Weighted Sobolev spaces and regularity for polyhedral domains

Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659 www.elsevier.com/locate/cma

Weighted Sobolev spaces and regularity for polyhedral domains Bernd Ammann a, Victor Nistor b

b,*

a Institut E´lie Cartan, Universite´ Henri Poincare´ Nancy 1, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France Pennsylvania State University, Mathematics Department, University Park, 305 McAllister Bldg., PA 16802, USA

Received 1 November 2005; accepted 30 October 2006 Available online 14 March 2007 Dedicated to Ivo Babusˇka on the occasion of his 80th birthday.

Abstract We prove a regularity result for the Poisson problem Du ¼ f , ujoP ¼ g on a polyhedral domain P  R3 using the Babusˇka–Kondratiev spaces Kma ðPÞ. These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of Kma —regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a ‘‘trace theorem’’ for the restriction to the boundary of the functions in Kma ðPÞ. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Poisson equation; Laplace operator; Well posedness on polyhedral domains; Regularity of solutions; Weighted Sobolev space; Trace theorem

1. Introduction Let X  Rn be a smooth, bounded domain. Then it is well known [6,16,26,47,53] that the equation Du ¼ f 2 H m1 ðXÞ;

u¼0

on oX;

ð1Þ

has a unique solution u 2 H mþ1 ðXÞ. In particular, u will be smooth on X if f is smooth on X. This well-posedness result is especially useful in practice for the numerical approximation of the solution u of Eq. (1), see for example [6,12,16] among many possible references. In practice, however, it is rarely the case that X is smooth. In fact, if oX is not smooth, then the smoothness of f on X does not imply that the solution u of Eq. (1) is also smooth on X. Therefore, there is a loss of regularity for elliptic problems on non-smooth domains. Wahlbin [55] (see also [5,35,56]) has shown that this leads to some inconvenience in numerical applications, namely that a quasi-uniform sequence of triangulations on X will not lead *

Corresponding author. Tel.: +1 814 865 7527; fax: +1 814 865 3735. E-mail addresses: [email protected] (B. Ammann), nistor@ math.psu.edu (V. Nistor). URL: http://www.differentialgeometrie.de/ammann (B. Ammann). 0045-7825/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2006.10.022

to optimal rates of convergence for the Galerkin approximations uh of the solution of (1). The loss of regularity can be avoided, however, if one removes the singular points by ‘‘sending them to infinity’’ by suitably changing the metric with a conformal factor. It can be proved then that the resulting Sobolev spaces are the ‘‘Sobolev spaces with weights’’ considered for instance in [6,11,12,33] and in several other papers. A related construction, leading however to countably normed spaces, was considered in [29]. Let f > 0 be a smooth function on a domain X, then the mth Sobolev space with weight f is defined by Kma ðX; f Þ :¼ fu; f jaja oa u 2 L2 ðXÞ; jaj 6 mg; m 2 Zþ ; a 2 R:

ð2Þ

The regularity result for Eq. (1) extends to polyhedral domains P in three dimensions with the usual Sobolev spaces replaced by the spaces Kma ðPÞ :¼ Kma ðP; #Þ, # being the distance to the edges. The spaces Kma ðoP; #Þ on the boundary are defined similarly for m 2 Zþ :¼ f0; 1; . . .g; for m 2 Rþ they are defined using interpolation. Theorem 1.1. Let P  R3 be a polyhedral domain. Let m 2 Zþ and a 2 R. Assume that u 2 K1aþ1 ðPÞ,

B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659 mþ1=2

Du 2 Km1 and ujoP 2 Kaþ1=2 ðoP; #Þ, then u 2 a1 ðPÞ, Kmþ1 aþ1 ðPÞ and there exists C > 0 independent of u such that  kukKmþ1 ðPÞ 6 C kDukKm1 ðPÞ þ kukK0 aþ1

a1

aþ1

 : þ kuj k mþ1=2 ðPÞ oP K ðoP;#Þ aþ1=2

The same result holds if we replace D with a strongly elliptic operator or system. Theorem 1.1 is well known in two dimensions, i.e., for polygonal domains, and for domains with conical points [15,33]. See also [23,34,38] where similar results were proved using a dyadic partition of unity technique. For the result in two dimensions, # is the distance to the vertices of the polygonal domain considered or to the conical points. In general, in d dimensions, one takes #ðxÞ to be the distance to the set of non-smooth boundary points of P. Significantly less papers have dealt with the case of three dimensions. Nevertheless, let us mention the following. A general and far reaching theory (valid also in higher dimensions) was developed by Dauge in [24]. Regularity estimates based on singular function expansions were proved by Apel and Nicaise [2] and Lubuma and Nicaise [36]. These results were then applied in these papers in order to obtain optimal rates of convergence in the Finite Element Method. In [39], Mazya and Rossmann have obtained similar results using estimates on Green functions. Buffa, Costabel, and Dauge [19] have proved or stated similar regularity and well-posedness results for polyhedral dimensions in three dimensions. Our modified weight rX was introduced in [22], where the above regularity theorem was proved for m ¼ 1. In [32], Kellogg and Osborn have obtained regularity results of a similar kind for the Stokes operator. Borsuk and Kondratiev established many regularity results for Dini–Liapunov regions in Rn , n P 3, in their recent monograph [37]. Note that the notion of a Dini–Liaponov region is a generalisation of a domain with C 1;a -boundary. See also [3,21,25,31,40,48,49], to mention just a few other papers. A regularity result valid in all dimensions was obtained in [1] using ‘‘Lie manifolds’’. We are grateful to one of the referees, who pointed out to us that Theorem 1.1 can also be obtained from the results of the monograph [46]. In this paper, we follow [1], but we use more elementary methods that lead to a short proof. We also introduce some ideas that are specific to polyhedral domains in three dimensions and may be useful in applications to Numerical Analysis. Moreover, our paper is self-contained and the references to [1] are only for comparison. We would like to stress that Theorem 1.1 does not constitute a Fredholm (or ‘‘normal solvability’’) result, because 0 the inclusion Kmþ1 aþ1 ðPÞ ! Kaþ1 ðPÞ is not compact for all m and a [1]. By contrast, if P is a polygon, then P ¼ D with Dirichlet boundary conditions is a Fredholm operator mþ1 m1 from Kaþ1 ðPÞ to Ka1 ðPÞ precisely when a is different from kp=a, where k 2 Z, k 6¼ 0, and a ranges through the angles of the polygon [33,34].

