ETNA
Electronic Transactions on Numerical Analysis.
Volume 35, pp. 69-87, 2009.
Copyright  2009, Kent State University.
ISSN 1068-9613.
Kent State University
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SPECTRAL APPROXIMATION OF VARIATIONALLY FORMULATED
EIGENVALUE PROBLEMS ON CURVED DOMAINS∗
ANA ALONSO† AND ANAHı́ DELLO RUSSO†‡
To the memory of Professor Jorge D. Samur
Abstract. This paper is concerned with the spectral approximation of variationally formulated eigenvalue problems posed on curved domains. As an example of the present theory, convergence and optimal error estimates are
proved for the piecewise linear finite element approximation of the eigenvalues and eigenfunctions of a second order
elliptic differential operator on a general curved three-dimensional domain.
Key words. spectral approximation, eigenvalue problems, curved domains
AMS subject classifications. 65N15, 65N25, 65N30
1. Introduction. In this paper we present an extension of the spectral approximation
theory for non-compact operators in Hilbert spaces. In particular, we consider the numerical approximation of the eigenvalues and eigenvectors of variationally formulated problems
posed over general curved domains. There are not many references about error estimates for
this kind of problems. In particular, the finite element approximations of the spectrum of the
Laplace operator on non-convex domains with curved boundaries have been studied only in
a few papers.
The first proof of the convergence for a Laplace eigenproblem, for simple eigenvalues
and Dirichlet boundary conditions, was given by Vanmaele and Ženı́šek [16] by using the minmax characterization; see [15]. The same authors generalized their results to include multiple
eigenvalues [17] and numerical integration effects [18]. Almost at the same time, Lebaud
[11] analyzed a similar problem posed on two-dimensional domains by using isoparametric
finite elements methods in the framework of the classical spectral approximation theory; see
[1]. She also considered simple eigenvalues and Dirichlet boundary conditions but assuming
exact integration. In this case, the known results (see [13]) give only an order O(hk+1 ) for
the eigenvalues, in contrast to O(h2k ) which would be achievable on the polygonal domains
if the eigenfunctions were smooth enough. Lebaud showed how to construct “a good approximation” of the boundary in order to obtain the optimal order of convergence for eigenvalues.
However, no direct extension of this method to three-dimensional domains seems to be possible.
More recently, Hernández and Rodrı́guez [8] considered finite element approximation of
the spectral problem for the Laplace equation with Neumann boundary conditions on curved
non-convex domains. By using the abstract spectral approximation theory, they proved optimal order error estimates for the eigenfunctions and a double order for eigenvalues. Later,
the same authors proved convergence results and error estimates for the Raviart-Thomas approximations of the spectral acoustic problem on a curved non-convex two-dimensional domain [9].
The goal of this paper is to prove some abstract results on spectral approximation that can
be applied to a wide variety of eigenvalue problems defined over curved domains. These results are obtained by introducing suitable modifications in the theory developed by Descloux,
∗ Received April 15, 2008. Accepted for publication December 18, 2008. Published online on May 1, 2009.
Recommended by O. Widlund.
† Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 172, 1900
La Plata, Argentina ({ana,anahi}@mate.unlp.edu.ar).
‡ Member of CIC, Provincia de Buenos Aires, Argentina.
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A. ALONSO AND A. DELLO RUSSO
Nassif and Rappaz [4, 5]. Our analysis adapts the theory presented there to the fact that we
are dealing with nonconforming discretizations because of the approximation of the given
domain by a polyhedral one.
The remainder of the paper is organized as follows. Section 2 is devoted to introducing
the notation. In Section 3, we give a precise statement of the eigenvalue problems and the
approximation methods we will consider. In Section 4, we prove the abstract results. Finally,
in Section 5, as an application of our results, we analyze the finite element approximation
of the spectral problem for the Lamé equation with boundary conditions of Dirichlet type on
a general curved three-dimensional domain. We prove convergence and optimal order error
estimates for standard piecewise linear continuous elements.
Let us remark that our analysis is suitable for studying numerical approximations of
operators with non-compact inverse. In particular, in a forthcoming paper we will apply this
theory to investigate the finite element approximation of the Maxwell eigenproblem on curved
Lipschitz polyhedral domains.
2. Notation. Throughout this paper Ω denotes a bounded open domain in Rn , n = 2
or 3, in general non-convex, with a Lipschitz continuous boundary ∂Ω. Let W (Rn ) be a
complex Hilbert function space with norm k · kRn . Given an open set O ⊂ Rn , let W (O)
denote a generic complex Hilbert space of functions defined in O and k · kO its norm.
First, we define the restriction operator Š by
Š : W (Rn ) → W (O)
f 7→ f |O .
We restrict our attention to Hilbert spaces such that the norm k · kRn satisfies
k · k2Rn = k · k2O + k · k2Rn \O .
Then, as an immediate consequence of this assumption, we obtain that Š is a bounded operator.
We will need to provide extensions for functions on O to Rn . With u ∈ W (O), we extend
it by zero from its original domain to Rn and we denote this extended function by ū. Now,
let W0 (O) be the space of all functions in W (O) defined in such a way that the extension
operator Ŝ, given by
(2.1)
Ŝ : W0 (O) → W (Rn )
u 7→ ū,
f (Rn ) := Ŝ(W0 (O))
is well defined and bounded. Finally, we can define the function space W
endowed with the norm k · kRn . In what follows, to simplify notation, we will write
k · kRn = k · k.
3. Statement of the eigenvalue problem. Let X(Ω) be a complex Hilbert function
space with norm | · |Ω . Let V (Ω) be a closed subspace of X(Ω), with norm k · kΩ , such
that the inclusion V (Ω) ֒→ X(Ω) is continuous. We denote by V0 (Ω) the subspace of V (Ω)
defined as in (2.1).
Consider the eigenvalue problem:
Find µ ∈ C, u 6= 0, u ∈ V0 (Ω), such that
(3.1)
a(u, v) = µb(u, v),
∀v ∈ V0 (Ω),
where a : V (Ω) × V (Ω) → C is a continuous and coercive sesquilinear form and
b : X(Ω) × X(Ω) → C is a continuous sesquilinear form.
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Let T be the linear operator defined by
T : X(Ω) → V0 (Ω) ֒→ X(Ω)
x 7→ u,
where u ∈ V0 (Ω) is the solution of
(3.2)
a(u, y) = b(x, y),
∀y ∈ V0 (Ω).
