Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast

0.71 MB·English

by  Leonid Berlyand

Flux norm approach to finite dimensional homogenization
approximations with non-separated scales and high contrast.
Leonid Berlyand∗ and Houman Owhadi†.
January 11, 2010
0
1
0
2
Abstract
n
a We consider linear divergence-form scalar elliptic equations and vectorial equa-
J tions forelasticitywithrough(L∞(Ω), Ω Rd)coefficientsa(x) that,inparticular,
1 ⊂
model media with non-separated scales and high contrast in material properties.
1
While the homogenization of PDEs with periodic or ergodic coefficients and well

] separated scales is now well understood, we consider here the most general case of
P
arbitraryboundedcoefficients. Forsuchproblems,weintroduceexplicitandoptimal
A
finite dimensional approximations of solutions that can be viewed as a theoretical
. Galerkin method with controlled error estimates, analogous to classical homoge-
h
t nization approximations. In particular, this approachallows one to analyze a given
a
medium directly without introducing the mathematical concept of an ǫ family of
m
2
media as in classical homogenization. We define the flux norm as the L norm of
[ 1
the potential part of the fluxes of solutions, which is equivalent to the usual H -

7 norm. We showthatinthe fluxnorm,the errorassociatedwithapproximating,ina
v properly defined finite-dimensionalspace,the set ofsolutions of the aforementioned
3
PDEs with rough coefficients is equal to the error associated with approximating
6
the setofsolutionsofthe sametypeofPDEswithsmoothcoefficientsinastandard
4
1 space (e.g., piecewise polynomial). We refer to this property as the transfer prop-
. erty. A simple application of this property is the construction of finite dimensional
1
approximation spaces with errors independent of the regularity and contrast of the
0
9 coefficients and with optimal and explicit convergencerates. This transfer property
0 also provides an alternative to the global harmonic change of coordinates for the
v: homogenization of elliptic operators that can be extended to elasticity equations.
i The proofs of these homogenization results are based on a new class of elliptic in-
X
equalities. These inequalities play the same role in our approach as the div-curl
r lemma in classical homogenization.
a
Contents
1 Introduction 3
∗Pennsylvania StateUniversity,Department of Mathematics
†Corresponding author. California Institute of Technology, Applied & Computational Mathematics,
Control & Dynamical systems, MC 217-50 Pasadena, CA 91125, [email protected]
1
2 The flux norm and its properties 4
2.1 Scalar case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Vectorial case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Application to theoretical finite element methods with accuracy inde-
pendent of material contrast. 11
3.1 Scalar divergence form equation . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Approximation with piecewise linear nodal basis functions of a
regular tessellation of Ω . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Approximation with eigenfunctions of the Laplace-Dirichlet oper-
ator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Vectorial elasticity equations. . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 A new class of inequalities 17
4.1 Scalar case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1 A brief reminder on the mapping using harmonic coordinates. . . . 22
4.2 Tensorial case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 ACordesConditionfortensorialnon-divergenceformellipticequa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Application of the flux-norm to theoretical non-conforming Galerkin. 27
5.1 Scalar equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Tensorial equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Relations with homogenization theory and other works. 33
7 Conclusions 38
8 Appendix 39
8.1 Extension to non-zero boundary conditions. . . . . . . . . . . . . . . . . . 39
8.2 Proof of lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Acknowledgements. Part of the research of H. Owhadiis supportedby the National
Nuclear Security Administration through thePredictive Science Academic Alliance Pro-
gram. The work of L. Berlyand is supported in part by NSF grant DMS-0708324 and
DOE grant DE-FG02-08ER25862. We would like to thank L. Zhang for the computa-
tions associated with figure 1. We also thank B. Haines, L. Zhang, and O. Misiats for
carefully reading the manuscript and providing useful suggestions. We would like to
thank Bjo¨rn Engquist, Ivo Babuˇska and John Osborn for useful comments and show-
ing us related and missing references. We are also greatly in debt to Ivo Babuˇska and
John Osborn for carefully reading the manuscript and providing us with very detailed
comments and references which have lead to substantial changes. We would also like to
thank two anonymous referees for precise and detailed comments and suggestions.