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The Poincare´ inequality kukK1 ðPÞ 6 CkrukL2 ðXÞ proved 1 in [14], gives that D is coercive on the space K11 ðPÞ and hence the map D : K11 ðPÞ \ fu ¼ 0 on oPg ! K1 1 ðPÞ is a continuous bijection. By combining this with Theorem 1.1 we obtain that m1 D : Kmþ1 aþ1 ðPÞ \ fu ¼ 0 on oPg ! Ka1 ðPÞ

ð3Þ

is a continuous bijection, for any m 2 Zþ and jaj < g, with g depending only on P. The same result holds if D is replaced with P þ cP , where P is a strongly elliptic system with smooth coefficients and cP > 0 and g > 0 are constants depending only on P [14]. To prove Theorem 1.1, we first introduce the weighted Sobolev spaces Kma ðoP; #Þ on the boundary of P. For m 62 Zþ , these spaces are defined by duality and interpolation. Then we provide an alternative definition of the spaces Kma ðPÞ :¼ Kma ðP; #Þ and Kma ðoP; #Þ using partitions of unity. This allows us to define a trace map m1=2 Kma ðPÞ ! Ka1=2 ðoP; #Þ, which extends the restriction map and is a continuous surjection, as in the case of a smooth domain. We also show that any differential operator P of order m with smooth coefficients induces a continuous map P : Ksa ðPÞ ! Ksm am ðPÞ. We need to introduce an enhanced space of smooth, bounded functions C1 ðRPÞ, which contains the cylindrical and spherical coordinates functions and is minimal with this property. In particular, C1 ðXÞ  C1 ðRPÞ  C1 ðXÞ. Let qP ðpÞ be the distance from p to the vertex P of P and re ðpÞ be the distance from p to the line determined by the edge e of P (for P non-convex we need to slightly change the definition of re). Then qP ; qe 2 C1 ðRPÞ, although they are not smooth functions on X in the usual sense. Let A and B be the end vertices of the edge e (i.e., ½AB). Q e ¼Q 1 We further define ~re :¼ q1 re  P qP . A qB r e and rP ¼ e~ Then ~re ; rP 2 C1 ðRPÞ. The functions in C1 ðRPÞ have the following strong boundedness property i

j

k

ðrP ox Þ ðrP oy Þ ðrP oz Þ u 2 C1 ðRPÞ  L1 ðPÞ 1

ð4Þ 1

for all u 2 C ðRPÞ. The consideration of C ðRPÞ and of the derivatives of the form rP ox , rP oy , and rP oz is a substitute for the results on Lie manifolds used in [1]. However, the results of [1] also apply to non-compact manifolds and to a larger class of singular domains. The methods of this paper are used for a general regularity and well-posedness result for anisotropic elasticity in general polyhedral domains (including cracks) in [43]. We do not include in this paper any concrete applications, but let us refer the reader to [2,3,9,18,19,22], where concrete applications of results similar to ours were provided. 2. Smooth functions and differential operators on P In this section, we shall introduce the space C1 ðRPÞ  C1 ðPÞ \ L1 ðPÞ and relate it to the differentials rP ox , rP oy , and rP oz mentioned in Section 1. Similar vector fields have appeared also in [17]. When only edges are

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B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659

involved (i.e., no vertices), the use of these vector fields goes back to [41,42]. See also [44,50]. 2.1. Polygons and polyhedral domains Let us fix some terminology to be used in what follows. A polygon P0 in a two-dimensional Euclidean space is an open, connected subset whose boundary consists of finitely many straight segments (possibly of infinite length) called sides and having at most the end points in common. For simplicity, we assume that oP0 ¼ oP0 , which means that no point of the boundary oP0 is in the interior of P0 (thus cracks are excluded). The points common to more than one straight segment of the boundary are called the vertices of P0 . We require that each vertex belongs to exactly two sides. We do not require the boundary of P0 to be connected. For simplicity, in this paper we also assume that the sides are maximally extended, so that they are not contained in larger segments contained in the boundary. This assumption is however not essential. Similarly, a polyhedral domain P  R3 is a connected, SN open subset whose boundary satisfies oP ¼ oP ¼ j¼1 Dj and (i) each Dj is a polygon contained in an affine 2-dimensional subspace of R3 ; (ii) the sets Dj are disjoint; (iii) a side of Dj is a side of exactly one other Dk . The vertices of P are the vertices of the polygonal domains Dj. The edges of P are the sides of the polygonal domains Dj. Hence, an edge belongs to exactly two faces of P. For each vertex P of P, we choose a small open ball VP centered in P. We assume that the neighborhoods VP are chosen to be disjoint. We stress that, in our convention, both the polygons and the polyhedra are open subsets. We do not require these sets to be bounded in general, although this assumption is needed for some of our results.

recall that we have denoted by re ðpÞ the distance from p to the line determined by the edge e of P and by Y Y 1 ~re  qP ; where ~re :¼ q1 for e ¼ ½AB: rP :¼ A qB r e e