Since a is elliptic, b is continuous, and V (Ω) ֒→ X(Ω), The Lax-Milgram Lemma allows us
to conclude that T is a bounded linear operator. It is simple to show that µ is an eigenvalue
of (3.1) if and only if λ = 1/µ is an eigenvalue of the operator T and the corresponding
associated eigenfunctions u coincide.
Now, we define the linear operator A by
A : X(Rn ) → Ve (Rn )
x 7→ ū = ŜTŠx.
It is clear that ū|Ω = u, where u ∈ V0 (Ω) is the solution of problem (3.2).
The curved domain Ω is approximated by a family of domains Ωh , h > 0, with polygonal
boundary ∂Ωh . Let Th be a standard partition of Ωh into n-simplices such that each vertex
of ∂Ωh also lies on ∂Ω. The index h denotes, as usual, the mesh size of Th . We assume that
the family {Th } is regular in the sense of the minimal angle condition, i.e., there is a constant
C independent of the choice of Th such that vol(T ) ≥ Cdiamn (T ) for all T ∈ Th , where
vol(T ) denotes the n-dimensional volume of T ; see [2], for instance.
Let Vh (Ωh ) be a finite-dimensional space on Ωh such that Vh (Ωh ) ⊂ V (Ωh ), for all h.
We denote by V0h (Ωh ) the space of all the functions in Vh (Ωh ) defined as in (2.1). Then, we
consider the following discretization of eigenvalue problem (3.1):
Find µh ∈ C, uh 6= 0, uh ∈ V0h (Ωh ), such that
(3.3)
ah (uh , v) = µh bh (uh , v),
∀v ∈ V0h (Ωh ).
In what follows we shall assume that the approximate sesquilinear forms ah and bh are
continuous on V (Ωh ) uniformly in h and that ah is coercive on V (Ωh ) uniformly in h. We remark that, since V0h (Ωh ) 6⊂ V0 (Ω), (3.3) represents a nonconforming approximation to (3.1).
Let us now define the function space Veh (Rn ) := Ŝ(V0h (Ωh )). Then, the discrete analogue of the operator A can be define as follows:
Ah : X(Rn )
x
where uh ∈ V0h (Ωh ) is the solution of
→ Veh (Rn )
7→ ūh : ūh |Ωh = uh ,
ah (uh , y) = bh (x, y),
∀y ∈ V0h (Ωh ).
Once again, because of the Lax-Milgram Lemma, the operator Ah is bounded uniformly in
h. As in the continuous case, it is simple to show that µh is an eigenvalue of problem (3.3) if
and only if λh = 1/µh is an eigenvalue of the operator Ah , and the corresponding associated
eigenfunctions are related by uh = ūh |Ωh .
We end this section by making other assumptions for the sesquilinear forms a and ah .
We assume that the form a(x, y) can be expressed as
(3.4)
a(x, y) = a1 (x|Ω∩Ωh , y|Ω∩Ωh ) + a2 (x|Ω\Ωh , y|Ω\Ωh ),
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A. ALONSO AND A. DELLO RUSSO
where a1 and a2 are continuous bilinear forms on V (Ω ∩ Ωh ) and V (Ω \ Ωh ), respectively.
We also assume that
(3.5)
ah (xh , yh ) = a1h (xh |Ω∩Ωh , yh |Ω∩Ωh ) + a2h (xh |Ωh \Ω , yh |Ωh \Ω ).
Finally, if x, y ∈ V (Rn ), we assume that
(3.6)
a1 (x|Ω∩Ωh , y|Ω∩Ωh ) = a1h (x|Ω∩Ωh , y|Ω∩Ωh )
holds.
4. Spectral approximation. In this section, we present several abstract results on the
approximation of eigenvalues and eigenvectors of non-compact operators defined over curved
domains. These results are obtained by suitable modifications of the theory presented in [4]
and [5]. As a consequence of these modifications, consistency terms arise in the error estimates.
First, we introduce some notation that will be used in the sequel. For further information
on eigenvalue problems we refer the reader to [1]. We denote by ρ(A) the resolvent set of
A and by σ(A) the spectrum of A. For any z ∈ ρ(A), Rz (A) = (z − A)−1 defines the
resolvent operator.
Let λ be a nonzero isolated eigenvalue of A with algebraic multiplicity m. Let Γ be a
circle in the complex plane centered at λ which lies in ρ(A) and which encloses no other
points of σ(A). The continuous spectral projector, E : V (Rn ) → Ve (Rn ), relative to λ, is
defined by
Z
1
E=
Rz (A) dz.
2πi Γ
We assume that the following properties are satisfied:
P1:
lim k(A − Ah )|Veh (Rn ) k = 0.
h→0
P2:
For each function x of E(V (Rn )),
lim kxkΩ\Ωh = 0.
h→0
P3:
For each function x of E(V (Rn )),
inf
lim
h→0
eh (Rn )
x h ∈V
kx − xh k = 0.
P4:
lim k(A − Ah )|E(V (Rn )) k = 0.
h→0
We are going to give an extension of the theory developed in [4] to deal with curved
domains. Most of the proofs of the results stated below are slight modifications of those
in [4]. From now on, C denotes a constant, not necessarily the same at each occurrence, but
always independent of h.
L EMMA 4.1. Let G be a closed subset of ρ(A). Under assumption P1, there exist
positive constants C and h0 , independent of h, such that
k(z − Ah |Veh (Rn ) )−1 k ≤ C,
∀z ∈ G,
∀h < h0 .
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Proof. The proof is identical to that of [4, Lemma 1].
T HEOREM 4.2. Let O ∈ C be a compact set not intersecting σ(A). There exist h0 > 0
such that, if h < h0 , then O does not intersect σ(Ah |Veh (Rn ) ).
Proof. The proof is a direct consequence of assumption P1, as it is shown in [4, Theorem 1].
Therefore, by virtue of the previous theorem, if h is small enough, Γ ⊂ ρ(Ah |Veh (Rn ) )
and the discrete spectral projector, Eh : V (Rn ) → Veh (Rn ), can be defined by
1
Eh =
2πi
Z
Γ
Rz (Ah |Veh (Rn ) ) dz.
