Inverse inequalities on non-quasi-uniform meshes and spplication to the mortar ...

Volume 73, Number 247, Pages 1107–1138 S 0025-5718(03)01583-7

Article electronically published on July 14, 2003

INVERSE INEQUALITIES ON NON-QUASI-UNIFORM MESHES AND APPLICATION TO THE MORTAR ELEMENT METHOD

W. DAHMEN, B. FAERMANN, I. G. GRAHAM, W. HACKBUSCH, AND S. A. SAUTER

Abstract. We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functionsudefined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the formkh^αukW^s,p(Ω) for positive and nega- tive s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved isN, the total number of degrees of freedom in the finite element space, and we avoid esti- mates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element anal- ysis to extend results—previously known only for quasi-uniform meshes—to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

1. Introduction

Many classical a priori estimates for the error in a finite element approxima- tion u to a solution v of an operator equation take the form kv−uk_Ht(Ω) ≤ C(hmax)^s−tkvkH^s(Ω), for some s > t, where hmax is the global maximum mesh diameter andH^s(Ω) is the usual Sobolev space of index son a domain Ω . Since modern applications produce meshes with significant local variation of mesh size, many authors avoid the introduction of hmax and use instead localised estimates, such as

(1.1) kv−ukH^t(Ω)≤C (X

τ∈T

h^2(s_τ ^−t)kvk²H^s(τ)

)1/2

,

withτ denoting a typical element of the meshT and hτ denoting the diameter of τ. Other authors work with estimates of the form

(1.2) h^−(s−t)(v−u)

H^t(Ω)≤CkvkH^s(Ω) ,

whereh(x) is some positive “mesh-size” function, designed to reflect the local mesh diameter near each point x. Estimates such as these allow quite general mesh refinement procedures including some produced by adaptive algorithms (e.g., [21]).

Received by the editor May 2, 2001 and, in revised form, January 10, 2003.

2000 Mathematics Subject Classification. Primary 65N12, 65N30, 65N38, 65N55, 41A17, 46E35.

Key words and phrases. Inequalities, mesh-dependent norms, inverse estimates, nonlinear ap- proximation theory, nonmatching grids, mortar element method, boundary element method.

c2003 American Mathematical Society 1107

Some finite element error analyses require, as an additional ingredient, certain fundamental inequalities which have to be satisfied by the finite element func- tions. Examples are inverse estimates, which bound the Sobolev norm of a finite element function in terms of its Sobolev norm of lower index, multiplied by a mesh-dependent constant. These are used in many classical analyses, for example uniform norm error estimates for finite elements ([8]) and stiffness matrix condi- tioning analysis in boundary elements (e.g., [24]). More recently, inverse estimates have appeared also in the analysis of the mortar element method for PDEs, where negative index spaces on lower dimensional manifolds appear naturally ([5]).

Classical inverse estimates (like classical a priori error estimates) are usually global: The mesh-dependent constant in the bound grows as an inverse power of hmin(the globalminimum mesh diameter). If the mesh is strongly locally refined, this bound is large and may not be useful. Thus analyses which make use of inverse estimates often make the additional mesh assumption of global quasi-uniformity (i.e.,hmin∼hmax), and so most interesting adaptive procedures are then ruled out.

In this paper we prove a range of inequalities, including localised inverse esti- mates which apply to both piecewise constant and continuous piecewise linear func- tions defined with respect to meshes of simplices on domains inR^d. The meshes are assumed shape-regular but need not be quasi-uniform. Our results include estimates of the form

(1.3) kh^αukW^s,p(Ω)≤Ckh^α−sukL^p(Ω), for a range of nonnegatives and

(1.4) kh^s+αukL^q(Ω)≤Ckh^αukW−s,q(Ω), for all nonnegatives,

whereα∈Randhis the mesh-size function mentioned above. If the mesh is quasi- uniform, then his bounded above and below in terms of eitherhmin or hmax and these bounds reduce to standard inverse estimates. In the locally refined case they represent localised inverse inequalities which may be considerably less pessimistic than the classical ones.

While some particular localised inverse estimates have been developed in the literature in connection with special applications (e.g., [1] proved special cases of (1.3) and [18] proved special cases of (1.4)), we know of no source in the literature for the general inequalities presented here. In fact (1.3) and (1.4) are only examples of a range of inequalities which we prove. Some of these are elementary and others depend on a rather more delicate analysis. A recurring elementary tool throughout the analysis is the use of mesh-size dependent discrete `<sup>p</sup> norms defined on the degrees of freedom of the finite element functions. Preliminary manipulations using such`² norms can be found in [18].