P

ð5Þ In the above formula, the products are taken over all vertices P and all edges e of P. The notation e ¼ ½AB means that e is the edge joining the vertices A and B. If e ¼ ½A; 1Þ, that is, if e is a half-line, then ~re :¼ q1 A re . Finally, if e is infinite in both directions (i.e., for a dihedral angle), we let ~re :¼ re . Choose for each edge e a plane Pe containing one of the faces Dj of P such that e  Dj . If x is not on the line defined by e, we define he to be the angle in a cylindrical coordinates system ðre ; he ; zÞ determined by the edge e and the plane Pe . More precisely, let q 2 e be the foot of the perpendicular from p to e. Then he ðpÞ is the angle between pq and Pe . Similarly, for each vertex P and edge e adjacent to P, we define /P ;e ðpÞ to be the angle between the segment pP and the edge e (except for p ¼ P , in which case /P ;e ðpÞ is not defined). If P is convex, then the functions he and /P ;e are defined and smooth on P (recall that P is an open subset). They will be part of the spherical coordinate system ðqP ; he ; /P ;e Þ centered at P. For P non-convex, this property will be enjoyed by the modified functions he and /P ;e introduced in Appendix. All the following definitions and constructions below are the same in the case of a non-convex domain, but using the modified h and / variables. We shall denote by h ¼ ðhe1 ; . . . ; her Þ the vector variable that puts together all the he functions, for e ranging through the set of all edges fe1 ; . . . ; er g. Similarly, let f/1 ; . . . ; /p g list all the functions /P ;e , for all vertices P and all edges e containing P we shall denote by / ¼ ð/1 ; . . . ; /p Þ the vector variable that puts together all the /P ;e functions. We then introduce the space W k;1 ðRPÞ as the space of functions u : P ! C of the form: uðx; y; zÞ ¼ f ðx; y; z; h; /Þ ¼ f ðx; y; z; he1 ; . . . ; her ; /1 ; . . . ; /p Þ; r

p

f 2 W k;1 ðP  ð0; 2pÞ  ð0; pÞ Þ: 2.2. Useful functions and other notation Assume, for the definition of re, he , and /P ;e in this subsection, that P is convex. If P is not convex, then we slightly change the definitions of these functions such that the new functions retain their behaviour around e, but will become smooth everywhere in space except on e [1]. The modified functions /P ;e and he will then be defined and smooth on P. We postpone the technical construction of the modified functions /P ;e and he for Appendix, in order not to interrupt the flow of the presentation. (Let us stress, however, that none of our results requires the assumption that P be convex.) Let us first recall from Section 1 that we have denoted by qP ðpÞ the distance from p to the vertex P of P. Also,

Thus f above T has k bounded weak derivatives. We let C1 ðRPÞ :¼ k W k;1 ðRPÞ. The point of this definition is that, for example, he is a smooth function on P that is not in W k;1 ðPÞ for k P 1. On the other hand he 2 C1 ðRPÞ, by definition. One can show as in [1,17] that there exists a canonical Riemannian manifold RðPÞ such that C1 ðRðPÞÞ ¼ C1 ðRPÞ, so our notation is justified. The construction of a space with this property is not very intuitive. However, at this point, we do not assign any significance to RP, which should be regarded in this paper just as a symbol. (Let us mention however, that, had we used curved boundaries, then the desingularizations RDj of the faces would have been necessary. See [43].)

B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659

2.3. Vector fields and C1 ðRPÞ We now establish several technical properties of the functions in C1 ðRPÞ, especially in relation to the vector fields (differentials) rP ox , rP oy , and rP oz . Let us notice first that it follows right away from the definition that C1 ðRPÞ is closed under addition and multiplication. Lemma 2.1. Let P be a vertex of P, then qP 2 C1 ðRPÞ. Similarly, let e ¼ ½AB be the edge of P joining the vertices A 1 1 1 and B, Q thenQ ~re :¼ qA1qB re 2 C ðRPÞ. In particular, rP :¼ e~re  P qP 2 C ðRPÞ. This is proved using polar coordinates. Assume P 1 belongs to the edge e, then qP ¼ ðsin /P ;e cos he Þ x, where this is defined (x stands for the first component variable). Similar formulas for qP in terms of y and z then combine, using a partition of unity on R3 n fP g with functions in C1 ðRPÞ, to define qP globally as an element in C1 ðRPÞ. Similarly, ~re ¼ qA sin /A;e , so ~re =qA is ‘‘smooth’’ near A. The same argument, together with a partition of unity, shows that re 2 C1 ðRPÞ. Our result then follows from the fact that C1 ðRPÞ is closed under products, by definition. Lemma 2.2. Let #ðpÞ be the distance from p to the union of the edges of P. Then there exists C > 0 such that C 1 #ðpÞ 6 rP 6 C#ðpÞ for all p 2 P. This lemma is proved using the homogeneity properties of the functions # and rP close to the vertices and edges of P. Using a compactness argument, it is enough to prove that the ratio rP =# is bounded and bounded away from zero in the neighborhood of each point. This allows us to assume that P is either a dihedral angle or an infinite cone. If P is the dihedral angle 0 < h < a, with a fixed, then rP =# ¼ 1. If P is a cone with center the origin, let at be the dilation with center the origin and ratio t. Then rP ðat ðpÞÞ ¼ trP ðpÞ and #ðat ðpÞÞ ¼ t#ðpÞ. This shows that the ratio rP ðpÞ=#ðpÞ depends only on p=jpj. Furthermore, rP is a continuous function on the compact set P \ S n1 , and the lemma follows from this. Lemma 2.3. We have that the functions re ox he , re oy he , re oz he , qP ox /P ;e , qP oy /P ;e , qP oz /P ;e , ox re , oy re , oz re , ox qP , oy qP , and oz qP are all in C1 ðRPÞ. To prove this, let us notice first that we can use any linear system of coordinates (x, y, z). In particular, for each of the above calculations, we can assume that our cylindrical or spherical coordinate system is aligned to the coordinate system (x, y, z). Then the result is simply an exercise in the calculation of the partial derivatives of the cylindrical coordinates h and r and of the spherical coordinates / and q. Corollary 2.4. We have ox rP ; oy rP ; oz rP 2 C1 ðRPÞ. Proof. Let us concentrate on ox . We use the product rule to compute the derivative of rP . A summand containing ox qP

3653

is in C1 ðRPÞ by Lemma 2.3. Let e ¼ ½AB. The other prod1 ucts are obtained by replacing ~re :¼ q1 A qB re with 1 1 re  q1 re : ox ð~re Þ ¼ q1 A qB ox ðre Þ  qA ox ðqA Þ~ B ox ðqB Þ~ 1 The factors of q1 A and qB then cancel out in the product defining rP and all the remaining factors are in C1 ðRPÞ by Lemma 2.3. h

Proposition 2.5. If u 2 C1 ðRPÞ, then the functions rP ox u, rP oy u, and rP oz u are in C1 ðRPÞ. Proof. This follows from re C1 ðRPÞ \ qP C1 ðRPÞ. h

Lemma

2.3

and

rP 2

Let us denote by Diff m0 ðPÞ the differential operators of order m on P linearly generated by differential operators of the form a

a

a

a

uðrP oÞ :¼ uðrP ox Þ 1 ðrP oy Þ 2 ðrP oz Þ 3 ; jaj :¼ a1 þ a2 þ a3 6 m; u 2 C1 ðRPÞ: m 1 We agree that S Diff 0mðPÞ :¼ C ðRPÞ and we shall denote 1 Diff 0 ðPÞ :¼ m Diff 0 ðPÞ. In case of edges (no vertices), similar algebras were considered also by Mazzeo [41,42]. Algebras more closely related to ours appear in [44]. To get more insight into the structure of Diff 1 0 ðPÞ, we shall need two simple calculations that we formalize in the following lemma, whose proof is based on the fact that oj rP 2 C1 ðRPÞ.

