introduction to te matematical teory of te Navier Stokes equations, vol 1, Linearized Steady problems, vol. 38, Springer racts in Natural Pilosopy, Springer Verlag, New York [12] M. Geissert, H. Heck, and M. Hieber, On te equation div u = g and Bogovskiĭ s operator in Sobolev spaces of negative order, Oper. eory Adv. Appl. 168 (2006), [13] V. Girault and P.-A. Raviart, Finite element metods for Navier-Stokes equations, Springer Series in Computational Matematics, vol. 5, Springer-Verlag, Berlin, 1986, eory and algoritms.

21 A UNIFORM INF SUP CONDIION 21 [14] M.-J. Lay and L.L. Scumaker, Spline functions on triangulations, vol. 110 of Encyclopedia of Matematics and its Applications, Cambridge University Press, [15] K.-A. Mardal and R. Winter, Preconditioning discretizations of systems of partial differential equations, Numer. Linear Alg. Appl. 18 (2011), [16] K.-A. Mardal and R. Winter, Uniform preconditioners for te time dependent Stokes problem, Numer. Mat. 98 (2004), no. 2, [17], Erratum: Uniform preconditioners for te time dependent Stokes problem [Numer. Mat. 98 (2004), no. 2, ], Numer. Mat. 103 (2006), no. 1, [18] M. A. Olsanskii, J. Peters, and A. Reusken, Uniform preconditioners for a parameter dependent saddle point problem wit application to generalized Stokes interface equations, Numerisce Matematik 105 (2006), [19] S. urek, Efficient solvers for incompressible flow problems, Springer-Verlag, Centre for Biomedical Computing at Simula Researc Laboratory, Norway and Department of Informatics, University of Oslo, Norway address: kent-and@simula.no URL: ttp://simula.no/people/kent-and Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria address: joacim.scoeberl@tuwien.ac.at URL: ttp:// Centre of Matematics for Applications and Department of Informatics, University of Oslo, 0316 Oslo, Norway address: ragnar.winter@cma.uio.no URL: ttp://folk.uio.no/rwinter