2
1 Introduction
Inthis paperweareinterested infinitedimensionalapproximations of solutions ofscalar
and vectorial divergence form equations with rough coefficients in Ω Rd, d 2. More
⊂ ≥
precisely, in the scalar case, we consider the partial differential equation
div a(x) u(x) = f(x) x Ω;f L2(Ω), a(x) = a L∞(Ω)
ij
− ∇ ∈ ∈ { ∈ } (1.1)
(u= 0(cid:16) on ∂Ω, (cid:17)
whereΩisaboundedsubsetofRd withasmoothboundary(e.g., C2)andaissymmetric
and uniformly elliptic on Ω. It follows that the eigenvalues of a are uniformly bounded
frombelowandabovebytwostrictlypositiveconstants,denotedbyλ (a)andλ (a).
min max
Precisely, for all ξ Rd and x Ω,
∈ ∈
λ (a)ξ 2 ξTa(x)ξ λ (a)ξ 2. (1.2)
min max
| | ≤ ≤ | |
In the vectorial case, we consider the equilibrium deformation of an inhomogeneous
elastic body under a given load b (L2(Ω))d, described by
∈
div(C(x) : ε(u)) = b(x) x Ω
− ∈ (1.3)
(u= 0 on ∂Ω,
where Ω Rd is a bounded domain, C(x) = C (x) is a 4th order tensor of elastic
ijkl
⊂ { }
modulus (with the associated symmetries), u(x) Rd is the displacement field, and for
∈
ψ (H1(Ω))d, ε(ψ) is the symmetric part of ψ, namely,
∈ 0 ∇
1 ∂ψ ∂ψ
i j
ε (ψ) = + . (1.4)
ij
2 ∂x ∂x
j i
(cid:16) (cid:17)
WeassumethatC isuniformlyellipticandC L∞(Ω). Itfollowsthattheeigenvalues
ijkl
∈
of C are uniformly bounded from below and above by two strictly positive constants,
denoted by λ (C) and λ (C).
min max
The analysis of finite dimensional approximations of scalar divergence form elliptic,
parabolic and hyberbolic equations with rough coefficients that in addition satisfy a
Cordes-type condition in arbitrary dimensions has been performed in [53, 54, 55]. In
these works, global harmonic coordinates are used as a coordinate transformation. We
also refer to the work of Babuˇska, Caloz, and Osborn[10, 8] in which a harmonic change
of coordinates is introduced in one-dimensional and quasi-one-dimensional divergence
form elliptic problems.
In essence, this harmonic change of coordinates allows for the mapping of the oper-
ator L := div(a ) onto the operator L := div(Q ) where Q is symmetric positive
a Q
∇ ∇
and divergence-free. This latter property of Q implies that L can be written in both
Q
a divergence form and a non-divergence form operator. Using the W2,2 regularity of
solutions of L v = f (for f L2), one is able to obtain homogenization results for the
Q
∈
3
operator L in the sense of finite dimensional approximations of its solution space (this
a
relation with homogenization theory will be discussed in detail in section 6).
This harmonic change of coordinates provides the desired approximation in two-
dimensional scalar problems, but there is no analog of such a change of coordinates
for vectorial elasticity equations. One goal of this paper is to obtain an analogous
homogenization approximation without relying on any coordinate change and therefore
allowing for treatment of both scalar and vectorial problems in a unified framework.