A more sophisticated tool which we need is the scale of Besov spacesB_q^s(L^p(Ω)) of smoothness sand primary index p, in which are embedded the Sobolev spaces W^s,p(Ω). In fact we prove (1.3) by obtaining its more general analogue in the Besov scale, which allows evenp <1 and a range ofsup to the regularity limit ofu. On the other hand, (1.4) is proved using a separate and nonstandard duality argument and holds forall negativesand all 1< q≤ ∞.

A range of elementary inequalities (involving relations between differentL^p and

`^p norms) are given in Sections 2, and 3, while the inverse estimates (1.3), (1.4) are proved in Section 4. In Section 5 we give localised versions of trace and extension

theorems for finite element functions using the sameL^pnorm on the ambient space and on the lower-dimension manifold.

The rest of the paper is devoted to applications of these inequalities. As men- tioned above, one application (in particular of (1.4)) is in the analysis of boundary element methods for operators on negative order spaces. Since this is described in some detail in [18, 19], here we restrict ourselves instead to describing two other applications.

Our first application is to the Jackson and Bernstein inequalities which arise in the theory of adaptive finite element approximation. Recall that in a quasi-uniform mesh we have hτ ∼N^−1/d, where N is the number of degrees of freedom in the finite element space. The estimate (1.1) then reads (e.g., in the caset= 0):

(1.5) kv−ukL²(Ω)≤CN^−s/dkvkH^s(Ω).

When v fails to be in H^s(Ω), but still possesses enough Besov regularity, then it is known (e.g., [13]) that adaptive processes exist which ensure that the rate of convergenceN^−s/d of best finite element approximation to v is still attained, but with kvkH^s(Ω) on the right-hand side of the error estimate replaced by the weaker smoothness measurekvkB^s_r(L^r(Ω))with 1/r < s/d+ 1/2. This variant of the classical “Jackson-type” approximation estimate is described in Section 6 (in fact for primary index 2 replaced by general p). The Jackson inequality is classically accompanied by the “Bernstein-type” inverse inequality:

kukB^s_q(L^p(Ω)) ≤CN^s/dkukL^p(Ω), for finite element functionsu.

This is proved as the first application of our inverse inequalities in Section 6.

Our second application (Section 7) concerns the stability of the mortar element method in the case of non-quasi-uniform meshes on the subdomains. The mortar method seems to be particularly well suited for problems with strong jumps in coefficients. Since one therefore expects to deal with possibly irregular solutions, the use of nonuniform meshes appears to be very desirable. The so-called dual basis mortar method has indeed recently been shown in [22] to lead to stable and accurate discretisations for the much more flexible class of shape-regular meshes provided that a certain weak matching condition on adjacent meshes holds along interface boundaries. Our objective here is to establish stability for this version of the mortar method without requiring this matching condition. Aside from the inverse estimates from Section 3 we also make essential use of the extension theorem in Section 5.

Acknowledgements. This work was supported in part by British Council/DAAD ARC Grant 869 and in part by the Max-Planck-InstitutMathematik in den Natur- wissenschaften, Leipzig through a visiting position for I. G. Graham. This support is gratefully acknowledged.

2. Preliminaries

2.1. Meshes and finite elements. Throughout this paper Ω denotes either an open Lipschitz polyhedron or (a connected subset of) the surface of a Lipschitz polyhedron. In both cases, the dimension/surface dimension of Ω is denoted by d∈ N. Although most of our results extend to general dimension d, we give the proofs for the casesd∈ {1,2,3}.

Amesh T on Ω is a decomposition Ω =[

{τ:τ∈ T },

where the elements τ are either intervals, triangles or tetrahedra. The elements have nodes and edges and also (when they are tetrahedra) they have faces. For convenience we will always consider the elements to be closed sets. We assume throughout that our meshes are conforming, i.e., if σ, τ ∈ T, with σ 6= τ, then σ∩τ is either empty or a vertex, an edge or a face. We identify two sets of points in Ω which are useful as index sets.

LetN0 denote the set of centroids of elements ofT. We identify cj ∈ N0 with its counting index j and we write j ∈ N0 as well ascj ∈ N0. The setsT andN0

are in one-one correspondence and forj ∈ N0, we denote the element of T which containscj byτj.