Lemma 2.6. Let k 2 R and let oj and ok stand for either of 1 k ox , oy , or oz . Then rk P ðr P oj ÞrP  r P oj ¼ koj ðr P Þ 2 C ðRPÞ; and ½rP oj ; rP ok  :¼ ðrP oj ÞðrP ok Þ  ðrP ok ÞðrP oj Þ ¼ oj ðrP ÞrP ok  ok ðrP ÞrP oj 2 Diff 10 ðPÞ: Then we have the following simple but basic result. kþm Proposition 2.7. We have Diff k0 ðPÞDiff m ðPÞ 0 ðPÞ  Diff 0 1 and hence Diff 0 ðPÞ is an algebra.

Proof. We shall prove by induction on k þ m that Diff k0 ðPÞDiff m0 ðPÞ  Diff kþm ðPÞ. Indeed, if k þ m ¼ 0, then 0 k ¼ m ¼ 0 and the statement is clearly true because C1 ðRPÞ is closed under products. Let us assume then that k þ m > 0. We need to show that uðrP oÞa vðrP oÞb 2 Diff kþm ðPÞ if u; v 2 C1 ðRPÞ and jaj :¼ a1 þ a2 þ a3 ¼ k, 0 jbj :¼ b1 þ b2 þ b3 ¼ m, where a ¼ ða1 ; a2 ; a3 Þ and b ¼ ðb1 ; b2 ; b3 Þ. If m ¼ 0, then the relation X a a0 a00 uðrP oÞ v ¼ uðrP oÞ ½rP oj ðvÞðrP oÞ for suitable a0 ; a00 with ja0 j þ ja00 j ¼ k  1, together with the induction hypothesis and with Proposition 2.5, shows that a uðrP oÞ v 2 Diff k0 ðPÞ. Let now m be arbitrary. We shall proceed by a second induction on m. The same argument as in the paragraph above allows us to assume that v ¼ 1. We can also assume that the monomial ðrP oÞa ðrP oÞb is already ordered in the

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B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659

standard way. Then, using Lemma 2.6, we commute rP oj , the last derivative in ðrP oÞa , with rP ok , the first derivative in ðrP oÞb . Induction on k þ m for the terms containing oj ðrP ÞrP ok and ok ðrP ÞrP oj and induction on m or k þ m for the term containing ðrP ok ÞðrP oj Þ then complete the kþm proof of the fact that Diff k0 ðPÞDiff m ðPÞ. h 0 ðPÞ  Diff 0 The above proposition gives the following useful corollary. Corollary 2.8. If P is a differential operator of order m with m smooth coefficients, then rm P P 2 Diff 0 ðPÞ. Proof. It is enough to show that rmP oa 2 Diff m0 ðPÞ if oa ¼ oax 1 oay 2 oaz 3 with jaj ¼ m. We shall again proceed by induction on m. The case m ¼ 1 is obvious. Let oj be the 0 first derivative in oa , so that oa ¼ oj oa . Then Lemma 2.6 and Corollary 2.4 give 0

a rmP oa  ðrP oj Þðrm1 P o Þ 0

¼ ðm  1Þoj ðrP ÞðrPm1 oa Þ 2 Diff 0m1 ðPÞ: Then Proposition 2.7 shows that Diff 10 ðPÞDiff 0m1 ðPÞ  Diff m0 ðPÞ. This and the induction hypothesis allows us to complete the proof. h

Eq. (6) and Lemma 2.2 then give immediately the following lemma. a 2 Lemma 3.1. We have Km a ðPÞ ¼ fu; # Pu 2 L ðPÞ; for all k m P 2 Diff 0 ðPÞg: A similar result holds for Ka ðoP; #Þ and for W k;p;a BK ðPÞ.

Next, Proposition 2.5 and Corollary 2.4, together with a straightforward calculation, show the following. Lemma 3.2. The multiplication map W m;1;b  Km a ðPÞ 3 BK m ðu; f Þ7!uf 2 Kaþb ðPÞ is continuous. We also have ðPÞ and rbP 2 W m;1;b ðPÞ, and hence the C1 ðRPÞ  W m;1;0 BK BK m map Ka ðPÞ 3 u7!rbP u 2 Km ðPÞ is a continuous isomoraþb phism of Banach spaces. From this lemma we obtain right away the following result. Proposition 3.3. Let k P m. Each P 0 2 Diff m 0 ðPÞ defines a continuous map P 0 : Kka ðPÞ ! Kkm ðPÞ. The family a k rk P r is a family of bounded operators Kka ðPÞ ! 0 P P km Ka ðPÞ depending continuously on k. Similarly, if P is a differential operator with smooth k coefficients on P, then rk P Pr P defines a continuous family of k bounded operators Ka ðPÞ ! Kkm am ðPÞ.

The proof of the above corollary also shows that rmP oax 1 oay 2 oaz 3

a

a

a

 ðrP ox Þ 1 ðrP oy Þ 2 ðrP oz Þ 3 2 Diff m1 ðPÞ; 0

jaj ¼ m:

ð6Þ

We define the spaces Kk a ðPÞ, k 2 Zþ , by duality. More

3. Function spaces on P



k precisely, let Kka ðPÞ be the closure of C1 c ðPÞ in Ka ðPÞ.

