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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1092-1095 (novembre 2016)
Doi : 10.1016/j.crma.2016.09.013
Received : 15 February 2016 ;  accepted : 3 October 2016
A converse to Fortin's Lemma in Banach spaces
Une réciproque au lemme de Fortin dans les espaces de Banach
 

Alexandre Ern a , Jean-Luc Guermond b
a Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée cedex 2, France 
b Department of Mathematics, Texas A&M University 3368 TAMU, College Station, TX 77843, USA 

Abstract

We establish the converse of Fortin's Lemma in Banach spaces. This result is useful to assert the existence of a Fortin operator once a discrete inf–sup condition has been proved. The proof uses a specific construction of a right-inverse of a surjective operator in Banach spaces. The key issue is the sharp determination of the stability constants.

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Résumé

On montre une réciproque au lemme de Fortin dans les espaces de Banach. Ce résultat est utile afin d'affirmer l'existence d'un opérateur de Fortin une fois qu'une condition inf–sup discrète a été prouvée. La preuve utilise une construction spécifique d'un inverse à droite d'un opérateur surjectif dans les espaces de Banach. Le point crucial est la détermination précise des constantes de stabilité.

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Introduction

Let V and W be two complex Banach spaces equipped with the norms   and  , respectively. We adopt the convention that dual spaces are denoted with primes and are composed of antilinear forms; complex conjugates are denoted by an overline. Let a be a sesquilinear form on   (linear w.r.t. its first argument and antilinear w.r.t. its second argument). We assume that a is bounded, i.e.
(1)‖a‖:=supv∈V⁡supw∈W⁡|a(v,w)|‖v‖V‖w‖W<∞, and that the following inf–sup condition holds:(2)α:=infv∈V⁡supw∈W⁡|a(v,w)|‖v‖V‖w‖W>0. Here and in what follows, arguments in infima and suprema are implicitly assumed to be nonzero.

Assume that we have at hand two sequences of finite-dimensional subspaces   and   with   and   for all  , where the parameter h typically refers to a family of underlying meshes. The spaces   and   are equipped with the norms of V and W , respectively. A question of fundamental importance is to assert the following discrete inf–sup condition:
(3)αˆh:=infvh∈Vh⁡supwh∈Wh⁡|a(vh,wh)|‖vh‖V‖wh‖W>0. The aim of this Note is to prove the following result.

Theorem 1

Under the above assumptions, consider the following two statements:

(i)
there exists a map   and a real number   such that  , for all  , and   for all  ;
(ii)
the discrete inf–sup condition ((3)) holds.
Then,   with  . Conversely,   with  , and   can be constructed to be idempotent. Moreover,   can be made linear if W is a Hilbert space.

The statement (i) ⇒ (ii) in Theorem 1 is classical and is known in the literature as Fortin's Lemma, see [[5]] and [[1]]. It provides an effective tool to prove the discrete inf–sup condition ((3)) by constructing explicitly a Fortin operator  . We briefly outline a proof that (i) ⇒ (ii) for completeness. Assuming (i), we have
supwh∈Wh⁡|a(vh,wh)|‖wh‖W≥supw∈W⁡|a(vh,Πhw)|‖Πhw‖W=supw∈W⁡|a(vh,w)|‖Πhw‖W≥γΠhsupw∈W⁡|a(vh,w)|‖w‖W≥γΠhα‖vh‖V, since a satisfies ((2)) and  . This proves (ii) with  .

The proof of the converse (ii) ⇒ (i) is the main object of this Note. This property is useful when it is easier to prove the discrete inf–sup condition directly rather than constructing a Fortin operator. Another application of current interest is the analysis framework for discontinuous Petrov–Galerkin methods (dPG) recently proposed in [[3]], which includes the existence of a Fortin operator among its key assumptions. The proof of the converse is not so straightforward if one wishes to establish a sharp stability bound for  , i.e. that indeed one can take  . Incidentally, we observe that there is a gap in the stability constant   between the direct and the converse statements, since the ratio of the two is equal to   (which is independent of the discrete setting).

This Note is organized as follows. In Section 2, we establish a sharp bound on the stability of the right-inverse of surjective operators in Banach spaces. Since this result can be of independent theoretical interest, we present it in the infinite-dimensional setting. Then in Section 3, we prove the converse of Fortin's Lemma. The proof is relatively simple once the sharp stability estimate from Section 2 is available.

Right-inverse of surjective Banach operators

Let Y and Z be two complex Banach spaces equipped with the norms   and  , respectively. Let   be a bounded linear map. The following result is a well-known consequence of Banach's Open Mapping and Closed Range Theorems, see, e.g., [[2]] or [[4]].

Lemma 2

The following three statements are equivalent:

(i)
  is surjective;
(ii)
  is injective and   is closed in  ;
(iii)
the following holds:
(4)infz′∈Z′⁡‖B⁎z′‖Y′‖z′‖Z′=infz′∈Z′⁡supy∈Y⁡|〈B⁎z′,y〉Y′,Y|‖z′‖Z′‖y‖Y=:β>0.

Let us now turn to the main result of this section. To motivate the result, assume that ((4)) holds; then B is surjective and thus admits a bounded right-inverse. The crucial question is whether the stability of this right-inverse can be formulated using precisely the constant   from ((4)).