Let us recall the definition of the gap δb between two closed subspaces, Y and Z, of
V (Rn ). We define
where
b Z) := max{δ(Y, Z), δ(Z, Y )},
δ(Y,
δ(Y, Z) :=
sup
y∈Y
kyk = 1
to 0.
inf ky − zk .
z∈Z
The following theorem implies uniform convergence of Eh |Veh (Rn ) to E|Veh (Rn ) as h goes
T HEOREM 4.3. Under assumption P1,
lim k(E − Eh )|Veh (Rn ) k = 0.
h→0
Proof. It follows combining Lemma 4.1 with assumption P1 and it is essentially identical
to that of [4, Lemma 2].
T HEOREM 4.4. Under the assumption P1, for all x ∈ Eh (V (Rn )) there holds
lim δ(x, E(V (Rn ))) = 0.
h→0
Proof. It is a direct consequence of Theorem 4.3.
T HEOREM 4.5. Under the assumptions P1 and P3, for all x ∈ E(V (Rn )) holds
lim δ(x, Eh (V (Rn ))) = 0.
h→0
Proof. The proof is identical to that of [4, Theorem 3].
T HEOREM 4.6. Under the assumptions P1 and P3,
b
lim δ(E(V
(Rn )), Eh (V (Rn ))) = 0.
h→0
Proof. It is direct consequence of Theorem 4.4 and Theorem 4.5.
As a consequence of the previous theorems, isolated parts of the spectrum of A are
approximated by isolated parts of the spectrum of Ah ; see [10] and [4]. More precisely, for
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A. ALONSO AND A. DELLO RUSSO
any eigenvalue λ of A of finite multiplicity m, there exist exactly m eigenvalues λ1h , ···, λmh
of Ah , repeated according to their respective multiplicities, converging to λ as h goes to zero.
Next we are going to give estimates which show how the eigenvalues of T are approximated by those of Th . To attain this goal, we extend the theory developed in [5] so that it can
be applied to more general situations where the original and the discrete domains do not coincide. By so doing, consistency terms arise in the error estimates. These consistency terms are
associated with the variational crime committed by approximating the curved boundary with
a polyhedral one. We shall give general expressions for these additional consistency terms.
We begin considering the bounded operator A∗ defined by
A∗ : X(Rn ) → Ve (Rn )
x 7→ ū : ū|Ω = u,
where u ∈ V0 (Ω) is the solution of
a(y, u) = b(y, x),
∀y ∈ V0 (Ω).
It is known that λ̄ is an eigenvalue of A∗ with the same multiplicity m as that of λ. We also
consider the bounded operator A∗h defined by
A∗h : X(Rn ) → Veh (Rn )
x 7→ ūh : ūh |Ωh = uh ,
where uh ∈ V0h (Ωh ) is the solution of
ah (y, uh ) = bh (y, x),
∀y ∈ V0h (Ωh ).
Here, λ̄1h , · · ·, λ̄mh are the eigenvalues of A∗h which converge to λ̄ as h goes to zero.
Let E∗ be the spectral projector of A∗ relative to λ̄. We also assume the following
properties for A∗ and A∗h :
P5:
lim k(A∗ − A∗h )|Veh (Rn ) k = 0.
h→0
P6:
For each function x of E∗ (V (Rn )),
lim kxkΩ\Ωh = 0.
h→0
P7:
For each function x of E∗ (V (Rn )),
inf
lim
h→0
eh (Rn )
x h ∈V
kx − xh k = 0.
P8:
lim k(A∗ − A∗h )|E(V (Rn )) k = 0.
h→0
We now need to introduce other operators. Let Πh : V (Rn ) → V (Rn ) be the projector
defined by the relations
(4.1)
ah (x − Πh x, y) = 0,
(Πh x)|Rn \Ωh = 0.
∀y ∈ V0h (Ωh )
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Because V0h (Ωh ) is a closed subset of V (Ωh ), (Πh x)|Ωh ∈ V0h (Ωh ). Hence, we have
Πh x ∈ Veh (Rn ). Analogously, we define the projector Π∗h : V (Rn ) → V (Rn ) with range
Veh (Rn ) by the relations:
ah (y, x − Π∗h x) = 0,
(Π∗h x)|Rn \Ωh = 0.
(4.2)
∀y ∈ V0h (Ωh )
Since ah is continuous and coercive on V (Ωh ), both uniformly in h, the operators Πh and
Π∗h are bounded uniformly in h. Let us remark that for conforming methods Ah = Πh A.
This is assumed in the spectral approximation theory in [5] and used in the proofs therein.
When variational crimes in the discretization of the domains are allowed, Ah and Πh A do
not coincide.
Let Bh := Ah Πh : V (Rn ) → V (Rn ). Notice that σ(Ah ) = σ(Bh ) and that, for any
non-null eigenvalue, the corresponding invariant subspaces coincide. Let Fh : V (Rn ) →
V (Rn ) be the spectral projector of Bh relative to its eigenvalues λ1h , · · ·, λmh . It can be
proved that kRz (Bh )k is bounded uniformly in h for z ∈ Γ; see [5, Lemma 1]. Consequently,
the spectral projectors Fh are bounded uniformly on h.
Finally, let B∗h := A∗h Π∗h : V (Rn ) → V (Rn ) and let F∗h be the spectral projector
of B∗h relative to λ̄1h , · · ·, λ̄mh . It is easy to show that B∗h is the actual adjoint of Bh with
respect to ah . In fact, for all x and y ∈ V (Rn ), we have
ah (Bh x, y) = ah (Ah Πh x, y) = ah (Ah Πh x, Π∗h y) = bh (Πh x, Π∗h y).
Similarly, we get
ah (x, B∗h y) = bh (Πh x, Π∗h y).
Therefore, the spectral projector F∗h is also the adjoint of Fh with respect to ah .
Let
γh := δ(E(V (Rn )), Veh (Rn )) +
sup
y ∈ E(V (Rn ))
kyk = 1
kykΩ\Ωh .
Properties P2 and P3 imply that γh → 0 as h → 0. Analogously, let
γ∗h := δ(E∗ (V (Rn )), Veh (Rn )) +
sup
y ∈ E∗ (V (Rn ))
kyk = 1
kykΩ\Ωh .
Here, because P6 and P7, γ∗h → 0 as h → 0.
L EMMA 4.7.
k(I − Πh )|E(V (Rn )) k ≤ Cγh ,
k(I − Π∗h )|E∗ (V (Rn )) k ≤ Cγ∗h .
Proof. For a x ∈ E(V (Rn )), we have
(4.3)
k(I − Πh )xk2 = k(I − Πh )xk2Ωh + kxk2Ω\Ωh .