We now recall and study the Babusˇka–Kondratiev spaces Kma ðPÞ :¼ Kma ðP; #Þ and Kma ðoP; #Þ on a 3-dimensional polyhedral domain P and its boundary oP. These spaces are weighted Sobolev spaces with weight given by #, the distance to the set of edges of P, as in Eq. (2). Note that we can replace # with rP , by Lemma 2.2 (we shall use this below). 3.1. The Babusˇka–Kondratiev spaces We let W

k;p;a BK ðPÞ

Proof. The first part follows from Lemma 3.1. The second part follows from the first part of this proposition and Lemma 3.2. h

jaja a

¼ fu : P ! C; rP

o u 2 Lp ðPÞ

for all jaj 6 kg;

ð7Þ for k 2 Zþ ; a 2 R; p 2 ½1; 1. If p ¼ 2, we denote Kka ðPÞ :¼ W k;2;a BK ðPÞ, which coincides with the definition in Section 1 (Eq. (2)). We similarly define m;p;a W BK ðoPÞ ¼ fu : oP ! C; rPka P ðujDj Þ 2 Lp ðDj Þ for all k 6 m and all differential operators P of order k on Dj ; k 6 mg; m 2 Zþ : m;2;a We let Kka ðoP; #Þ :¼ W BK ðoPÞ. Thus, Kka ðoP; #Þ ’ k Ka ðDj ; #Þ is a direct sum of weighted Sobolev spaces. Note that we require no compatibility conditions for the resulting functions on the faces Dj.



k Then we define Kk a ðPÞ to be the dual of Ka ðPÞ, the duality pairing being an extension of the bilinear form R ðu; vÞ7! P uv dvol. With this definition, we can drop the requirement that k P m in Proposition 3.3. Let us also note that the resulting weighted Sobolev spaces on the polygons Dj are different from the weighted Sobolev spaces obtained by using the distance to the vertices of these polygons. A regularity theorem on Dj would involve the latter weight (as in Kondratiev’s theorem [33] mentioned in Section 1). A consequence of this is that the spaces Kka ðoP; #Þ behave more like the Sobolev spaces defined on a smooth manifold without boundary than like the Sobolev spaces defined on a bounded domain with smooth boundary. In particular, we define Kk a ðoP; #Þ as the dual of Kka ðoP; #Þ. The spaces Ksa ðoPÞ, s 62 Z, can be defined by interpolation, although in this paper we shall use a different definition using partitions of unity (see the following subsection; the two definitions are equivalent, although we shall not need a proof of this fact in this article).

3.2. Definition of Sobolev spaces using partitions of unity As in [1], it is important to define the spaces Kam ðPÞ using partitions of unity. Similar constructions were used in [20,51,52,54]. This construction is possible because the

B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659

spaces Km3=2 ðPÞ are the Sobolev spaces associated to the metric r2 P g E , where gE is the usual Euclidean metric. We shall need the following lemma. Recall that #ðpÞ denotes the distance from p to the edges of P. In view of Lemma 2.2, in all estimates involving #, we can replace # with rP , although not the other way around, because # is not smooth. Let osing P be the union of the edges of P and P0 :¼ P n osing P. Lemma 3.4. There is 0 2 ð0; 1Þ, an integer j, and a sequence C m > 0 of constants such that, for any  2 ð0; 0 , there is a sequence of points fxj g  P0 :¼ P n osing P and a partition of unity /j 2 C1c ðP0 Þ with the following properties: (i) either Bðxj ; #ðxj Þ=4Þ is contained in P or xj 2 oP, #ðxj Þ > 0, and the ball Bðxj ; #ðxj ÞÞ intersects only the face Di to which xj belongs; (ii) suppð/j Þ  Bðxj ; #ðxj Þ=2Þ if xj 2 oP and suppð/j Þ  Bðxj ; #ðxj Þ=8Þ otherwise; a (iii) /j ðxj Þ ¼ 1 and kðrP oÞ /j kL1 ðPÞ 6 C jaj jaj ; and (iv) a point x 2 P can belong to at most j of the sets Bðxj ; #ðxj ÞÞ. Let us notice that Bðxj ; #ðxj ÞÞ does not intersect any edge of P because  < 1. Moreover, the conditions that krP r/j kL1 6 C 1 1 and /j ðxj Þ ¼ 1 guarantee that the support of /j is comparable in size with #ðxj Þ. This is reminiscent of the conditions appearing in the definition of the Generalized Finite Element spaces [7,8,10]. A proof of this lemma will be given in Appendix. It is essentially a result that, in the case of non-compact manifolds, goes back to Aubin. It was subsequently used by Gromov and in [1,51,52,54]. We shall fix  ¼ 0 in what follows and a sequence xj and a partition of unity /j as in the lemma. P Lemma 3.5. Let uk ¼ kj¼1 /j u, for u 2 Km a ðPÞ. Then uk ! u in Km ðPÞ. a Pk Proof. Let Uk :¼ j¼1 /j . We have that the sequence a ðrP oÞ Uk is bounded in the ‘sup’-norm and converges to 0 pointwise everywhere if a 6¼ 0. Similarly, Uk is bounded and converges to 1 pointwise everywhere. The result then follows from this using also the Lebesgue dominated convergence theorem. h Denote by aj ðxÞ ¼ xj þ #ðxj Þðx  xj Þ be the dilation of center xj and ratio #ðxj Þ. Let J be the set of indices j such that xj 2 oP. Below, by H m we shall mean either H m ðR3 Þ or H m ðR3þ Þ. Also, denote by X 2 32a 2 mm;a ðuÞ :¼ #ðxj Þ kð/j uÞ  aj kH m j

:¼

X

#ðxj Þ32a kð/j uÞ  aj k2H m ðR3 Þ

j62J

þ

X j2J

#ðxj Þ

32a

2

kð/j uÞ  aj kH m ðR3þ Þ :