In section 2, we introduce a new norm, called the flux norm, defined as the L2-norm
of the potential component of the fluxes of solutions of (1.1) and (1.3). We show that
thisnormisequivalent totheusualH1-norm. Furthermore,thisnewnormallows forthe
transfer of error estimates associated with a given elliptic operator div(a ) and a given
∇
approximation space V onto error estimates for another given elliptic operator div(a′ )
∇
with another approximation space V′ provided that the potential part of the fluxes of
elements of V and V′ span the same linear space. In this work, this transfer/mapping
property will replace the transfer/mapping property associated with a global harmonic
change of coordinates.
In section 3, we show that a simple and straightforward application of the flux-norm
transfer property is to obtain finite dimensional approximation spaces for solutions of
(1.1) and (1.3) with “optimal” approximation errors independent of the regularity and
contrast of the coefficients and the regularity of ∂Ω.
Another application of the transfer property of the flux norm is given in section 5 for
controlling the approximation error associated with theoretical discontinuous Galerkin
solutionsof(1.1)and(1.3). Inthiscontext,forelasticityequations,harmoniccoordinates
are replaced by harmonic displacements. The estimates introduced in section 5 are
based on mapping onto divergence-free coefficients via the flux-norm and a new class of
inequalitiesintroducedinsection4. Webelievethattheseinequalitiesareofindependent
interest for PDE theory and could be helpful in other problems.
Connections between this work, homogenization theory and other related works will
be discussed in section 6.
2 The flux norm and its properties
In this section, we will introduce the flux-norm and describe its properties when used
as a norm for solutions of (1.1) (and (1.3)). This flux-norm is equivalent to the usual
H1(Ω)-norm(or(H1(Ω))d-normforsolutionstothevectorialproblem),butleadstoerror
0 0
estimates that are independent of the material contrast. Furthermore, it allows for the
transfer of error estimates associated with a given elliptic operator div(a ) and a given
∇
approximation space V onto error estimates for another given elliptic operator div(a′ )
∇
with another approximation space V′ provided that the potential part of the fluxes of
elementsofV andV′ spanthesamelinearspace. In[53],approximationerrorshavebeen
obtainedfortheoreticalfiniteelementsolutionsof(1.1)witharbitrarilyroughcoefficients
a. These approximation errors are based on the mapping of the operator div(a )
− ∇
onto an non-divergence form operator Q ∂ ∂ using global harmonic coordinates as
i,j i j
−
4
a change of coordinates. It is not clear how to extend this change of coordinates to
elasticity equations, whereas the flux-norm approach has a natural extension to systems
of equations and can be used to link error estimates on two separate operators.
2.1 Scalar case.
Definition 2.1. Fork (L2(Ω))d,denotebyk andk thepotentialanddivergence-
pot curl
∈
free portions of the Weyl-Helmholtz decomposition of k. Recall that k and k are
pot curl
orthogonal with respect to the L2-inner product. k is the orthogonal projection of
pot
k onto L2 (Ω) defined as the closure of the space f : f C∞(Ω) in (L2(Ω))d.
pot {∇ ∈ 0 }
k is the orthogonal projection of k onto L2 (Ω) defined as the closure of the space
curl curl
ξ : ξ (C∞(Ω))d div(ξ) = 0 in (L2(Ω))d
{ ∈ }
For ψ H1(Ω), define
∈ 0
kψka-flux := k(a∇ψ)potk(L2(Ω))d. (2.1)
Motivations for the flux norm
The () in the a-flux-norm is explained by the fact that in practice, we are inter-
pot
• ·
ested in fluxes (of heat, stress, oil, pollutant) entering or exiting a given domain.
Furthermore, for a vector field ξ, ξ nds = div(ξ)dx = div(ξ )dx, which
∂Ω · Ω Ω pot
means the flux entering or exiting is determined by the potential part of the vector
R R R
field. Thus, as with the energy norm, u 2 := ( u)Ta u, the flux norm has a
k ka Ω ∇ ∇
natural physical interpretation. An error bound given in the flux-norm shows how
R
well fluxes (of heat or stresses) are approximated.