Using that ah is coercive on V (Ωh ) uniformly in h, we have
k(I − Πh )xk2Ωh ≤ Cah ((I − Πh )x, (I − Πh )x) = Cah ((I − Πh )x, x − yh ), ∀yh ∈ V0h (Ωh ),
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where the last equality results from the definition of Πh . Now, taking into account that ah is
continuous on V (Ωh ) uniformly in h, we obtain
k(I − Πh )xkΩh ≤ C
inf
eh (Rn )
y h ∈V
kx − yh k,
which together (4.3) allows us to conclude the proof of the first estimation. An analogous
proof is valid for the second one.
L EMMA 4.8.
k(E − Fh )|E(V (Rn )) k ≤ Ck(A − Bh )|E(V (Rn )) k,
k(E∗ − F∗h )|E∗ (V (Rn )) k ≤ Ck(A∗ − B∗h )|E∗ (V (Rn )) k.
Proof. The proof is identical to that of [5, Lemma 3].
Let
δh := γh + k(A − Ah )|E(V (Rn )) k.
From properties P2, P3 and P4 it is easily seen that δh → 0 as h → 0. Analogously, let
δ∗h := γ∗h + k(A∗ − A∗h )|E(V (Rn )) k.
Given P6, P7 and P8 δ∗h → 0 as h → 0.
L EMMA 4.9.
k(A − Bh )|E(V (Rn )) k ≤ Cδh ,
k(A∗ − B∗h )|E(V (Rn )) k ≤ Cδ∗h .
Proof. Let x ∈ E(V (Rn )) with kxk = 1. We have
k(A − Bh )xk
≤ k(A − Ah )xk + kAh (I − Πh )xk
≤ k(A − Ah )|E(V (Rn )) k + kAh k k(I − Πh )|E(V (Rn )) k
≤ (k(A − Ah )|E(V (Rn )) k + γh ,
where the last inequality follows from Lemma 4.7 and the fact that kAh k is uniformly
bounded with respect to h. An analogous proof is valid for the second estimate of the
Lemma.
Let
Λh := Fh |E(V (Rn )) : E(V (Rn )) → Fh (V (Rn )).
L EMMA 4.10. For h small enough, Λh is a bijection and kΛ−1
h k is bounded uniformly
in h.
Proof. See the proof of [5, Theorem 1].
T HEOREM 4.11.
b h (V (Rn )), E(V (Rn ))) ≤ Cδh .
δ(F
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Proof. The proof is identical to that of [5, Theorem 1].
Let us now define the operators  := A|E(V (Rn )) : E(V (Rn )) → E(V (Rn )) and
n
n
B̂h := Λ−1
h Bh Λh : E(V (R )) → E(V (R )). From these definitions, it follows that  has a
unique eigenvalue λ of algebraic multiplicity m and that B̂h has the eigenvalues λ1h , ···, λmh .
Let us consider the following consistency terms:
Mh =
M∗h =
sup
sup
x ∈ E(V (Rn ))
kxk = 1
y ∈ E∗ (V (Rn ))
kyk = 1
sup
sup
x ∈ E(V (Rn ))
kxk = 1
y ∈ E∗ (V (Rn ))
kyk = 1
Nh =
|ah (Ax, Π∗h y − y) − bh (x, Π∗h y − y)|,
|ah (Πh x − x, A∗ y) − bh (Πh x − x, y)|,
sup
sup
x ∈ E(V (Rn ))
kxk = 1
y ∈ E∗ (V (Rn ))
kyk = 1
|ah (Ax, y) − bh (x, y)|.
T HEOREM 4.12.
k − B̂h k ≤ C δh δ∗h + Mh + M∗h + Nh .
Proof. We have
k − B̂h k
=
sup
x ∈ E(V (Rn ))
kxk = 1
≤ C
(4.4)
= C
= C
≤ C
k(Â − B̂h )xk =
sup
sup
x ∈ E(V (Rn ))
kxk = 1
e (Rn )
y∈V
kyk = 1
sup
sup
x ∈ E(V (Rn ))
kxk = 1
e (Rn )
y∈V
kyk = 1
sup
sup
x ∈ E(V (Rn ))
kxk = 1
e (Rn )
y∈V
kyk = 1
sup
x ∈ E(V (Rn ))
kxk = 1
k(Â − B̂h )xkΩ
a((Â − B̂h )x, y)
a(E(Â − B̂h )x, y)
a((Â − B̂h )x, E∗ y)
sup
sup
x ∈ E(V (Rn ))
kxk = 1
y ∈ E∗ (V (Rn ))
kyk = 1
a((Â − B̂h )x, y).
Since (Â − B̂h )x, y ∈ Ve (Rn ), we can use (3.4) and (3.6) to get
a((Â − B̂h )x, y)= a1h ((Â − B̂h )x|Ω∩Ωh , y|Ω∩Ωh ) + a2 ((Â − B̂h )x|Ω\Ωh , y|Ω\Ωh )
(4.5)
= ah ((Â − B̂h )x, y) + a2 ((Â − B̂h )x|Ω\Ωh , y|Ω\Ωh ).
Now, using that (Λ−1
h Fh − I)A|E(V (Rn )) = 0 and that Bh commutes with its spectral
projector Fh , we obtain
(4.6)
 − B̂h = (A − Bh )|E(V (Rn )) + (Λ−1
h Fh − I)(A − Bh )|E(V (Rn )) .
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Let x ∈ E(V (Rn )) and y ∈ E∗ (V (Rn )), with kxk = kyk = 1. Since Fh (Λ−1
h Fh − I) = 0
and F∗h is the adjoint of Fh with respect to ah , we have
|ah ((Λ−1
h Fh − I)(A − Bh )x, y)|
−1
= |ah ((Λ−1
h Fh − I)(A − Bh )x, y)| − |ah (Fh (Λh Fh − I)(A − Bh )x, y)|
−1
= |ah ((Λ−1
h Fh − I)(A − Bh )x, y)| − |ah ((Λh Fh − I)(A − Bh )x, F∗h y)|
= |ah ((Λ−1
h Fh − I)(A − Bh )x, (I − F∗h )y)|
(4.7)
≤ C kΛ−1
h Fh − Ik k(A − Bh )|E(V (Rn )) k k(I − F∗h )|E∗ (V (Rn )) k ≤ Cδh δ∗h .