Also we let N1 denote the set of nodal points of T. Similarly, we writei∈ N1

as well asxi∈ N1.

Each elementτ has a diameter denotedhτ and a volume|τ|=R

τdx.

We are concerned with inequalities for piecewise polynomial functions onT in the two most important cases:

S0(T) =

v: Ω→R:v|int(τ)is constant,τ ∈ T , (2.1)

S1(T) ={u: Ω→R:uis continuous andu|τ is affine, τ ∈ T }. (2.2)

For eachj ∈ N0, letχj∈ S0(T) denote the characteristic function ofτj and for eachi∈ N1, we defineφi∈ S1(T) to be the “hat” function with valuesφi(xj) =δi,j, fori, j∈ N1. Eachu∈ Sk(T) then has the familiar expansion:

u= X

j∈N0

ujχj, withuj =u(cj), u∈ S0(T), (2.3)

u= X

i∈N1

uiφi, withui=u(xi), u∈ S1(T).

(2.4)

If Ω is a d-dimensional surface in R^d+1, the surface derivatives of a sufficiently smooth function u : Ω → R on (plane) surface elements τ ∈ T are defined by introducinglocal (d+ 1)-dimensional Cartesian coordinates so that the firstdco- ordinates lie in the tangential plane through τ. Letκτ denote the mapping from local to global coordinates and put ˆuτ :=u◦κτ. Then, for anyα∈N^d0, we define

∂^αu|_τ:= (∂^αuˆτ)◦κ⁻_τ¹:τ→R.

Similarly, we put

∇u:= (∇ˆuτ)◦κ⁻_τ¹:τ→R^d.

Note that, by using this definition, Leibniz’ rule for differentiation of products of functions holds as usual. At various places in the text, we consider polynomials on elementsτ ∈ T. In the case of surfaces this notation always has to be understood in the sense that the function, inlocal coordinates, is a polynomial.

2.2. Mesh regularity.

Definition 2.1. Two verticesxi, xj are calledneighbouring if there is an element τ ∈ T such that xi, xj ∈τ (i.e.,xi andxj are connected by an edge of the mesh).

Two elementsτ, σ∈ T are called neighbouring ifτ∩σ6=∅.

We shall consider classes of meshes which satisfy the following weak regularity assumptions.

Definition 2.2. ForK≥1 andε >0, MK,ε denotes the class of meshesT which satisfy

hτ ≤Khσ, for all neighbouring elements τ, σ∈ T (2.5)

and

|τ| ≥εh^dτ, for allτ ∈ T. (2.6)

Whend≥2, it may be shown that for the conforming meshes considered here, the “shape-regularity” assumption (2.6) implies the “local quasi-uniformity” or “K- mesh” condition (2.5), and so in this case the meshes could be characterised by the single parameterε. However in other situations it is of interest to consider locally quasi-uniform meshes which are not shape regular, and so we choose to keep the parametersK andεseparate in our analysis.

Notation 2.3. Throughout the paper, if A(T) andB(T) are two mesh-dependent properties, then the inequality

A(T).B(T)

will mean that there is a constantCdepending onK andε such that A(T)≤CB(T), for allT ∈ MK,ε.

Also the notation

A(T)∼B(T) will mean that A(T).B(T) andB(T).A(T).

In some situations we will be considering other parameters as well. If the con- stants in the estimates are independent of another parameter, say α ∈ [α, α], we shall write explicitly, “A(T) . B(T) uniformly in α ∈ [α, α]” or “A(T) ∼ B(T) uniformly inα ∈ [α, α]”, as appropriate. This means that the equivalence constants may depend onαandαbut not onα∈[α, α].

For our later estimates we need to introduce a mesh-dependent functionhon Ω, such that the value ofhon any τ∈ T is proportional to the size of elements ofT nearτ. To this end, for i∈ N1, we introduce the subsets ofT:

(2.7) T(xi) :={τ∈ T : τ has a corner atxi}.

Then we set

hi:= max{hτ :τ ∈ T(xi)},

and we defineh∈ S1(T) to be the interpolant of these values, i.e.,

(2.8) h= X

i∈N1

hiφi.