While the energy norm is natural in many problems, we argue that this is no
•
longer the case in the presence of high contrast. Observe that in [22], contrast
independent error estimates are obtained by renormalizing the energy norm by
λ (a). In [14], the error constants associated with the energy norm are made
min
independent of the contrast by using terms that are appropriately and explicitly
weighted by a. These modifications on the energy norm or on the error bounds
(expressed in the energy norm) have to be introduced because, in the presence of
highcontrast inmaterial properties,theenergy normblows up. Even inthesimple
case where a is a constant (a =αI with α >0), the solution of (1.1) satisfies
d
1
( u)Ta u= ∆−1f 2 . (2.2)
∇ ∇ α ∇ (L2(Ω))d
ZΩ
(cid:13) (cid:13)
Hence the energy norm squared of the sol(cid:13)ution of ((cid:13)1.1) blows up like 1/α as α 0
↓
whereas its flux-norm is independent of α (because (a u) = ∆−1f)
pot
∇ ∇
kuka-flux = ∇∆−1f (L2(Ω))d. (2.3)
(cid:13) (cid:13)
Equation (2.3) remains valid even wh(cid:13)en a is no(cid:13)t a constant (this is a consequence
of the transfer property, see Corollary 2.1). In reservoir modeling, fluxes of oil
5
and water are the main quantities of interest to be approximated correctly. The
energy norm is less relevant due to high contrast and has been modified (in [22]
for instance) in order to avoid possible blow up.
Similar considerations of convergence in terms energies and fluxes are present in
•
classicalhomogenizationtheory. Indeed,theconvergenceofsolutionsof div(aǫ uǫ) =
− ∇
f canbeexpressedintermsofconvergenceofenergiesinthecontextofΓ-convergence
[32, 17] (and its variational formulation) or in the terms (of weak) convergence of
fluxes in G or H-convergence [48, 31, 59, 58, 47] (aǫ uǫ a0 u0). Here, weak
∇ → ∇
L2 convergence of fluxes is used and no flux norm is necessary unlike in our study,
where it arises naturally.
Proposition 2.1. . is a norm on H1(Ω). Furthermore, for all ψ H1(Ω)
kka-flux 0 ∈ 0
λmin(a)k∇ψk(L2(Ω))d ≤ kψka-flux ≤ λmax(a)k∇ψk(L2(Ω))d (2.4)
Proof. The proof of the left hand side of inequality (2.4) follows by observing that
( ψ)Ta ψ = ( ψ)T(a ψ) (2.5)
pot
∇ ∇ ∇ ∇
ZΩ ZΩ
from which we deduce by Cauchy-Schwarz inequality that
( ψ)Ta ψ ψ ψ . (2.6)
L2(Ω) a-flux
∇ ∇ ≤ k∇ k k k
ZΩ
The proof of the main theorem of this section will require
Lemma 2.1. Let V be a finite dimensional linear subspace of H1(Ω). For f L2(Ω),
0 ∈
let u be the solution of (1.1). Then,
sup inf ku−vka-flux = sup inf k(∇w−a∇v)potk(L2(Ω))d (2.7)
f∈L2(Ω)v∈V kfkL2(Ω) w∈H2(Ω)∩H01(Ω)v∈V k∆wkL2(Ω)
Proof. Since f L2(Ω), it is known that there exists w H2(Ω) H1(Ω) such that
∈ ∈ ∩ 0
∆w = f x Ω
− ∈ (2.8)
(w = 0 on ∂Ω.
We conclude by observing that for v V,
∈
k(∇w−a∇v)potk(L2(Ω))d = k(a∇u−a∇v)potk(L2(Ω))d. (2.9)
6
For V, a finite dimensional linear subspace of H1(Ω), we define
0
(diva V) := span div(a v) : v V . (2.10)
∇ { ∇ ∈ }
Note that (diva V) is a finite dimensional subspace of H−1(Ω).
∇
The following theorem establishes the transfer property of the flux norm which is
pivotal for our analysis.