The last inequality in (4.7) follows from Lemmas 4.8, 4.9, and 4.10, the fact that ah is continuous on V (Ωh ) independently of h and that Fh is bounded uniformly in h. On the other
hand,
(4.8) ah ((A − Bh )x, y) = ah ((A − Bh )x, Π∗h y) + ah ((A − Bh )x, (I − Π∗h )y).
To bound the second term in the right-hand side of this equation, we use Lemmas 4.7 and 4.9.
We thus obtain
|ah ((A − Bh )x, (I − Π∗h )y)|≤ C k(A − Bh )|E(V (Rn )) k k(I − Π∗h )|E∗ (V (Rn )) k
(4.9)
≤ C δh γ∗h .
For the first term, we have
(4.10) ah ((A − Bh )x, Π∗h y) = ah ((A − Ah )x, Π∗h y) + ah ((Ah − Bh )x, Π∗h y).
Now,
(4.11)
|ah ((A − Ah )x, Π∗h y)| = |ah (Ax, Π∗h y) − bh (x, Π∗h y)| ≤ Mh + Nh ,
and
ah ((Ah − Bh )x, Π∗h y)= ah (Ah (I − Πh )x, Π∗h y) = bh ((I − Πh )x, Π∗h y)
(4.12)
= bh ((I − Πh )x, y) − bh ((I − Πh )x, (I − Π∗h )y).
The first term in the right-hand side of (4.12) can be written as
bh ((I−Πh )x, y) = [ah ((Πh −I)x, A∗ y)−bh ((Πh −I)x, y)]−ah ((Πh −I)x, (I−Π∗h )A∗ y).
(4.13)
Now, the last term of the right-hand side above can be easily bounded by
|ah ((Πh − I)x, (I − Π∗h )A∗ y)| ≤ C k(Πh − I)|E(V (Rn )) k k(I − Π∗h )|E∗ (V (Rn )) k kA∗ k.
(4.14)
Then, Lemma 4.7, (4.13), and (4.14) immediately yield
(4.15)
|bh ((I − Πh )x, y)| ≤ C(M∗h + γh γ∗h ).
Finally, we estimate the last term in (4.5). Using that (Λ−1
h Fh − I) is bounded uniformly
in h, we obtain from (4.6) and Lemma 4.9
|a2 ((Â − B̂h )x|Ω\Ωh , y|Ω\Ωh )| ≤ Ck(A − Bh )|E(V (Rn )) k kykΩ\Ωh
kykΩ\Ωh ≤ Cδh γ∗h .
≤ Cδh
sup
(4.16)
n
y ∈ E∗ (V (R ))
kyk = 1
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Now, the theorem is a consequence of formulae (4.4) to (4.16).
By using the previous theorem, we deduce the following result about the approximation
of the eigenvalue λ:
T HEOREM 4.13.
m
1 X
i)
λih ≤ C δh δ∗h + Mh + M∗h + Nh
λ −
m i=1
1/α
max |λ − λih | ≤ C δh δ∗h + Mh + M∗h + Nh
ii)
i=1,···,m
where α is the ascent of the eigenvalue λ of Â.
Proof. Taking into account that σ(Â) = λ and that λ1h , · · ·, λmh are the eigenvalues of
Pm
B̂h , we have tr(Â) = mλ and tr(B̂h ) = i=1 λih . Then, from the continuity of the traces
m
1
1 X
λih =
|tr(Â) − tr(B̂h )| ≤ Ck − B̂h k.
λ −
m i=1
m
On the other hand, it is known that,
|λ − λih |α ≤ Ck − B̂h k,
for any 1 ≤ i ≤ m. Therefore, we can conclude i) and ii) directly from Theorem 4.12.
R EMARK 4.14. In many applications, the operator A is self-adjoint. In this case, if µ
is a nonzero eigenvalue of A, the ascent α of (µ − A) is one. So, the space of generalized
eigenvectors E(Rn ) coincide with the space of the actual eigenvectors corresponding to µ;
see [1].
5. Example. Let Ω be a bounded three-dimensional domain with a Lipschitz continuous
boundary ∂Ω. We assume that ∂Ω is piecewise smooth, more precisely, is piecewise of class
C 2 . To avoid additional technical difficulties, we will assume that the set of points where the
condition of C 2 - smoothness of ∂Ω is not satisfied consists of a finite number of straight lines
and single points.
Let (·, ·) be the scalar product in L2 (Ω) and let | · | denote the corresponding L2 norm.
Further, H σ (Ω) denotes the standard Sobolev spaces with the usual norms k · kσ and H01 (Ω)
denotes the subspace of functions in H 1 (Ω) satisfying a zero Dirichlet boundary conditions.
We consider the spectral problem:
Given s > 0, find λ ∈ R and u 6= 0 such that
s grad ( div u) − curl curl u = λu
in Ω,
(5.1)
u=0
on ∂Ω.
Let X(Ω) := (L2 (Ω))3 , V (Ω) := (H 1 (Ω))3 and V0 (Ω) := (H01 (Ω))3 . Let a0 and b be
the symmetric bilinear forms defined by
Z
a0 (u, v) :=
curl u · curl v + s div u div v, ∀u, v ∈ V (Ω),
Ω
b(u, v) :=
Z
Ω
u · v,
∀u, v ∈ X(Ω).
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The bilinear form a0 is coercive on V0 (Ω) but is not coercive on V (Ω). However, a := a0 +b
can be used in our problem and it turns out to be coercive on V (Ω). Furthermore, a is
continuous on V (Ω).
λs + 2µs
, the bilinear form a0 (u, v) is associated to the elasR EMARK 5.1. When s =
µs
ticity system for a material of Lamé coefficients λs and µs . Denoting
the material density by
s
λµs
ρs , problem (5.1) gives the vibration eigenfrequencies ω =
of an elastic, homogeρs
neous and isotropic three-dimensional body fixed along its boundary.
The variational formulation of problem (5.1) associated with a is given by:
Find λ ∈ R and u ∈ V0 (Ω), u 6= 0, such that
(5.2)
a(u, v) = (λ + 1) b(u, v),
∀v ∈ V0 (Ω).
It is well known that problem (5.2) has an increasing sequence of finite multiplicity eigenvalues λn > 0, n ∈ N . There is no finite accumulation point. The corresponding L2 (Ω)orthonormal eigenfunctions un belong to V0 (Ω). Now, as in Section 2, we consider the
bounded linear operator T : X(Ω) → X(Ω) defined by Tf = u ∈ V0 (Ω) and
(5.3)
a(u, y) = b(f , y),
∀y ∈ V0 (Ω).
