Singularities of Corner Problems and
Problems of Corner Singularities
MoniqueDAUGE
Resume.
Dans le cadre des problemes aux limites elliptiques du second ordre, nous presentons l'essentiel de la structure des solutions singulieres aupres des sommets et ar^etes d'un polyedre, et nous evoquons les dicultes inherentes a la presence de ces singularites.Mots cles.
Coin, Ar^ete, Sommet, Probleme aux limites elliptique, Singularite.Abstract.
In the framework of second order elliptic boundary value problems, we describe the main features of the singular solutions near vertices and edges of a polyhe-dron, and we discuss some diculties arising from the presence of these singularities.Keywords.
Corner, Edge, Vertex, Elliptic Boundary Value Problem, Singularities.AMS Subject Classication.
35J, 65N, 73C1 Non-smooth domains and domains with corners
At our natural observation scale, \corners" are everywhere: they belong to sharp or reen-trant edges, to cracks fronts, they are also the meeting points of several edges. At a micro industrial scale, the constituting parts of electronic components have many edges and sharp corners, combined moreover with very large (or small) aspect ratios. At the material structure scale, alloys and ceramic defaults look like micro-cracks. But, at
Cracks
Figure 1: Polyhedral domains and cracks
A mathematical domain is an open subset inRnwhere partial dierential equations
are set, and represent an idealized subset of the physical world. A smooth domain is then a domain whose boundary can be described by a nite number of smooth maps. Thus a non-smooth domain can have innitely various descriptions. Here, we will concentrate on polyhedral geometries, letting aside purely Lipschitz domains and domains with cusps. A Lipschitz domain is described by bi-Lipschitz maps and can be very pathologic, containing an innite number of sides and corners for example. For cuspidal domains (where there are piecewise smooth faces, but tangent in certain points), we refer to 35], 17], 37, 38].
So, to x ideas, we consider three-dimensional polyhedral domains . To such a
domain belong faces
F
, edgesE
and verticesv
, where is locally dieomorphic to ahalf-space R2R+, a wedge R;E or a polyhedral cone ;
v respectively. The wedge is
the product ofRby a plane sector ;E and the polyhedral cone ;
v is an innite cone with
vertex 0 and plane faces.
We end this section by the following remark: in general, polyhedral domains are the limit case of real domains where edges and corners are replaced with neighboring regions where the curvature of the boundary is very large. Though being in principle smooth domains, these more realistic geometries will create at least similar diculties
than polyhedra, see Maz'ya, Nazarov, Plamenevskii33].
2 Elliptic boundary value problems
Here, we do not consider time-dependent problems and we only quote as selection in the
literature Kozlov25] for parabolic problems, and Lebeau 30] for hyperbolic problems.
We concentrate on steady or periodic states, governed for example by continuum mechanics, uid mechanics or electromagnetics. In a rst approximation, the equations arising from physical laws in these situations can be written as a linear elliptic boundary value problem, and, even, as a coercive variational problem.
Thus, to x ideas, we consider elliptic boundary value problems generated by a variational formulation based on a subspace
V
ofH
1() and a coercive integro-dierential forma
of order 1:u
2V
8v
2V a
(uv
) =f
(v
):
(1)
Here are our basic assumptions
V
=fv
2H
1()v
= 0 on@
Dg, where@
D is a part of@
. We assume that theintersection of
@
D with each faceF
of@
is either empty, or the whole face or a polygonal subset ofF
.The form
a
can be written asa
(uv
) = X jj1X
jj1 Z
a
(x
)@
xu @
xv dx
where the coecients
a
are \polyhedral piecewise smooth" in the sense that there exists a nite polyhedral partition (j)j2J of such that alla
areC 1(
The right hand side
f
is a continuous bilinear form onV
.The assumption of coerciveness is that there exists
c >
0 such that for allx
2 and all 2R3there holdsX
jj1 X
jj1
a
(x
)c
jj2
:
Thus the boundary value problem corresponding to (1) is a transmission problem (if we
make the more usual assumption that the coecients
a
are smooth on , we obtain theboundary value problem consisting in the rst two lines of (2))
8 < :
Lu
=f
in jj
2J
Bu
= 0 on@
u
] =Nu
] = 0 on@
jn@
,(2)
where
the operator
L
is equal to;@
a
(x
)@
x in each j,the boundary operator
B
is the Dirichlet operator (simple trace) in@
D and theNeumann operator
N
associated witha
in@
n@
D,the expressions
u
] andNu
] are the jumps ofu
andNu
across the interior interfacescontained in
@
j n@
. Note that the expressions ofN
are in general dierent ateach side of an interface. Examples for such problems are
1. The equation for an electric potential in an heterogeneous material (
"
j is the electric permittivity of its homogeneous part j) associated with the bilinear forma
(uv
) =Xj
Z
j
"
jru
rv dx
foruv
2H
10():
Here
N
="
j@
n, withn
a unit normal derivative onF
.2. The equation for the displacement of an elastic multi-material structure (
A
j is the rigidity matrix of the material j) associated with the bilinear forma
(uv
) =Xj
Z
j
A
je
(u
) :e
(v
)dx
foruv
2V
with
e
the strain tensor and : indicates the contraction of tensors. HereN
is the traction operator jn
, where j =A
je
(u
). The part@
D is the clamped part ofthe structure.