Theorem 2.1. (Transfer property of the flux norm) Let V′ and V be finite-
dimensional subspaces of H1(Ω). For f L2(Ω) let u be the solution of (1.1) with
0 ∈
conductivity a and u′ be the solution of (1.1) with conductivity a′. If (diva V) =
∇
(diva′ V′), then
∇
u v u′ v
a-flux a′-flux
sup inf k − k = sup inf k − k . (2.11)
f∈L2(Ω)v∈V kfkL2(Ω) f∈L2(Ω)v∈V′ kfkL2(Ω)
Remark 2.1. The usefulness of (2.11) can be illustrated by considering a′ = I so that
diva′ = ∆. Then u′ H2 and therefore V′ can be chosen as, e.g., the standard
∇ ∈
piecewise linear FEM space with nodal basis ϕ . The space V is then defined by its
i
{ }
basis ψ determined by
i
{ }
div(a ψ )= ∆ϕ (2.12)
i i
∇
with Dirichlet boundary conditions (see details in section 3.1.1). Furthermore, equation
(2.11) shows that the error estimate for a problem with arbitrarily rough coefficients is
equal to the well-known error estimate for the Laplace equation.
Remark 2.2. Equation (2.11) remains valid without the supremum in f. More precisely
writing u and u′ the solutions of (1.1) with conductivities a and a′ and the same right
hand side f L2(Ω), one has
∈
inf u v = inf u′ v . (2.13)
a-flux a′-flux
v∈V k − k v∈V′k − k
Equation (2.13) is obtained by observing that
u v = ∆−1(f +div(a v)) (2.14)
k − ka-flux ∇ ∇ L2(Ω)
(cid:13) (cid:13)
Corollary 2.1. Let X and V be finit(cid:13)e-dimensional subspace(cid:13)s of H1(Ω). For f L2(Ω)
0 ∈
let u be the solution of (1.1) with conductivity a. If (diva V) = (div X) then
∇ ∇
sup inf ku−vka-flux = sup inf k∇w−∇vk(L2(Ω))d (2.15)
f∈L2(Ω)v∈V kfkL2(Ω) w∈H01(Ω)∩H2(Ω)v∈X k∆wkL2(Ω)
Equation(2.15)canbeobtainedbysettinga′ = I intheorem2.1andapplyinglemma
2.1.
Theorem2.1 isobtained fromthefollowing propositionbynotingthattherighthand
side of equation (2.16) is the same for pairs (a,V) and (a′,V′) whenever div(a V) =
∇
div(a′ V′).
∇
7
Proposition 2.2. For f L2(Ω) let u be the solution of (1.1). Then,
∈
sup inf ku−vka-flux = sup kzkL2(Ω) , (2.16)
f∈L2(Ω)v∈V kfkL2(Ω) z∈(diva∇V)⊥ k∇zk(L2(Ω))d
where
(diva V)⊥ := z H1(Ω) : v V,( z,a v) = 0 . (2.17)
∇ { ∈ 0 ∀ ∈ ∇ ∇ }
Proof. For w H2(Ω), define
∈
J(w) := vi∈nVf k(∇w−a∇v)potk(L2(Ω))d. (2.18)
Observe that
J(w) = v∈V,ξ∈(L2(Rind)f)d:div(ξ)=0k∇w−a∇v−ξk(L2(Ω))d. (2.19)
Additionally, observing that the space spanned by z for z (diva V)⊥ is the orthog-
∇ ∈ ∇
onal complement (in (L2(Ω))d) of the space spanned by a v+ξ, we obtain that
∇
( w, z)
J(w) = sup ∇ ∇ . (2.20)
z∈(diva∇V)⊥ k∇zk(L2(Ω))d
Integrating by parts and applying the Cauchy-Schwarz inequality yields
z
L2(Ω)
J(w) ∆w sup k k . (2.21)
≤ k kL2(Ω)z∈(diva∇V)⊥ k∇zk(L2(Ω))d
which proves
sup inf ku−vka-flux sup kzkL2(Ω) , (2.22)
f∈L2(Ω)v∈V kfkL2(Ω) ≤ z∈(diva∇V)⊥ k∇zk(L2(Ω))d
Dividingby ∆w ,integratingbyparts,andtakingthesupremumoverw H2(Ω)
L2(Ω)
k k ∈ ∩
H1(Ω), we get
0
J(w) (∆w,z)
sup = sup sup .