By virtue of the Lax-Milgram Lemma, we have
kukΩ ≤ C kf k0,Ω .
As a consequence of the classical a priori estimates, for any f ∈ X(Ω), u = Tf is known to
satisfy some further regularity. In fact, u ∈ (H 1+r (Ω))3 for r ∈ (1/2, 1] (see [3]) and there
holds
(5.4)
kuk1+r,Ω ≤ C kf k0,Ω .
Now, we consider the bounded linear operator A : X(R3 ) → Ve (R3 ) defined by
Af = ŜTŠf , where Ŝ and Š are the extension and the restriction operators, respectively,
defined in Section 2. Since a and b are symmetric, T is self-adjoint with respect to a. Clearly,
A is also self-adjoint with respect to a. Notice that (λ, u) is a solution of problem (5.2) if
1
1
, u) is an eigenpair of T which, in its turn, is equivalent to ( λ+1
, ū) being
and only if ( λ+1
an eigenpair of A, where ū = Ŝ(u).
Let the curved domain Ω be approximated by a polyhedron Ωh with vertices on ∂Ω. Let
Th be a partition of Ωh , i.e., a set of a finite number of closed tetrahedra T , which has the
following properties:
• each vertex of Ωh is a vertex of a T ∈ Th ,
• each T ∈ Th has at least one vertex in the interior of Ωh ,
′
• any two tetrahedra, T, T ∈ Th share at most a vertex, a whole side, or a whole face.
Let Nh and Eh denote the set of all vertices and the set of all edges in Th , respectively.
We assume that
• Nh ⊂ Ω̄,
• Nh ∩ ∂Ωh ⊂ ∂Ω,
• Eh contains all the points where the boundary ∂Ω is not C 2 ,
• for all T ∈ Th , at most one face of T lies on ∂Ωh .
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We also assume that the family {Th } is regular.
In what follows we will use some notation and definitions introduced in [6]. Consider a
T ∈ Th which has a face ShT ⊂ ∂Ωh , called a boundary tetrahedra. We enumerate the vertices
of T such that the vertices of ShT are numbered first and we denote them by P1T , P2T , P3T , and
P4T , in local notation. Let ΣTh be the part of ∂Ω which is approximated by the face ShT . We
denote by T id the closed tetrahedra with three plane sides, having P4 as a common vertex, and
with one curved side, coinciding with ΣTh , and we call it the ideal tetrahedra associated with
T ∈ Th . For the sake of simplicity, we assume that the partitions Th are such that for each
boundary tetrahedra T , either T ⊂ T id or T ⊃ T id . If we replace all boundary tetrahedra in
Th by their associated ideal tetrahedra T id , we obtain the so-called ideal partition Thid of the
domain Ω.
With the triangulation Th , we consider the finite element spaces
X(Ωh ) := (L2 (Ωh ))3 ,
V (Ωh ) := (H 1 (Ωh ))3 ,
Vh (Ωh ) := {vh ∈ V (Ωh ) : vh |T ∈ (P1 (T ))3
∀T ∈ Th },
and
V0h (Ωh ) := {vh ∈ Vh (Ωh ) : vh |∂Ωh = 0}.
Let ah and bh be the symmetric bilinear forms defined by
Z
Z
u · v,
curl u · curl v + s div u div v +
ah (u, v) :=
∀u, v ∈ V (Ωh ),
Ωh
Ωh
bh (u, v) :=
Z
u · v,
∀u, v ∈ X(Ωh ).
Ωh
Notice that the bilinear form ah is coercive and continuous on V (Ωh ) uniformly in h. Then,
the discretization of the spectral problem (5.2) is given by
Find λh ∈ R and uh ∈ V0h (Ωh ), uh 6= 0, such that
ah (uh , vh ) = (λh + 1) bh (uh , vh ),
∀vh ∈ V0h (Ωh ).
Now, we can define a discrete analogue of A. Let Ah : X(R3 ) → Veh (R3 ) be the
bounded linear operator defined by Ah f ∈ Veh (R3 ) and
ah (Ah f , vh ) = bh (f , vh ),
∀vh ∈ V0h (Ωh ).
It remains to show that the bilinear forms a and ah satisfy the assumptions (3.4), (3.5),
and (3.6). To that end, let ω be a closed subset of Ω ∪ Ωh and consider the bilinear form
Z
Z
u · v, ∀u, v ∈ (H 1 (ω))3 ,
curl u · curl v + s div u div v +
aω (u, v) :=
ω
ω
1
3
Thus, noting that aω is continuous on (H (ω)) uniformly in h, it suffices to take
a1 (u|Ω∩Ωh , v|Ω∩Ωh ) = a1h (u|Ω∩Ωh , v|Ω∩Ωh ) = aΩ∩Ωh (u, v),
a2 (u|Ω\Ωh , v|Ω\Ωh ) = aΩ\Ωh (u, v),
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a2h (u|Ωh \Ω , v|Ωh \Ω ) = aΩh \Ω (u, v).
In order to prove properties P1, P2, P3, and P4 for this problem, we establish the following lemmas and definitions.
L EMMA 5.2. There exists a positive constant C such that:
kvk0,Ω\Ω̄h ≤ Chσ kvkσ,Ω
∀v ∈ (H σ (Ω))3 ,
kvk0,Ωh \Ω̄ ≤ Chσ kvkσ,Ωh
∀v ∈ (H σ (Ωh ))3 ,
0 ≤ σ ≤ 1,
0 ≤ σ ≤ 1.
Proof. By adapting the arguments used in the proof of [6, Lemma 3.3.11] for the threedimensional case, the inequalities can be proved for σ = 1. Since the two inequalities are
clearly true for σ = 0, they follow for 0 < σ < 1 from standard results on interpolation in
Sobolev spaces.
D EFINITION 5.3. Let wh ∈ V0h (Ωh ). A function ŵ ∈ V0 (Ω) is called associated with
wh if it has the following properties:
• ŵ ∈ C 0 (Ω̄),
• ŵ(Pi ) = wh (Pi ), ∀Pi ∈ Nh ,
• ŵ is linear on each tetrahedra T ∈ Th ∩ Thid ,
• if T ⊂ T id , ŵ = 0 on T id \ T and ŵ = wh on T ,
• if T id ⊂ T , ŵ|∂T id ⊂∂Ω = 0.