ð8Þ

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We agree that kð/j uÞ  aj kH m ¼ 1 if ð/j uÞ  aj 62 H m ðR3 Þ (or if ð/j uÞ  aj 62 H m ðR3þ Þ, respectively). Note that the functions ð/j uÞ  aj will all have support contained in a fixed ball, namely, the ball Bð0; 0 =2Þ of radius 0 =2 and center the origin. Moreover, all derivatives oa ð/j  aj Þ are bounded for each fixed a and arbitrary j by Lemma 3.4. Proposition 3.6. We have u 2 Km a ðPÞ, m 2 Z, if, and only if, mm;a ðuÞ < 1. Moreover, mm;a ðuÞ defines an equivalent norm on Km a ðPÞ. The proof of this Proposition is standard [Lemma 2.4] (see [1,13] or [54]); for m < 0 one also has to check that both definitions are compatible with duality. We include a brief sketch below. Proof. Let us also introduce X 2 ~mm;a ðuÞ2 :¼ kð/j uÞkKma ðPÞ : j

Then the fact that #ðxÞ=#ðxj Þ and #ðxj Þ=#ðxÞ are bounded 1 by ð1  Þ on the ball Bðxj ; #ðxj ÞÞ, for  2 ð0; 1Þ and a change of variables shows that ~mm;a and mm;a define equivalent norms. It is then enough to prove that ~mm;a ðuÞ defines an equivalent norm on Kma ðPÞ. For m ¼ 0, this follows from the inequalities 2

2

kra m20;a 6 jkra P ukL2 ðPÞ 6 j~ P ukL2 ðPÞ : For m, we use induction on m and the fact that P arbitrary a h j jðrP oÞ /j ðpÞj is bounded uniformly in p 2 P for all a. We proceed in the same way to study the spaces Ksa ðoP; #Þ, s 2 R. Let us identify the plane containing each face Dk of P with a copy of R2 . Then let X 2 22a 2 ls;a ðuÞ :¼ #ðxj Þ kð/j uÞ  aj kH s ðR2 Þ ; s 2 Rþ : ð9Þ j2J

Note that only the indices j for which xj 2 oP are used above. Also, note that the power of #ðxj Þ was changed from 3  2a to 2  2a. Then we have an analogous description of the spaces Ksa ðoP; #Þ, s 2 Z. Proposition 3.7. We have u 2 Ksa ðoP; #Þ if, and only if, ls;a ðuÞ < 1. Moreover, ls;a ðuÞ defines an equivalent norm on Ksa ðoP; #Þ, s 2 Z. We can then define Ksa ðoP; #Þ, s 2 R, as the space of functions u for which ls;a ðuÞ < 1 with the induced norm. From this we obtain, by reducing to the Euclidean case, the following Trace Theorem. Let o sing P be the union of the edges of P and P0 :¼ P n osing P, as above. 0 Theorem 3.8 (Trace theorem). The space C1 c ðP Þ is dense m in Ka ðPÞ, m 2 Zþ . The restriction to the boundary extends m1=2 to a continuous, surjective map Km a ðPÞ ! Ka1=2 ðoP; #Þ for m P 1. For m ¼ 1, the kernel of this map is the closure of 1 C1 c ðPÞ in Ka ðPÞ. m 0 Proof. Clearly, C1 c ðP Þ  Ka ðPÞ, for any m 2 Zþ and any a 2 R. To prove that it is a dense subspace, let u 2 Kma ðPÞ.

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By Lemma 3.5, we may assume that the support of u does not intersect osing P (replace u with uk for some k large). Then we use the fact that C1 ðXÞ is dense in H m ðXÞ for X a smooth, bounded domain and the fact that the H m -norm is equivalent to the norm on Kma ðPÞ when restricted to functions with support in a fixed compact K such that K does not intersect any edge of P (i.e., K \ osing P ¼ ;). We have X lm1=2;a1=2 ðujoP Þ2 :¼ #ðxj Þ32a kð/j uÞ  aj joP k2H m1=2 ðR2 Þ j2J

6C

X

#ðxj Þ

32a

2

kð/j uÞ  aj kH m ðR3 Þ

j2J

6 Cmm;a ðuÞ; m1=2

and hence the restriction map Kma ðPÞ ! Ka1=2 ðoP; #Þ is defined and continuous for m P 1, by Propositions 3.6 and 3.7. To prove that this map is continuous, let us fix a continuous extension operator E : H m1=2 ðR2 Þ ! H m ðR3 Þ. By rotation and translation, we extend this definition to an extension operator E : H m1=2 ðV Þ ! H m ðR3 Þ, for any two dimensional subspace V  R3 . m1=2 Let then v : oP ! C be a function in Ka1=2 ðoP; #Þ. Let 3 us fix a function w 2 C1 c ðR Þ with support in the ball Bð0; 0 Þ of radius 0 and center at the origin such that w ¼ 1 on Bð0; 0 =2Þ. Let vj ðpÞ ¼ /j ðaj ðpÞÞuðaj ðpÞÞ, which is defined on a subspace of R3 of dimension 2. We define X ðwEðvj ÞÞ  a1 u¼ j :

The constant Cr in the two equations above depends only on 0 . (In fact, Eq. (10) implies Eq. (11), by taking v to be odd with respect to the reflection in the boundary of the half space R3þ .) We shall proceed by induction on m P 0. For m ¼ 0, the result is tautologically true, because of the term kukK0 ðPÞ aþ1 on the right-hand side of the regularity estimate of Theorem 1.1. Let now f/j g be the partition of unity and aj be dilations appearing in Eq. (8). In particular, the partition of unity /j satisfies the conditions of Lemma 3.4, which implies that suppð/j Þ  Bðxj ; 0 #ðxj Þ=2Þ if xj 2 oP and suppð/j Þ  Bðxj ; 0 #ðxj Þ=8Þ otherwise. We also have that all derivatives of order 6 k of the functions /j  aj are bounded. This implies in turn that the commutator P j :¼ ½D; /j  aj  :¼ Dð/j  aj Þ  ð/j  aj ÞD is a differential operator all of whose coefficients have bounded derivatives. Let kvkH m denote either kvkH m ðR3 Þ or kvkH m ðR3þ Þ , depending on where the function v is defined. Let gj ¼ w  a1 j , 3 where w 2 C1 ðR Þ has support in Bð0;  Þ and is equal to 1 0 c on Bð0; 0 =2Þ, as before. Then Eqs. (10) and (11) and the above remarks give X #ðxj Þ32a kð/j uÞ  aj k2H mþ2 mmþ2;a ðuÞ2 :¼ j