w∈H2(Ω)∩H01(Ω) k∆wkL2(Ω) z∈(diva∇V)⊥w∈H2(Ω)∩H01(Ω)−k∇zk(L2(Ω))dk∆wkL2(Ω)
(2.23)
we conclude the theorem by choosing ∆w = z.
−
The transfer property (2.11) for solutions can be complemented by an analogous
property for fluxes. To this end, for a finite dimensional linear subspace (L2(Ω))d
V ⊂
define
(diva ) := div(aζ) : ζ . (2.24)
V { ∈ V}
Observe that (diva ) is a finite dimensional subspace of H−1(Ω). The proof of the
V
following theorem is similar to the proof of theorem 2.1.
8
Theorem 2.2. (Transfer property for fluxes) Let ′ and be finite-dimensional
V V
subspaces of (L2(Ω))d. For f L2(Ω) let u be the solution of (1.1) with conductivity a
∈
and u′ be the solution of (1.1) with conductivity a′. If (diva ) = (diva′ ′) then
V V
sup inf k(a(∇u−ζ))potk(L2(Ω))d = sup inf k(a′(∇u′−ζ))potk(L2(Ω))d (2.25)
f∈L2(Ω)ζ∈V f L2(Ω) f∈L2(Ω)ζ∈V′ f L2(Ω)
k k k k
Theorem 2.2 will be used in section 5 for obtaining error estimates on theoretical
non-conforming Galerkin solutions of (1.1).
Corollary 2.2. Let be a finite-dimensional subspace of (L2(Ω))d and X a finite-
V
dimensional subspace of H1(Ω). For f L2(Ω) let u be the solution of (1.1) with
0 ∈
conductivity a. If (diva ) = (div X) then
V ∇
sup inf k(a(∇u−ζ))potk(L2(Ω))d = sup inf k∇w−∇vk(L2(Ω))d (2.26)
f∈L2(Ω)ζ∈V kfkL2(Ω) w∈H01(Ω)∩H2(Ω)v∈X k∆wkL2(Ω)
Remark 2.3. The analysis performed in this section and in the following one can be nat-
urally extended to other types of boundary conditions (nonzero Neumann or Dirichlet).
To support our claim, we will provide this extension in the scalar case with non-zero
Neumann boundary conditions. We refer to subsection 8.1 for that extension.
2.2 Vectorial case.
For k (L2(Ω))d×d, denote by k the potential portion of the Weyl-Helmholtz decom-
pot
∈
position of k (the orthogonal projection of k onto the closure of the space f : f
{∇ ∈
(C∞(Ω))d in (L2(Ω))d×d). Define
0 }
kψkC-flux := k(C :ε(ψ))potk(L2(Ω))d×d. (2.27)
Remark 2.4. BecauseofthesymmetriesoftheelasticitytensorC,onehas f (C∞(Ω))d
∀ ∈ 0
f,(C :ε(ψ)) = ε(f),(C :ε(ψ)) (2.28)
pot pot
∇ (L2(Ω))d×d (L2(Ω))d×d
(cid:16) (cid:17) (cid:16) (cid:17)
from which it follows that definition 2.27 would be the same if the projection was made
on the space of symmetrized gradients.