The definition above is due to Feistauer and Ženı́šek; see [6]. The construction of such
a function follows basically from the interpolation theory developed to Zlámal [19] for twodimensional curved finite elements. The extension of his ideas to the three-dimensional case
is relatively straightforward so we do not include the details here.
L EMMA 5.4. Let ŵ ∈ V0 (Ω) be associated with wh ∈ V0h (Ωh ). Let T id ∈ Thid lie
along ∂Ω and let T ∈ Th be its approximation. If T id ⊂ T , then
kŵ − wh kT id ≤ C hkwh kT ,
where C is a constant independent of h.
Proof. The proof is a consequence of Definition 5.3 and a suitable extension of [19,
Theorem 2].
In what follows, we will use smooth extensions of functions originally defined in Ω. We
denote by ϕe an extension of ϕ from H σ (Ω), σ > 0, into H σ (R3 ) satisfying ϕe ∈ H σ (R3 )
and
kϕe kσ,R3 ≤ Ckϕkσ,Ω ;
(5.5)
see [7], for instance.
Let f ∈ Veh (R3 ) and define ū := Af and ūh := Ah f .
L EMMA 5.5. There exists a positive constant C such that
kū − ūh k ≤ C
inf
kvh − ue kΩh
vh ∈V0h (Ωh )
+
|ah (ue − uh , wh )|
+ kukΩ\Ωh + kue kΩh \Ω .
kwh kΩh
wh ∈V0h (Ωh )
sup
Proof. We have
kū − ūh k2 = kū − ūh k2Ω∪Ωh = ku − uh k2Ω∩Ωh + kuk2Ω\Ωh + kuh k2Ωh \Ω .
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Now, let vh be an arbitrary element in the space V0h (Ωh ). We can write
ku − uh k2Ω∩Ωh ≤ 2 (ku − vh k2Ω∩Ωh + kvh − uh k2Ω∩Ωh ),
and
kuh kΩh \Ω ≤ kvh − uh kΩh \Ω + kvh kΩh \Ω .
By using the uniform coerciveness and continuity of the bilinear form ah , we obtain
Z
Z
2
2
2
|vh − uh |2
| curl (vh − uh )| + s| div (vh − uh )| +
αkvh − uh kΩh ≤
Ωh
ZΩh
curl (vh − ue ) · curl (vh − uh ) + s div (vh − ue ) div (vh − uh )
≤
Z Ωh
curl (ue − uh ) · curl (vh − uh ) + s div (ue − uh ) div (vh − uh )
+
Z
ZΩh
e
(ue − uh )(vh − uh )
(vh − u )(vh − uh ) +
+
Ωh
Ωh
≤ C kvh − ue kΩh kvh − uh kΩh + ah ((ue − uh ), (vh − uh )) ,
from which we deduce
kvh − uh kΩh ≤ C
e
kvh − u kΩh
|ah (ue − uh , wh )|
+
sup
kwh kΩh
wh ∈V0h (Ωh )
!
.
On the other hand,
ku − vh kΩ∩Ωh = kue − vh kΩ∩Ωh ,
and
kvh kΩh \Ω ≤ kvh − ue kΩh \Ω + kue kΩh \Ω .
Combining the above inequalities, we conclude the proof.
We now estimate the terms appearing in the right-hand side of the inequality in Lemma 5.5.
In the sequel, we shall assume that r is the constant appearing in equation (5.4).
L EMMA 5.6. There exists a positive constant C such that
inf
vh ∈V0h (Ωh )
kue − vh kΩh ≤ C hr kuk1+r,Ω .
Proof. Since ue ∈ (H 1+r (R3 ))3 , ue ∈ (C 0 (R3 ))3 . Therefore, Lue , the Lagrange linear
interpolant of ue |Ωh , is well defined; see [2], for instance. By using standard interpolation
results, we have
kue − Lue kΩh ≤ Chr kue k1+r,Ωh .
Observe that Lue ∈ V0h (Ωh ) although ue |∂Ωh 6= 0. Then, using the estimate (5.5), we
conclude the proof.
L EMMA 5.7. There exists a positive constant C such that
|ah (ue − uh , wh )|
≤ C hr kf k.
kw
k
h Ωh
wh ∈V0h (Ωh )
sup
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A. ALONSO AND A. DELLO RUSSO
Proof. For any function wh ∈ V0h (Ωh ), we have
ah (ue − uh , wh ) =
=
Z
ΩZh
+
Z
curl (ue − uh ) · curl wh + s div (ue − uh ) div wh
(ue − uh ) · wh
Ωh
curl ue · curl w̄h + s div ue div w̄h + ue · w̄h −
Z
ZΩ∪Ωh
curl u · curl w̄h + s div u div w̄h + u · w̄h
=
Z
ZΩ
e
e
e
curl u · curl wh + s div u div wh + u · wh −
+
Ωh \Ω
f · wh
Ωh
f · wh .
Ωh
The last three terms can be easily bounded. In fact, by using the Cauchy-Schwarz inequality,
Lemma 5.2 and estimate (5.5), we obtain
Z
curl ue · curl wh ≤ C k curl ue k0,Ωh \Ω k curl wh k0,Ωh \Ω
Ωh \Ω
(5.6)
≤ C hr kue k1+r,Ωh kwh kΩh ≤ C hr kuk1+r,Ω kwh kΩh ,
Z
div ue div wh ≤ C k div ue k0,Ωh \Ω k div wh k0,Ωh \Ω ≤ C hr kue k1+r,Ωh kwh kΩh
Ωh \Ω
≤ C hr kuk1+r,Ω kwh kΩh ,
Z
ue · wh ≤ C kue k0,Ωh \Ω kwh k0,Ωh \Ω ≤ C h2 kue k1+r,Ωh kwh kΩh
Ωh \Ω
(5.7)
≤ C h2 kuk1+r,Ω kwh kΩh .