Indeed, this framework can be widely extended (general Agmon-Douglis-Nirenberg's type 1] transmission problem of arbitrary order, with a coerciveness assumption) so that there holds similar results to those we will describe. See also Grisvard20] for oblique
deriva-tives problems in polygons, to Nicaise 41] for polygonal transmission problems and to
3 Problematics
In any subdomain 0such that 0is contained in one of the
j, the elliptic regularity holds for solutions of problem (1) (shift theorem). That means that if
f
belongs toH
s;1()with
s >
0, thenu
belongs toH
s+1(0). Such a regularity result still holds if 0intersects@
and the@
j only in their smooth parts. Then, iff
is piecewiseH
s;1, i.e. iff
j j2
H
s ;1(j)
8j
2J
denoted byf
2PH
s;1
()
(3)
then
u
2PH
s+1(0). Now, if 0 intersects any edge or vertex of or of one of the
j, such a result does not hold anymore in general. We will see later on that it holds for small enough
s
, the upper bound being explicit in a certain sense.Let us point out that for Lipschitz domains and the Dirichlet or Neumann boundary value problems associated with the standard Laplace operator the general bound is known: this is 12, see 22, 23]. Of course such a result is coherent with what can be obtained for polyhedral domains. But the corresponding bound can be much smaller as soon as transmission problems are concerned.
The determination of the upper bound for regularity is closely related to the spec-trum of what we call Mellin symbols along edges and at vertices. Moreover, the eigenfunc-tions of these symbols generate the singular funceigenfunc-tions which are precisely the default for the shift theorem to hold.
4 Mellin symbols
For the transmission problem (2) associated with the partition (j)j2J the set of vertices
is the union of the vertices of all j and the set of edges is the union of the edges of all j. Of course these sets reduce to the vertices and edges of respectively if there is no transmission condition inside and if there is no transition of boundary conditions inside the faces of .
4.1 Vertices
Let
v
be a vertex in the previous sense. We assume that there exists a local chart which transforms in a neighborhood ofv
, and the j havingv
as vertex into polyhedral cones ;v and ;vj. By this local chart, problem (2) is transformed into a similar problem in thecone ;v. In polar coordinates (
#
), the operatorsL
,B
andN
can be formally expandedin increasing powers of
according to:8 > > > > > > > < > > > > > > > :
L
=;2L(
#
@
@
#) + Xp1
;2+pLp(
#
@
@
#)B
=;degBB(
#
@
@
#) + Xp1
;degB+pBp(
#
@
@
#)N
=;1N(
#
@
@
#) + Xp1
;1+pNp(
#
@
@
#):
X G
E
G G
Figure 2: Associated solid angles To the operatorsL,B and N are associated their Mellin symbols
8 > > < > > :
C 3
7;! L(#
@
#) =:L() C 3 7;! B(#
@
#) =:B() C 3 7;! N(#
@
#) =:N():
(5)
The denomination \Mellin symbols" comes from the Mellin transform
M
u
]() = Z1
0
;
u
(
#
)d
which satises
ML
u
]() =L()Mu
]():
Let
G
v be the solid angle intercepted by ;v on the sphereS2 (so that there holds ; v = f
x
2R3#
2G
v
g). Then the operator valued Mellin symbol (also called operator pencil
in the Russian literature) associated with problem (2) at the vertex
v
is the underlying operator to problemP
v
]8 < :
L(
)U = F inG
vjj
2
J
vB(
)U = 0 on@G
v
U] = N(
)U] = 0 on@G
vj n
@G
v
:
Here
G
vj is the part ofG
intercepted by the sub-cone ;vj corresponding to thesub-domain j for any
j
such thatv
is a vertex of j (which denes the subsetJ
v ofJ
).Figure 3: Associated solid angle to a non-Lipschitz corner
Lemma 4.1
Problem Pv
] is uniquely solvable inH
1(G
v) for all
2C nS(
v
) where Sv
] is a discrete set (the spectrum). The set Sv
] has the following property: in all stripof the form Re
2ab
], withab
2R, the number of elements ofSv
] is nite. Moreoverthe solution operator R