Our aim in this paper is to create methods of analysis which are relevant to locally refined meshes and to exploit them in several applications. To this end, information about the mesh being used will be contained in the function hdefined above and this plays a key role in most of our estimates. In some situations we also need to include a global mesh parameter. For this we avoid using the maximum or minimum mesh diameters, given byhmax = max{hτ :τ ∈ T } andhmin= min{hτ :τ ∈ T }, but choose instead to use the cardinality of the mesh N := #N1. Note that

asymptotically it is unimportant whether N is defined as the number of nodes or elements since it follows from the conformity of the meshes that

(2.9) #N0 ∼#N1.

Since adaptive techniques try to construct a good approximation for a minimalN, estimates involving N are more natural in the context of adaptivity than those involvinghminorhmax.

Remark 2.4. Since minx∈Ωh(x) = mini∈N1hi>0, any powerh^sis well defined for s∈R.

Throughout the paper we will frequently use the estimates in the following propo- sition without further reference.

Proposition 2.5. Let T ∈ MK,ε. Then

(a)hτ ≤hi≤Khτ for all τ∈ T(xi), i∈ N1.

(b)K⁻¹hi⁰ ≤hi≤Khi⁰, for all pairs of neighbouring vertices xi andxi⁰ ∈ N1. (c)For all j∈ N0,

hτj ≤ min

i∈τj∩N1

hi≤h(cj)≤ max

i∈τj∩N1

hi≤Khτj. (d)For allj ∈ N0,

h^d_τj ≤ |τj| ≤h^d_τj.

(e) For any two points x, y ∈Ω, let |Λ (x, y)| denote the length of the minimal path in Ω connecting xand y and let CΩ denote the minimal constant such that, for allx, y∈Ω,|Λ (x, y)| ≤CΩ|x−y|. The function his Lipschitz continuous with Lipschitz constant satisfying

L≤C(K−1)/ε, whereC depends only on CΩandd.

Proof. The proofs of (a)–(d) are trivial. To obtain (e), first observe that by the definition ofhwe have, for anyj∈ N0 and anyx∈τj,

∇h(x) =∇h(cj) =∇(h−h(cj))(cj) = X

i∈τ_j∩N1

(hi−h(cj))∇φi(cj).

Now using the shape-regularity property (2.6), it follows that|∇φi(cj)| ≤C(hτj)⁻¹ whereC only depends ond. Also, fori∈τj∩ N1, we have

(1−K)hτ_j ≤ min

i∈τj∩N1

hi− max

i∈τj∩N1

hi≤hi−h(cj)

≤ max

i∈τj∩N1

hi− min

i∈τj∩N1

hi≤(K−1)hτ_j, and thus

k∇hkL^∞(Ω)≤C(K−1)/ε.

Now take any x, y ∈ Ω and recall that Ω is as specified at the beginning of Sec- tion 2.1. Let Λ denote a shortest path in Ω connectingxandy. Since the elements in T are simplices, the restriction of Λ to anyτ is either empty or a straight line

(possibly degenerated to a single point). Choose a minimal sequence (τi)^m_i=1 in T such thatτi∩Λ =:AiAi+1 for 1≤i≤m−1 andA1=x,Am=y. Then,

|h(x)−h(y)| ≤

mX−1 i=1

|h(Ai)−h(Ai+1)|

≤

mX−1 i=1

k∇hk_L∞(τi)|Ai−Ai+1| ≤ k∇hk_L∞(Ω)|Λ|

≤CΩk∇hkL^∞(Ω)|x−y| ≤CCΩ

K−1

ε |x−y|.

2.3. The Sobolev norms. For 1≤p≤ ∞, we introduce the usual space L^p(Ω) with norm k · kL^p(Ω). Extending this definition to 0 < p <1, we obtain a quasi- norm which satisfies the modified triangle inequalityku+vkL^p(Ω)≤C[kukL^p(Ω)+ kvkL^p(Ω)] withC:= 2^1/p−1.For further details onL^pspaces withp <1, see [13, 14]

and the references therein. The casep <1 will become important in Section 6.

We introduce the usual Sobolev spacesW^s,p(Ω) and we note that fors∈(0,1), these are equivalently defined using the Slobodeckij norm :

kvkW^s,p(Ω):=



kvk^p_Lp(Ω)+ Z Z

Ω×Ω

|v(x)−v(y)|^p

|x−y|^d+ps dxdy





1/p

(see, e.g., [20, Section 6.2.4]). In the special case p = 2, we write k·k_Hs(Ω) :=

k·kW^s,2(Ω). For negative−s, W^−s,q(Ω) is the dual spaceW^−s,q(Ω) := (W^s,p(Ω))⁰ with ¹_p +¹_q = 1, p < ∞, endowed with the dual norm. The norms can also be used when Ω is a d-dimensional manifold in R^d+1, d = 1,2 (see, e.g., [18], [19]).