Proposition 2.3. . is a norm on (H1(Ω))d. Furthermore, for all ψ (H1(Ω))d
kkC-flux 0 ∈ 0
λmin(C)kε(ψ)k(L2(Ω))d×d ≤ kψkC-flux ≤ λmax(C)kε(ψ)k(L2(Ω))d×d. (2.29)
Proof. The proof of the left hand side of inequality (2.29) follows by observing that
(ε(ψ))T : C : ε(ψ) ≤ kε(ψ)k(L2(Ω))d×dkψkC-flux. (2.30)
ZΩ
The fact that ψ is a norm follows from the left hand side of inequality (2.29) and
C-flux
k k
Korn’s inequality [39]: i.e., for all ψ (H1(Ω))d,
∈ 0
k∇ψk(L2(Ω))d×d ≤ √2kε(ψ)k(L2(Ω))d×d. (2.31)
9
For V, a finite dimensional linear subspace of (H1(Ω))d, we define
0
(divC : ε(V)) := span div(C :ε(v)) : v V . (2.32)
{ ∈ }
Observe that (divC : ε(V)) is a finite dimensional subspace of (H−1(Ω))d. Similarly for
X, a finite dimensional linear subspace of (H1(Ω))d, we define
0
∆X := span ∆v : v X . (2.33)
{ ∈ }
Theorem 2.3. Let V′ and V be finite-dimensional subspaces of (H1(Ω))d. For b
0 ∈
(L2(Ω))d let u be the solution of (1.3) with elasticity C and u′ be the solution of (1.3)
with elasticity C′. If (divC : ε(V)) =(divC′ :ε(V′)) then
u v u′ v
C-flux C′-flux
sup inf k − k = sup inf k − k (2.34)
b∈(L2(Ω))dv∈V kbk(L2(Ω))d b∈(L2(Ω))dv∈V′ kbk(L2(Ω))d
Corollary 2.3. Let X and V be finite-dimensional subspaces of (H1(Ω))d. For b
0 ∈
(L2(Ω))d let u be the solution of (1.3) with elasticity tensor C. If (divC : ε(V)) = ∆X
then
sup inf ku−vkC-flux = sup inf k∇w−∇vk(L2(Ω))d×d (2.35)
b∈(L2(Ω))dv∈V kbk(L2(Ω))d w∈(H01(Ω)∩H2(Ω))dv∈X k∆wk(L2(Ω))d
The proof of theorem 2.3 is analogous to the proof of theorem 2.1.
For a finite dimensional linear subspace of (L2(Ω))d×d we define
V
(divC : ) := span div(C :ζ) : ζ . (2.36)
V { ∈ V}
Observe that (divC : ) is a finite dimensional subspace of (H−1(Ω))d. The proof of
V
the following theorem is analogous to the proof of theorem 2.1.
Theorem 2.4. Let ′ and be finite-dimensional subspaces of (L2(Ω))d×d. For b
V V ∈
(L2(Ω))d let u be the solution of (1.3) with conductivity C and u′ be the solution of (1.3)
with conductivity C′. If (divC : )= (divC′ : ′) then
V V
sup inf k(C : (ε(u)−ζ))potk(L2(Ω))d×d = sup inf k(C′ : (ε(u)−ζ))potk(L2(Ω))d×d
b∈(L2(Ω))dζ∈V kbk(L2(Ω))d b∈(L2(Ω))dζ∈V′ kbk(L2(Ω))d
(2.37)
Corollary 2.4. Let be a finite-dimensional subspace of (L2(Ω))d×d and X a finite-
V
dimensional subspace of (H1(Ω))d. For b (L2(Ω))d let u be the solution of (1.3) with
0 ∈
elasticity C. If (divC : ) = (∆X) then
V
sup inf k(C :(ε(u)−ζ))potk(L2(Ω))d×d = sup inf k∇w−∇vk(L2(Ω))d×d
b∈L2(Ω)ζ∈V kbk(L2(Ω))d w∈H01(Ω)∩H2(Ω)v∈X k∆wk(L2(Ω))d
(2.38)
10