v
] of problem Pv
] is meromorphic in 2 C, the set of itspoles is S
v
] and its polar part in any 2Sv
] has a nite range.4.2 Edges
Let
e
be a point in an edgeE
. We assume that there exists a local chart which transformsin a neighborhood of
e
, and the j havinge
in their boundaries into wedges R;e
and R;
ej. By this local chart, problem (2) is transformed into a similar problem in the
wedge R;
e. In cylindrical coordinates (
rz
), withz
2Rand (
r
) polar coordinatesin the sector ;e, the operators
L
,B
andN
can be formally expanded in increasing powersof
r
according to:8 > > > > > > > < > > > > > > > :
L
=r
;2L(
r@
r@
) + Xp1
r
;2+pLp(
r@
r@
@
z)B
=r
;degBB(
r@
r@
) + Xp1
r
;degB+pBp(
r@
r@
@
z)N
=r
;1N(
r@
r@
) + Xp1
r
;1+pNp(
r@
r@
@
z):
(6)
To the operatorsL,BandN are associated their Mellin symbols in
r
, similarly to (5):r@
ris changed into
. The important point is that the tangential derivative@
z along the edge does not appear in these symbols. WithG
eandG
ej the angular intervals correspondingdierential operator)
P
e
]8 < :
L(
)U = F inG
ejj
2
J
eB(
)U = 0 on@G
e
U] = N(
)U] = 0 on@G
ej n
@G
e
:
And similarly to the vertices, there holds
Lemma 4.2
The solution operator Re
] of problem Pe
] is meromorphic, the set ofits poles is a discrete subset S
e
] ofC and its polar part in any2Se
] has a nite range.The intersection of the set S
e
] with any strip of the form Re2ab
] is nite.5 Singularities and regularity at conical points
In order to explain the basic ideas which lead to regularity and expansion results, let us, only in this section, consider the case where Rnhas only one singular point (vertex)
in
v
= 0, in the neighborhood of which it coincides with a cone ;v with (smooth) solidangle
G
v. We assume moreover thatL
is homogeneous with constant coecients andB
is the Dirichlet operator. Let us consider
u
2H
10() such thatLu
=f
2H
s ;1().By a cut-o procedure, we can restrict ourselves to the case when
u
inH
10(;v) withcompact support is such that
Lu
2H
s;1(;
v). Applying the Mellin transform to the
equation
2Lu
=2
f
=:g
, we obtainR
v
](Mg
]()) =Mu
]() 8 Re1;n
2
:
(7)Both sides are analytic functions for Re
<
1;n2. Since, because of the regularity off
,the Mellin transform of
g
is dened (as a meromorphic function) for Res
+ 1; n2,formula (7) yields a meromorphic extension U of M
u
] for Re21;n2s
+ 1;n2].IfU has no pole on the straight line Re
=s
+ 1;n2, letu
0 be the inverse Mellintransform ofU on this line:
u
0() = 12Z + 1 ;1
+i U(+i
)d
with =s
+ 1;n
2
:
(8)An extension of the residue formula to the unbounded contour Re
= 1;n2 and Re=s
+ 1;n2 yields, see Kondrat'ev24]:u
0;u
=X
1; n 2<Re
0<s+1 ;
n 2
Res
=0
Rv
](Mg
]()):
(9)
Note that the proof of this formula relies on a priori elliptic estimates for the operator
L
with its boundary conditions in a smooth part of the cone ;v (away from the corner).
If
is a smooth cut-o function which is equal to 1 in a neighborhood of 0 and equal to 0 away from a larger neighborhood where coincides with ;v, as a consequence of (8)It remains to analyze the residues in formula (9). The
0 occurring in this formula are either poles of the solution operator Rv
] (0 2Sv
]) or poles of the Mellin transformof the right hand side M
g
](): this latter poles are located on the only integersd
2and the corresponding residues are the Taylor part (homogeneous of degree
d
) ofg
in0. As a consequence of Lemma 4.1, the right hand side of (9) involves a nite set in C
(
0 2Sv
]N2, with N2 the set of integers 2). Moreover, for each0 2Sv
]N2, thecorresponding residue lives in a nite dimension space. We have to investigate the nature of its elements.
The residues in
0 have the general form QX
q=0
0logq
Uq(#
)(11)
with, in the particular situation of this section, Uq 2
H
10(G
v)\C 1(
G
v).