Throughout the paper we restrict the range of Sobolev indices in the case of only piecewise smooth surfaces tos∈[−1,1].

2.4. The `^p norms. Let v= (vi)i∈I ∈R^I be a vector with I denoting its index set. Then, as usual, we write

kvk_`p(I)=X

i∈I|vi|^p1/p

forp∈(0,∞), and kvk_`∞(I)= max{|vi|:i∈ I}.

If w = (wi)i∈I, then we define the `² inner product and the pointwise product, respectively, by

(v,w)`²(I)=X

i∈I

viwi, vw= (viwi)i∈I.

Iff is any function on Ω, we introduce the discrete norm off onN0 and onN1

defined by kfk`^p(Nk)=

( P

i∈N0|f(ci)|^{p 1/p} where f = (f(ci))_i∈N

0, whenk= 0, P

i∈N1|f(xi)|^{p 1/p} where f = (f(xi))_i∈N1, whenk= 1, when these quantities are well defined.

3. Estimates in`^p andL^p norms 3.1. Relations between discrete and continuous norms.

Proposition 3.1. Let p0>0 andα < α be given. Then, fori= 0or1, (3.1)

kh^αukL^p(Ω)∼ kh^α+d/puk`^p(Ni), uniformly in u∈Si(T), p∈[p0,∞]andα∈[α, α].

Proof. Throughout this proof the relations . and ∼ will hold uniformly in p ∈ [p0,∞] andα ∈ [α, α]. First consider p0 ≤ p < ∞, and observe that for any f ∈L^p(Ω), we can write

(3.2) kh^αfkL^p(Ω)∼



 X

j∈N0

h^αp_τj Z

τj

|f|^p





1/p

.

Thus if we consider anyu∈ S0(T), we have, kh^αukL^p(Ω)∼



 X

j∈N0

h^αpτ_j |τj| |u(cj)|^p





1/p

∼



 X

j∈N0

h(cj)^αp+d|u(cj)|^p





1/p

=kh^α+d/puk`^p(N0).

This estimate is uniform in u∈ S0(T),p∈[p0,∞) andα∈[α, α].

On the other hand, supposeu∈ S1(T). Since, for allj∈ N0,u|τ_j is a polynomial of degree 1, it follows by a scaling argument that

(3.3)

|τj|^−1/p (Z

τ_j

|u(x)|^pdx )1/p

∼



 X

i∈τj∩N1

|u(xi)|^p





1/p

, uniformly inj, p∈[p0,∞).

More precisely, the equivalence (3.3) is obtained by first transferring the left-hand side to a unit simplex ˆτ (with nodes ˆxj), via the usual affine map [10]. This yields equivalence tokukˆ L^p(τ), where ˆuis the pullback ofuonto ˆτ. Since the map (ˆu, p)7→ kˆukL^p(τ)is continuous on the compact set

{(ˆu, p) : ˆuis affine on ˆτ , Xd j=1

|u(ˆˆ xj)|^p= 1, andp∈[p0,1]},

the assertion (3.3) follows forp∈[p0,1]. An easier argument based on the Riesz- Thorin interpolation theorem forL^p spaces establishes it for the full range ofp.

Hence, inserting (3.3) in (3.2), we get, uniformly inu∈ S1(T) andp∈[p0,∞), kh^αukL^p(Ω)∼



 X

j∈N0

h^αp_τj |τj| X

i∈τ_j∩N1

|u(xi)|^p





1/p

∼ (X

i∈N1

h^αp+d_i |u(xi)|^p )1/p

=kh^α+d/puk`^p(N1),

as required. (In the second to the last step we have used the fact that the mesh is conforming and shape regular and so the number of elements attached to any given node is bounded over all meshes in the classMK,ε.)

The remaining case ofp=∞follows by similar arguments.

The following corollary identifies two simple special cases of Proposition 3.1.

Corollary 3.2. It holds that X

j∈N0

h(cj)^d∼ X

i∈N1

h^di ∼1, h^−d/p

L^p(Ω)∼N^1/p, uniformly inp∈[p0,∞).