(5.8)
We are going to estimate the remainder terms. To this end, we need to introduce some notation. We denote by ωT the domain bounded by ΣTh and ShT , with ΣTh ⊂ ∂Ω being the curved
side of an ideal tetrahedra and with ShT ⊂ ∂Ωh being the corresponding side of the associated
tetrahedra T ∈ Th . Now, we consider a function ŵ = (ŵ1 , ŵ2 , ŵ3 ), with ŵi , i = 1, 2, 3, as
defined in Definition 5.3. Since ŵ ∈ V0 (Ω), we may take it as a test function in (5.3). Then,
we can obtain
Z
Z
curl u · curl (ŵ − w̄h ) + s div u div (ŵ − w̄h ) + u · (ŵ − w̄h )
f ŵ −
Ω Z
Ω f · wh −
Z
Z
ΩhX
curl u · curl (ŵ − w̄h ) + s div u div (ŵ − w̄h )
f ŵ −
=
T ∈ Thid
∂T
Z ∩ ∂Ω 6= ∅
T
u · (ŵ − w̄h ) −
+
T
T
X
T ∈ Th
∂T ∩ ∂Ωh 6= ∅
Z
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=
+
Z
X
f · (ŵ − wh ) −
T
T ∈ Thid
Z∂T ∩ ∂Ω 6= ∅
s div u div (ŵ − w̄h ) −
T
≤C
≤C
h
h
X
T ∈ Th
T
curl u · curl (ŵ − w̄h )
T
u · (ŵ − w̄h ) −
X
T ∈ Th
∂T ∩ ∂Ωh 6= ∅
Z
ωT
f · wh i
kf k0,T kŵ − w̄h k0,T + kukT kŵ − w̄h kT + kf k0,Ωh \Ω kwh k0,Ωh \Ω
T ∈ Thid
∂T ∩ ∂Ω 6= ∅
X
Z
Z
i
hkf k0,T kwh kT + hkukT kwh kT + h2 kf k1,Ωh kwh k1,Ωh
i
h ∂T
∩ ∂Ωh 6= ∅ ≤ C h kf k + kukΩ + h2 kf k kwh kΩh ,
where we have used Lemmas 5.2 and 5.4. We conclude the proof combining the previous
inequality with (5.6), (5.7), (5.8), and the estimate (5.4).
T HEOREM 5.8. (P1) There exists a positive constant C such that
k(A − Ah )f k ≤ C hr kf k ∀f ∈ Veh (R3 ).
Proof. It is an immediate consequence of the previous lemmas.
T HEOREM 5.9. (P2) For each eigenfunction ū of A associated to λ, there exists a strictly
positive constant C such that
kūkΩ\Ωh = kukΩ\Ωh ≤ C hr kuk1+r,Ω .
Proof. It is an immediate consequence of the regularity of the eigenfunctions u of the
operator T and Lemma 5.2.
T HEOREM 5.10. (P3) For each eigenfunction ū of A associated to λ, there exists a
strictly positive constant C such that
inf
vh ∈V0h (Ωh )
kū − v̄h k ≤ C hr kuk1+r,Ω .
Proof. We have
kū − v̄h k2
= kū − vh k2Ωh + kuk2Ω\Ωh ≤ kū − ue k2Ωh + kue − vh k2Ωh + kuk2Ω\Ωh
≤ kue k2Ωh \Ω + kue − vh k2Ωh + kuk2Ω\Ωh ≤ h2r kuk21+r,Ω ,
which follows because the regularity of the eigenfunctions u of the operator T, estimate (5.5)
and Lemma 5.6. Then, taking the infimum with respect to vh ∈ V0h (Ωh ), we can conclude
the proof.
By virtue of the previous theorems, the spectrum of Ah furnishes the approximations of
the spectrum of A as we stated in Section 3.
T HEOREM 5.11. (P4) There exists a positive constant C such that
k(A − Ah )|E(V (R3 )) k ≤ C hr .
Proof. For x ∈ E(V (R3 )), ū = Ax ∈ H 1+r (R3 ). Then, the proof runs identically to
that of Theorem 5.8 .
Observe that, since A and Ah are self-adjoint, properties P5, P6, P7 and P8 are equally
valid.
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Now, we are going to estimate the consistency terms appearing in Theorem 4.13. Notice
that Mh and M∗h also coincide because of the symmetry of ah and bh .
L EMMA 5.12. There exists a positive constant C such that
Mh =
|ah (Ax, Πh y − y) − bh (x, Πh y − y)| ≤ C h2r ,
sup
sup
3
x ∈ E(V (R ))
kxk = 1
3
y ∈ E(V (R ))
kyk = 1
with Πh being the projection onto Veh (R3 ) with respect to ah , defined by equation (4.1).
Proof. Let x ∈ E(V (R3 )), with kxk = 1, and put w = ŠAx. According to the
definition of A, we have w = ŠŜTŠx = TŠx. From (5.4) we know that w ∈ (H 1+r (Ω))3
and
kwk1+r,Ω ≤ C kxk = C.
Now, taking v ∈ D(Ω) as test functions in (5.3), it can be shown that w is the solution of the
following strong problem
s grad ( div w) − curl curl w + w = x
w=0
in Ω,
on ∂Ω.
Let us denote by w̄ the extension of w by zero from Ω to R3 . Let y ∈ E(V (R3 )) with
kyk = 1, and take vh = Πh y − y. Integrating by parts, we obtain
Z
|ah (w̄, vh ) − bh (x, vh )| = curl w · curl vh + s div w div vh + w · vh
ZΩ\Ωh
x · vh +
Ω\Ω
h
≤ C |w|1,Ω\Ωh |vh |1,Ω\Ωh + kwk0,Ω\Ωh kvh k0,Ω\Ωh
+kxk0,Ω\Ωh kvh k0,Ω\Ωh
≤ C h2r kwk1+r,Ω kyk1+r,Ω + h2r kwkr,Ω kykr,Ω
+h2r kxkr,Ω kykr,Ω .
In the last inequality, we have used that vh |Ω\Ωh = −y|Ω\Ωh and the estimate in Lemma 5.2.
Finally, we can conclude the proof using estimate (5.4).
L EMMA 5.13. There exists a positive constant C such that
Nh =
|ah (Ax, y) − bh (x, y)| ≤ C h2r .
sup
sup
3
x ∈ E(V (R ))
kxk = 1
3
y ∈ E(V (R ))
kyk = 1
Proof. It is identical to that of the previous lemma by substituting vh by y.
T HEOREM 5.14. There exists a positive constant C such that
max |λ − λih | ≤ Ch2r .
i=1,···,m
Proof. It is a immediate consequence of the properties P2, P3, P4, and the previous
lemmas.
ETNA
Kent State University
http://etna.math.kent.edu
SPECTRAL APPROXIMATION OVER CURVED DOMAIN
87
Acknowledgments. The authors wish to thank Rodolfo Rodrı́guez for many valuable
comments on this paper.
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