If
0 62 N, as Re0< s
+ 1; n2, any non-zero termu
v0 (11) satises
u
v 0
62
H
s+1() (although satisfyingL
(u
v 0)
2 C
1() and the boundary conditions). All
these terms are singular functions and contribute to the singular part.
If
0 2 N, some of the residues can be homogeneous polynomials (in cartesianvariables of course) of degree
0 and do not contribute to the singular part. That is the reason for the introduction of the following denitions.Denition 5.1
i) For any
02Sv
]N2, we denote byu
kv
0
fork
= 1:::K
v0
a basis of non-polynomial residues Res=0
R
v
]F() when F spansmeromor-phic functions with poles at integers and corresponding Taylor polar part, with the convention that
K
v0 = 0 if all residues are polynomials.
ii) We denote by e
S
v
] the subset of 0 2Sv
]N2 such thatK
v0
6
= 0. iii) We denote by
v the least real part>
1;n2 of
0 2 e Sv
].Thus,
v is the least real part of active singular functions. The main result is thenTheorem 5.2
Letu
2H
10() be such thatLu
=f
2H
s;1(). If the straight line
Re
=s
+ 1;n2 does not meet the pseudo-spectrum eS
v
], there holds:a) If
s <
v, thenu
2
H
s+1().b) If
s >
v, thenu
2
H
+1() for all<
v and there exist coecients
k
v such that
u
;X
2 e Sv]
1; n 2<Re<s+1 ; n 2 Kv X k=1
kv
u
k
v
2
H
s+1():
With the strongest assumption \the straight line Re
0 =s
+ 1; n2 does not meet Sv
]N2", the previous statement is not too dicult to prove. But the assumption ofthe theorem hides serious diculties, which have to be overcome since this assumption is exactly the right one (necessary and sucient condition to have the result) and allows to recover integer Sobolev exponents in general in dimension
n
= 2, see 16].Denition 5.1 seems very complicate, due to the possible interaction between poly-nomials and poles of the solution operator R
v
] of the operator valued symbol Pv
].But, in a generic way, the situation is much simpler. We have rst to explain what we mean by generic.
The setsS
v
] or eS
v
] depend onL
, the nature of boundary conditions and the shapeof
G
v. Indeed, they depend continuously on a family of parameters which are living in anite (if
n
= 2) or innite dimensional domain. We will say that a situation is generic if it occurs only in a closed subset of measure zero in the parameter domain. In this sense:Remark 5.3
In a generic way there holds1. If
0 2 Sv
] and 0 62N, the pole of Rv
] in 0 is of order 1 and the polar partis an operator of rank 1, whose range is generated by a basis Uv
0 of the kernel
of P
v
0]. The associated singularity space is also of dimension 1, generated byu
v0 =
0U v
0(
#
).2. If
n
= 2, Sv
] = eS
v
] (this fails only when the opening ofG
v is 2
, i.e. if has acrack).
3. If
n
3,e
S
v
] = Sv
]N2, and for 0 2N2 the associated singular function spacehas the same dimension as the space of homogeneous polynomials of degree
0;2and all its elements have the form
0U(#
).6 Regularity results
Let us go back to the situation of section 4. From the solution operators R
v
] andR
e
] we dene the pseudo-spectrae
S
v
] and eS
e
] exactly as in Denition 5.1 and wedenote (
v the least real part>
;12 of
0 2 e Sv
], e the least real part>
0 of02 e S
e
].(13)
Then there holds
Theorem 6.1
If for all verticesv
,s <
v+1
2 and for all points
e
in any closed edgeE
,s <
e, then there holds the following regularity result: iff
2
PH
s;1(), see (3), then the
solution
u
of problem (1) belongs toPH
s+1().We refer to Maz'ya, Plamenevskii 35, 34] for sharp regularity results in
L
pThis statement can be extended to any three-dimensional domain (possibly decom-posed into sub-domains) having edges in the sense of section 4 and also vertices which are conical points. The requirement is that there exists a smooth local map which transforms a neighborhood of such a vertex into homogeneous cone(s) and that the transformed oper-ators can be expanded like in (4). In the situation of Figure 4 (an ordinary cone, a hollow
Figure 4: Conical points
cone or two materials lling two nested cones), Dirichlet or Neumann conditions associated with a general integro-dierential form as in section 2 satisfy the required conditions.
Thus it can be important to have a lower bound for
v and e. ForL
= ;,v is
always larger than 0, see 16, Ch. 19] for instance. For Dirichlet boundary conditions and
L
the Lam e system or the Stokes system, we have v>
0 also, see Kozlov, Maz'ya,Schwab28, 29], and for mixed boundary conditions, seeKozlov, Maz'ya, Rossmann
27, 26].