6 Cr

X

#ðxj Þ32a ðkD½ð/j uÞ  aj k2H m

j

þ kð/j uÞ  aj k2L2 Þ X 6C #ðxj Þ32a ðkð/j  aj ÞDðu  aj Þk2H m

j

Then u 2 Kma ðPÞ and ujoP ¼ v. Finally, let u 2 K1a ðPÞ such that ujoP ¼ 0. Let uk be as in Lemma 3.5. Then uk joP ¼ 0. Using again the equivalence of the H 1 and K1a ðPÞ—norms on functions with support in a fixed compact set K such that K \ osing P ¼ ;, we see that each uk can be approximately in K1a ðPÞ as well as we want 0 by a function vk 2 C1 Then we can take our approxc ðP PÞ. imation of u to be v ¼ Nk¼1 vk , for N large enough. h

j 2

2

þ kP j ðu  aj ÞkH m þ kð/j uÞ  aj kL2 Þ X 32a 4 2 6C #ðxj Þ ð#ðxj Þ kð/j DuÞ  aj ÞkH m j 2

2

þ kðgj uÞ  aj kH mþ1 þ kð/j uÞ  aj kL2 Þ X 2 2 2 6 Cðmm;a2 ðDuÞ þ mmþ1;a ðgj uÞ þ m0;a ðuÞ Þ: j

4. Proof of the regularity theorem We include in this section the proof of Theorem 1.1. Its proof is reduced to the Euclidean case using a partition of unity /j satisfying the conditions of Lemma 3.4 for  ¼ 0 , as in the previous section. Proof. (of Theorem 1.1.) The trace theorem, Theorem 3.8 allows us to assume that ujoP ¼ 0. We then notice that, locally, Theorem 1.1 is a well-known statement. Namely, let us consider a function v with support in the ball of radius 0 . We assume that either v 2 H 1 ðR3 Þ or v 2 H 10 ðR3þ Þ (that is, v ¼ 0 on R2 , the boundary of R3þ ¼ fz P 0g). Then there exists a constant C > 0 such that, for all m P 0, 2

2

2

kvkH mþ1 ðR3 Þ 6 C r ðkDvkH m1 ðR3 Þ þ kvkL2 ðR3 Þ Þ

ð10Þ

or, respectively, 2

2

2

kvkH mþ1 ðR3þ Þ 6 C r ðkDvkH m1 ðR3þ Þ þ kvkL2 ðR3þ Þ Þ:

ð11Þ

Since no more than j of the functions gj u are non-zero at a any given point of P and all the derivatives P ðrP @Þ gj 2are bounded for all fixed jaj, we obtain that j mmþ1;a ðgj uÞ 6 Cmmþ1;a ðuÞ2 . This then gives 2

2

2

mmþ2;a ðuÞ 6 Cðmm;a2 ðDuÞ þ mmþ1;a ðuÞ Þ: By induction on m we then obtain 2

2

2

mmþ2;a ðuÞ 6 Cðmm;a2 ðDuÞ þ m0;a ðuÞ Þ: The result then follows from Proposition 3.6, which states that the norms k  kKta ðPÞ and mt;a are equivalent. h See [14] for applications of these results, especially of the above theorem. By contrast, it is known that in the framework of the usual Sobolev spaces H m ðPÞ, the smoothness of the solution of (1) is limited [24,27,28,30,45].

B. Ammann, V. Nistor / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3650–3659

Acknowledgements We thank Ivo Babusˇka, Constantin Bacuta, Alexandru Ionescu, Robert Lauter, Anna Mazzucato, and Ludmil Zikatanov for useful discussions. The first named author wants to thank MSRI, Berkeley, CA, USA, for its hospital´ lie Cartan, ity, the last named author thanks Institut E Nancy, France, where part of the work has been completed. Appendix A. Additional constructions In this appendix, we explain how to modify the constructions of the functions he and /P ;e introduced in Section 2 when P is not convex and how to construct a partition of unity satisfying the conditions of Lemma 3.4. A.1. The modified functions h, /, and re We continue to denote by qP ðpÞ the distance from p to the vertex P. By a dilation, we can assume that each edge of P has length at least 4. Let us first modify the functions /A;e . We can find d > 0 small enough so that for any vertex P, the sets /P ;e > p  2d do not intersect (e ranges through the set of edges containing P). Let e ¼ ½AB and w1 : ½0; p ! ½0; 1  d be a smooth, non-decreasing function such that w1 ðxÞ ¼ x for 0 6 x 6 p  2d and w1 ðxÞ ¼ p  d for x P p  d. Also, let w2 : ½0; 1Þ ! ½0; 1 be a smooth, nonincreasing function such that w2 ðxÞ ¼ 1 for 0 6 x 6 1 and w2 ðxÞ ¼ 0 for 2 6 x. Then we replace /A;e with w1 ð/A;e Þw2 ðqA Þ. This modifies the function /A;e to make it smooth everywhere except on e. We now modify the functions he . They will be modified in two ways. Let us fix an edge e ¼ ½AB. To understand these modifications, it is useful to think of the spherical domain xA associated to the vertex A. The old function he served the purpose of both desingularizing xA close to the vertex associated to e and of providing global coordinates on xA away from the vertices (together with the functions /A;e ). These two purposes of the old he will be accomplished by two modified functions h. Let w3 : ½0; p ! ½0; 1 be a smooth, non-increasing function such that w3 ðxÞ ¼ 1 for x 2 ½0; a and w3 ðxÞ ¼ 0 for x P 2a. We then similarly modify he by replacing it with w3 ð/A;e Þw3 ð/B;e Þw2 ðre Þhe . This will make he defined and smooth everywhere in space except on e (if c is large enough). The resulting function he serves the purpose of desingularizing xA near the vertex corresponding to A. Let next w4 : ½0; 2p ! ½0; 2p be a smooth function such that w4 ðtÞ ¼ t for t 2 ½2; 1  2 and w4 ðtÞ ¼ 0 for t 2 ½0;  [ ½1  ; 1. The second kind of functions hA will be obtained by considering w4 ðhe Þw4 ð2/A;e Þw4 ð2/B;e Þ for  > 0 small enough and all choices of faces passing through e. (To define the old functions he , we first chose a plane through e and containing one of the faces of P. This plane was the plane where he ¼ 0. For the new functions he , we