Remark 3.3. The special case α= 0, p = 2 of (3.1) states that the mass matrix M(defined byhMu,vi`²(N1)=hu, viL²(Ω)) is spectrally equivalent to the diagonal matrixD= diag{h^d}. HenceDis an optimal preconditioner toM(whose condition number is bounded above by max{h^di/h^d_j : i, j∈ N1}).

3.2. Estimates between different `<sup>p</sup> norms. First we recall some inequalities satisfied by`^p norms.

Proposition 3.4. (a) Let0< p≤p⁰≤ ∞.Then, for any index setI, (3.4) kuk_{`</sub><sup>p0</sup><sub>(I)</sub>≤ kuk<sub>`}p(I).

(b)Let 0< α≤β≤ ∞andα≤p≤β. Then (3.5)

kuk_{`</sub>p(I)≤ kuk<sup>γ</sup><sub>`}α(I)kuk¹_`β⁻(^γI) withγ= _α

p β−p

β−α = 1−^β_pβ^p−₋^α_α if β <∞,

α/p if β=∞.

Proof. To prove (3.5), we use H¨older’s inequality to obtain (for finiteβ):

kuk^p<sub>`</sub>p(I)= (|u|<sup>αβ−αβ−p</sup>,|u|<sup>ββ−αp−α</sup>)`²(I)≤ k|u|^{αβ−αβ−p}k

`^β−αβ−p(I) k|u|^{βp−αβ−α}k

`^β−αp−α(I). On the other hand, ifβ=∞, we can write

(3.6) kuk`<sup>p</sup>(I)≤ kuk<sup>α/p</sup><sub>`</sub>α(I)kuk¹_`∞^−α/p(I) , and together these two estimates prove (3.5).

It can be easily checked thatkuk`<sup>∞</sup>(I)≤ kuk`^α(I)for any 0< α≤ ∞.Inserting this inequality into (3.6), one obtainskuk`<sup>p</sup>(I)≤ kuk`^α(I)and hence (3.4) follows.

The next two propositions contain inverses of inequality (3.4). These can only be obtained at a cost of either anN-dependent factor (Proposition 3.5) or a weighting by a negative power of h (Proposition 3.6). The exponent ^p_pp⁰⁻0^p appearing below should be understood as 1/p,ifp⁰=∞.

Proposition 3.5. Let 0< p≤p⁰ ≤ ∞. Then, ifi= 0 or1, (3.7) kuk`<sup>p</sup>(Ni)≤N<sup>p0−ppp0</sup> kuk`^p0(Ni). Proof. Recalling (2.9), we have

kuk^p<sub>`</sub>p(Ni)= (1,|u|<sup>p</sup>)`²(Ni)≤ k1k

`

p0 p0−p(Ni)

k|u|^pk

`

p0 p(Ni)

=N^p0−p0pkuk^p_`p0(Ni). Proposition 3.6. Fori= 0,1,

kuk`<sup>p</sup>(Ni).kh<sup>−d(p0−p)pp0</sup> uk`^p0(Ni), uniformly inu∈R^Ni andp0≤p≤p⁰≤ ∞.

(3.8)

Proof. We give the proof for i = 0. The case i = 1 is analogous. Take any u∈R^N0 and define u∈ S0(T) by requiringu(cj) =uj, j∈ N0. Then, by Propo- sition 3.8 below, we havekukL^p(Ω) .kukL^p0(Ω). Using Proposition 3.1, we obtain kh^d/puk`<sup>p</sup>(N0).kh<sup>d/p0</sup>uk`^p0(N0).Then (3.8) follows by replacingubyh^−d/pu.

From this we have the immediate corollary:

Corollary 3.7.

kh^αuk`<sup>p</sup>(Ni).kh<sup>α−d(p0−p)pp0</sup> uk`^p0(Ni),

uniformly inu∈R^Ni, α∈R andp0≤p≤p⁰≤ ∞. 3.3. Estimates between different L^p norms. The following result is obtained directly from H¨older’s inequality.

Proposition 3.8.

(3.9) kukL^p(Ω).kukL^p0(Ω), uniformly in p0≤p≤p⁰≤ ∞andu∈L^p0(Ω).

In the following generalisation of Proposition 3.8, we balance powers ofhinside the right-hand norm with an appropriate power of the global parameterN outside.