Lemma 3

Assume that ((4)) holds and that Y is reflexive. Then there is a right-inverse map   such that
(5)∀z∈Z,(B∘B†)(z)=zandβ‖B†z‖Y≤‖z‖Z. Moreover, this right-inverse map   is linear if Y is a Hilbert space.

Proof

Parts of this result can be found in [[4]]; for completeness, we present a proof. Owing to Lemma 2,   is injective and   is closed in  . Since the operator   is injective, it admits a left-inverse linear map   such that   for all  . Moreover, the inf–sup condition ((4)) implies that   for all  . Consider now the adjoint  . Let   be the Hahn–Banach extension operator that extends antilinear forms over   into antilinear forms over   (see [[2]]);   maps from   to  . Let   (resp.,  ) be the canonical isometry from Y to   (resp., Z to  ), and observe that   is an isomorphism since Y is assumed to be reflexive. Let us set
(6)B†:=JY−1∘ER′Y″hb∘B⁎‡⁎∘JZ:Z→Y, and let us verify that   satisfies the expected properties. We have, for all  ,〈z′,B(B†(z))〉Z′,Z=〈B⁎z′,B†(z)〉Y′,Y=〈JY(B†(z)),B⁎z′〉Y″,Y′‾=〈ER′Y″hb(B⁎‡⁎(JZz)),B⁎z′〉Y″,Y′‾=〈B⁎‡⁎(JZz),B⁎z′〉R′,R‾=〈JZz,B⁎‡B⁎z′〉Z″,Z′‾=〈JZz,z′〉Z″,Z′‾=〈z′,z〉Z′,Z, where we have used that   to pass from the first to the second line. This shows that  . Moreover, since   is an isometry and the extension operator   preserves the norm, we observe that, for all  ,‖B†z‖Y=‖B⁎‡⁎(JZz)‖R′=supz′∈Z′⁡|〈B⁎‡⁎(JZz),B⁎z′〉R′,R|‖B⁎z′‖Y′=supz′∈Z′⁡|〈JZz,z′〉Z″,Z′|‖B⁎z′‖Y′≤supz′∈Z′⁡‖z′‖Z′‖B⁎z′‖Y′‖z‖Z. We conclude from ((4)) that  . Finally, if Y is a Hilbert space, we can consider the orthogonal complement of R in   (recall that R is a closed subspace of  ) and write  . Then, the Hahn–Banach extension operator   in ((6)) can be replaced by the linear map   such that, for all  ,   for all   with  ,  ,  .  □

Proof of the converse in Theorem 1

Let   be the operator defined by   for all  . We identify   with   and   with   (since these spaces are finite-dimensional). We consider the adjoint operator  , and identify   with  . We apply Lemma 3 to  ,  , and  . Owing to the discrete inf–sup condition ((3)), we infer that ((4)) holds with  . Therefore, there exists a right-inverse map   such that, for all  ,   and  . Let us now set
(7)Πh:=Ah⁎†∘Θ:W→Wh, with the linear map   such that, for all  ,   for all  . We then infer thata(vh,Πh(w))=〈Ahvh,Ah⁎†(Θ(w))〉Wh′,Wh=〈Ah⁎(Ah⁎†(Θ(w))),vh〉Vh′,Vh‾=〈Θ(w),vh〉Vh′,Vh‾=a(vh,w), which establishes that   for all  . Moreover,αˆh‖Πh(w)‖W=αˆh‖Ah⁎†(Θ(w))‖W≤‖Θ(w)‖Vh′≤‖a‖‖w‖W, which proves that  . In addition, we observe that〈Θ(Ah⁎†(θh)),vh〉Vh′,Vh=〈Ahvh,Ah⁎†(θh)〉Wh′,Wh‾=〈Ah⁎(Ah⁎†(θh)),vh〉Vh′,Vh=〈θh,vh〉Vh′,Vh, for all  , which proves that   for all  . As a result,  , i.e.,   is idempotent. Finally, if W is a Hilbert space, the right-inverse map   is linear by Lemma 3, and so is the operator   defined from ((7)).

Remark 1

Without the use of Lemma 3, one only knows that   has a stable right-inverse, but a stability bound for this right-inverse is not available. Here, we obtain that, provided the discrete inf–sup condition ((3)) holds uniformly with respect to h , i.e. if there is   such that   for all  , then a uniform stability bound holds for   since   for all  .

Remark 2

Even in the case of Banach spaces, the linearity of the map   can be asserted if one has at hand a stable decomposition   such that there is   such that the induced projector   satisfies   for all   (this property holds in the Hilbertian setting with  ). Then, one can adapt the reasoning at the end of the proof of Lemma 3 to build a stable, linear right-inverse map  . The mild price to be paid is that the stability constant of   now becomes  .

Remark 3

Whether Lemma 3 holds true when Y is not reflexive seems to be an open question.

References

Boffi D., Brezzi F., Fortin M. Mixed Finite Element Methods and Applications  Heidelberg, Germany: Springer (2013). 
Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations  New York: Springer (2011). 
Carstensen C., Demkowicz L., Gopalakrishnan J. A posteriori error control for DPG methods SIAM J. Numer. Anal. 2014 ;  52 (3) : 1335-1353 [cross-ref]
Ern A., Guermond J.-L. Theory and Practice of Finite Elements  New York: Springer-Verlag (2004). 
Fortin M. An analysis of the convergence of mixed finite element methods RAIRO Anal. Numér. 1977 ;  11 : 341-354



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