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consider all the planes through e and containing one of the faces of P.) These new functions will be smooth on xA near its vertices, but provide global coordinates away from the vertices. Finally, let w3 and e ¼ ½AB be as in the above paragraph. We then replace re with w3 ð/A;e Þw3 ð/B;e Þre þ ð1  w3 ð/A;e ÞÞqA þ ð1  w3 ð/B;e ÞÞqB . Let us notice that one can define RP directly, which would provide the definition of C1 ðRPÞ as the space of smooth functions on RP [1]. The advantage of the approach in [1] is that it makes no distinction between the cases when P is convex or non-convex. The approach in this paper has the advantage that it is much simpler and more intuitive in the convex case. Further intuition in the construction of the spaces C1 ðRPÞ can be obtained from the paper [22, p. 254], by Costabel and Dauge, where various regions and subregions of a polyhedral domain were analyzed. See also [2,19,36]. A.2. The partition of unity Our partition of unity will depend on parameters (a, b, c) that will be specified below. First of all, let us denote by B(P, n) the open ball of center P and radius 2n a. By choosing a small enough, we can assume that the balls B(P,1) do not intersect. Then let Ee;n be the set of points p 2 P that do not belong to any B(P, 2) and are at distance 6 2n ab to the edge e. By choosing b small enough, we can assume that the sets Ee;1 do not intersect. Let X1 be obtained from P by removing the sets B(P,2) and Ee;2 . For each edge e, let Ne be the plane normal to e. Project Ee;1 n Ee;2 onto N e . The projection will be the intersection of an annulus with an angle. Denote this projection by Ce. We shall cover Ce with disks of radius c=2 and with disks of radius c=8. The disks of radius c=2 have the center on the straight sides of Ce (the ones obtained from the angle) and the disks of radius c=8 that have centers in the interior of Ce at distance at least c=4 to the angle defining Ce. This yields the disks D1 ; . . . ; DN with centers q1 ; . . . ; qN . Let z be the variable along the line containing e. Then we cover Ee;kþ1 n Ee;kþ2 , k 2 Zþ , with balls of radius 2k c and centers of the form ð2k qj ; 2k3 cÞ, if 2k qj is on one of the faces of P and is inside Ee;1 . Otherwise, we consider the ball of radius 2k2 c with centers of the form ð2k qj ; 2k3 cÞ as long as the center is still inside Ee;1 . Let us cover ! [ [ X1 :¼ P n BðP ; 2Þ [ Ee;2 P

e

with finitely many balls of radius c=2 or radius c=8 with centers in X1 such that the balls of radius c=2 have the centers on the faces of P and the balls of radius c=8 are at distance at least c=4 to the faces of P. Let DP ;1 ; . . . ; DP ;N ; . . . be the balls already constructed with centers in BðP ; 1Þ n BðP ; 2Þ. Then consider also the

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balls 2k DP ;1 ; . . . ; 2k DP ;N ; . . . obtained by dilations of ratio 2k and center P. We repeat this construction for all vertices P and all k 2 Zþ . We consider all the balls D1 ; D2 ; . . . ; constructed so far (relabeled into a sequence) from the coverings of Ee;1 , X1 , and from the dilations of ratio 2k for all the vertices P, as already explained. If we choose c small enough (after the choices of a and b have been made as explained above), then the sequence of these balls is locally finite, the center of each ball is either on the faces of P or the closure of the ball is inside P. Moreover, for any such ball D with center p and radius r, we have that r=#ðpÞ is bounded from above and bounded from below from zero, say r=#ðpÞ 2 ½0 ; 1 0 , for some 0 2 ð0; 1Þ. There is an integer j such that no j þ 1 of the balls constructed have a common point. To any ball D of center q and radius r we associate the bump function wD ðpÞ :¼ wðj p  q j =rÞ, where w : ½0; 1Þ ! ½0; 1 is smooth, is equal to 1 in a neighborhood of 0, is equal to 0 in a neighborhood of ½1; 1Þ, and is > 0 on P ½0; 1Þ. Then we let g ¼ wDj and /j ¼ wDj =g. By further decreasing c, if necessary, we see that our partition of unity (together with the points xj obtained as the centers of our balls) satisfies the conditions of Lemma 3.4 for the 0 chosen above. References [1] B. Ammann, A. Ionescu, V. Nistor, Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math. (electronic) 11 (2006) 161–206. [2] T. Apel, S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci 21 (1998) 519–549. [3] D. Arnold, R. Falk, Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials, Arch. Ration. Mech. Anal. 98 (1987) 143–165. [4] I. Babusˇka, Finite element method for domains with corners, Computing (Arch. Elektron. Rechnen) 6 (1970) 264–273. [5] The rate of convergence for the finite element method, SIAM J. Numer. Anal 8 (1971) 304–315. [6] I. Babusˇka, A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Proceedings of the Symposium, Univ. Maryland, Baltimore, MD, 1972, Academic Press, New York, 1972, pp. 1–359 (With the collaboration of G. Fix and R.B. Kellogg). [7] I. Babusˇka, U. Banerjee, J. Osborn, Survey of meshless and generalized finite element methods: a unified approach, Acta Numer. 12 (2003) 1–125. [8] I. Babusˇka, G. Caloz, J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994) 945–981. [9] I. Babusˇka, R.B. Kellogg, J. Pitka¨ranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33 (1979) 447–471. [10] I. Babusˇka, J.M. Melenk, The partition of unity method, Int. J. Numer. Methods Engrg. 40 (1997) 727–758. [11] I. Babusˇka, M. Rosenzweig, A finite element scheme for domains with corners, Numer. Math. 20 (1972/73) 1–21. [12] I. Babusˇka, S.L. Sobolev, Optimization of numerical processes, Appl. Math. 10 (1965) 96–129. [13] I. Babusˇka, V. Nistor, Boundary value problems in spaces of distributions and their numerical investigation. IMA Preprint 2006.

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