Proposition 3.9. Fori= 0,1 andα < α the estimate kh^αukL^p(Ω).N^θkh^α+dθukL^p0(Ω)

holds uniformly in u∈ Si(T), p0≤p≤p⁰ ≤ ∞,α∈[α, α] and 0≤θ≤^p_pp⁰⁻0^p. Proof. We give the proof fori= 0. The casei= 1 is very similar.

(a) Letu∈ S0(T). Then using (3.7) withi= 0 and withureplaced byh^α+d/pu and then (3.1), we obtain the required result in the caseθ=^p_pp⁰⁻₀^p.

(b) More generally, consider 0≤θ < ^p_pp⁰⁻₀^p.Then we can choosep⁰⁰∈(p, p⁰] such thatθ= ^p_p⁰⁻00p^p000. Then by (3.9) and part (a), we havekh^αukL^p(Ω).kh^αukL^p00(Ω)≤

N^θkh^α+dθukL^p0(Ω), as required.

Finally we obtain an inverse to the inequality in Proposition 3.9. As in Propo- sition 3.6, we pay the penalty of a negative power ofhon the right-hand side.

Proposition 3.10. Fori= 0,1 the estimates kh^αukL^p0(Ω).

h^{α−d(p0−p)pp0} u L^p(Ω)

hold uniformly in u∈ Si(T), α∈[α, α]andp0≤p≤p⁰≤ ∞.

Proof. Combine (3.1) and (3.4).

4. Inverse estimates in Sobolev norms

In this section we prove two types of inverse estimate in Sobolev norms. The first two subsections concern upper bounds for theW^s,pnorm of a function (s >0) in Si(T), i= 0,1, in terms of theL^p norm of an appropriately weighted function.

The range ofsis naturally restricted by the regularity of the spacesSi(T). In the case ofS1(T) our results are a generalisation of those already given in [18].

In this section we restrict to inverse estimates in Sobolev norms, wherep≥1.A more general result in Besov spaces—which also allowsp <1 —is given in Theorem A.1, and includes Theorems 4.1 and 4.2 as special cases. However, since the inverse estimates in Sobolev spaces are those which are mostly used by the numerical analysis community, we include, in this section, Besov-free proofs of these in order to maximise the usefulness of the paper. With this motivation we also restrict to p= 2 in Theorem 4.2.

In the third subsection we obtain lower bounds for theW^−s,qnorm (fors >0) of a function inSi(T) in terms of theL^q norm of an appropriately weighted function.

These are obtained by direct estimation of negative norms. The range of negative

−swhich can be reached is unlimited and again the argument generalises that in [18], in which only the casesu∈ S1(T) and 0≤s≤1 were covered.

4.1. Inverse estimates for u∈ S1(T) in W^s,p(Ω), s≥0.

Theorem 4.1. Suppose that 1 ≤ p ≤ ∞ and that 0 ≤ s < 1 + 1/p. Then the estimate

(4.1) kh^αukW^s,p(Ω).h^α−su

L^p(Ω)

holds uniformly in u∈ S1(T),α∈[α, α]and1≤p≤ ∞.

Proof. Throughout this proof the inequality.will be uniform in u∈ S1(T),α∈ [α, α] and 1≤p≤ ∞. We give the proof only for 0≤s≤1 here, since it can be based on elementary arguments. The proof for the missing range of s is given in the proof of Theorem A.1 of the Appendix.

Suppose s = 1 and τ ∈ T. The product rule yields ∇(h^αu) = αh^α−1u∇h+ h^α∇u on τ. Since Proposition 2.5(e) shows that |(∇h)|τ| . 1, it follows that αh^α−1u∇h

L^p(τ) . h^α−1u

L^p(τ). Moreover a simple scaling argument shows thatkh^α∇ukL^p(τ).h⁻_τ¹kh^αukL^p(τ).h^α−1u

L^p(τ). Summing thepth powers of these inequalities over allτ∈ T and taking thepth root, we obtain

(4.2) kh^αukW^1,p(Ω).h^α−1u

L^p(Ω).

Interpolating this result with the trivial estimatekh^αukL^p(Ω).kh^αukL^p(Ω)(for s= 0), we obtain (4.1) for generals∈[0,1]. (Note that here we have used the fact that the norm interpolating kh^α−1ukL^p(Ω) and kh^αukL^p(Ω) is kh^α−sukL^p(Ω) (see

Triebel [23, (1.15.2/4)]).

4.2. Inverse estimates for u∈ S0(T) in W^s,p(Ω), s >0. Analogously to Theo- rem 4.1, we have the following estimate for piecewise constant functions.

Theorem 4.2. Suppose that 1≤p≤ ∞and 0≤s <1/p. Then (4.3) kh^αukW^s,p(Ω) . kh^α−sukL^p(Ω), uniformly inu∈ S0(T),α∈[α, α], and1≤p≤ ∞.

It is a corollary of Theorem 4.2 that S0(T) ⊂ W^s,p(Ω) for all p and s in the ranges specified in the assumptions. Thus S0(T) ⊂W^1/2−ε,2(Ω) for ε > 0. The stated range ofsis maximal, since, as is well known,S0(T)6⊂W^1/2,2(Ω).

We give the proof forp= 2 only since it can be based on the localisation of the Slobodeckij norm given in the following lemma.

Lemma 4.3. Let Ω⊆R^d be a bounded domain and let T be any conforming mesh on Ω. Any function v∈H^s(Ω), s∈(0,1), satisfies

(4.4) kvk²H^s(Ω) ≤ X

τ∈T

h

c δ_τ^−2skvk²L²(τ) + X

τ0∩τ6=∅τ0∈T

Z

τ

Z

τ⁰

|v(x)−v(y)|²

|x−y|^d+2s dx dy i

,

where δτ := dist(τ, Dτ) and Dτ := S

{τ⁰ ∈ T : τ⁰ ∩τ = ∅}. For a domain Ω ⊂ R^d, the constant c is explicitly given by c = 1 + ⁴_s for d = 1 and by c= 1 + ^4π_s for d∈ {2,3}, provided that δτ ≤1. The constant c is more involved for a d-dimensional surface Ω, but still independent of v,T, K, ε.

The proof of Lemma 4.3 can be found in [16] ford= 1 and in [17] ford= 2. The 3-dimensional case can be shown analogously to the 2-dimensional case. Note that (4.4) holds for any conforming triangulation and the requirement thatT ∈ MK,ε

is not needed for Lemma 4.3.

Proof of Theorem 4.2 (restricted case). As mentioned above, we restrict to the case p= 2. In addition we describe here only the cased= 2. The proofs ford= 1 and d= 3 are similar. So, let u∈ S0(T). Since we are considering meshes from the classMK,ε, (4.4) implies that

(4.5)

kh^αuk²H^s(Ω) . kh^α−suk²L²(Ω) + X

j,j0 ∈N0 τj0 ∩τj6=∅

Z

τ_j

Z

τ_j0

|(h^αu)(x)−(h^αu)(y)|²

|x−y|^2+2s dxdy.

Observe that by elementary arguments|(h^αu) (y)|.h^α_τj|u(cj)|, fory ∈τj. Using this in (4.5), we obtain

kh^αuk²H^s(Ω) . kh^α−suk²L²(Ω) + X

j,j0 ∈N0,j6=j0 τj0 ∩τj6=∅

n

h^2α_τj0|u(cj0)|²+h^2α_τj|u(cj)|²o Jτ_j,τ_j0

(4.6)

+ X

j∈N0

|u(cj)|²Hτj

with (4.7)

Jτ,τ⁰:=

Z

τ

Z

τ⁰

|x−y|^−2−2sdx dy and Hτ :=

Z

τ

Z

τ

|h^α(x)−h^α(y)|²

|x−y|^2+2s dxdy.

Since the summand in (4.6) is symmetric with respect toj, j⁰, we may write (4.8)

kh^αuk²H^s(Ω) . kh^α−suk²L²(Ω)+ X

j∈N0

h^2α_τj|u(cj)|² X

j0 ∈N0,j6=j0 τj0 ∩τj6=∅

Jτ_j,τ_j0+X

j∈N0

|u(cj)|²Hτ_j.

We begin with the estimate

Hτ ≤ k∇(h^α)k²L^∞(Ω)

Z

τ

Z

τ

|x−y|^−2sdxdy.

Now, using polar coordinates with respect toy∈τ, it follows thatR

τ|x−y|^−2sdx. h²_τ^−2s. This, together with k∇(h^α)kL^∞(τ).h^α_τ⁻¹ yields Hτ .h^2(ατ ^−s)|τ| and the