Concerning
e, forL
=; with Dirichlet or Neumann boundary conditions, we have
e>
1
2 for any Lipschitz sector, and
e=1
2 if ;e is the sector of opening 2
. Combiningthis with the previous estimates on
vwe obtain that for a Lipschitz polyhedron the upperbound on
s
for the regularity result to hold is>
12, compare with the Lipschitz theory Jerison, Kenig 22, 23]. For Lam e and Stokes systems with Dirichlet conditions, we still have e
1 2.
But the situation can be very dierent when there are transmission conditions. Let us give the example of the equation of the electrostatic potential associated with the form
a
(uv
) = Pj
R
j
"
jr
u
rv dx
, with"
j>
0 and exterior Dirichlet conditions. We givein the following Table 1 the optimal minima for
e according to the number of materialsjoining at
e
and according to the opening of the full sector ;e (exterior angle), see 15] forthe proofs. The mention \none" means that the cone ;e lls the whole space R2.
Exterior Angle
1 material 2 materials 3 materials 4 materials2 2 1 0 0
Convex 1 12 0 0
Any 12 14 0 0
None 1 12 14 0
In Figure 5 we indicate the situations which produce the lower bound: two dierent values for the
"
j (materialized by the two colors in the drawings) are enough but the ratio between these two values has to be very large or very small.Figure 5: Limit situations for 1, 2, 3 or 4 materials
7 Singularities
We already gave in Theorem 5.2 the expansion into regular and singular parts at a conical point. Let us go back to the framework of section 4. At vertices
v
the singular functions(
#
)7;!u
k vare dened in the sense of Denition 5.1 (with
n
= 3) and at any edge pointe
, for anybelonging to the pseudo-spectrum e
S
e
] of the operator valued Mellin symbol Pe
], thesingular functions
(
r
)7;!u
k eare also dened in the sense of Denition 5.1 (with
n
= 2).7.1 Straight polyhedra and constant coecients
In this subsection, we will present (in a simplied framework) the results of 16] concerning the expansion into regular and singular parts in a polyhedron. So, we assume
8 > < > :
i)The domains and j are polyhedra (i.e. with plane faces),
ii)The form
a
has piecewise constant coecients (constant in each j), iii)The singular functionsu
kv and
u
k
e have no logarithmic term.
(14)
As a consequence of the points i) and ii) of the assumption (14), neither the pseudo
spectrum e
S
e
] nor the singular functionsu
ke depend on the xed point
e
2
E
: they onlydepend on
E
and can be denoted eS
E
] andu
kE respectively. From the expansion (12)we can guess that to each vertex and each edge will be associated a part of the expansion and that the edge parts will involve coecients along the edges.
In order to give a precise statement, we need weighted Sobolev spaces for these edge
singularity coecients and a smoothing operator. Let us x an edge
E
: we recall that(
rz
) are cylindrical coordinates associated to this edge and letbe a smooth function on
E
, equivalent to the distance to the endpoints ofE
: if for exampleE
=fx
jr
= 0z
2(;1+1)g, we can take(z
) = 1;z
2.For
m
2N and 2R, letV
m(E
) be the weighted Sobolev space dened asV
m(E
) =f2L
2(E
)j +k@
kz2L
2(E
)k
= 01:::m
g(16)
and by interpolation for non-integer
m
.The smoothing operatorK] acts like a lifting of functions on
E
into : in order todene it, we need two dierent stretched variables. Firstly ~
r
=r
:
We call the triple (~
rz
) the stretched cylindrical coordinates. The inequalities ~r <
r
~0 dene a sort of sectorial neighborhood ofE
.And secondly
~
z
=Z z
z0
1
()d
where
z
0 corresponds to an interior point ofE
. The change of variablez
7!z
~ is one toone
E
!Rand for any function dened onE
, we set ~(~z
) =(z
).Then K
](~rz
) is dened as the convolution operator with respect to ~z
:K
](~rz
) = ZR
1 ~
r '
t
~
r
~
(t
;z
~)dt
where
'
is a smooth function in S(R) such that RR
'
= 1. Thus for any xed ~r
06
= 0, the function
z
7!K](~r
0z
) is smooth insideE
, and tends to as ~r
0 !0.Finally, we denote, with a smooth cut-o
equal to 1 in a neighborhood of 0 andwith small enough support
U
kv(
x
) =()u
k
v(
#
) andU
kE(x
) =(~r
)u
kE(~r
):
It is important to note that there exists ~
r
0>
0 such that for all ~r
r
~0 and all 2G
E,the point of stretched cylindrical coordinates (~
rz
) belongs to .Theorem 7.1
We assume(14). Letu
be the solution of problem (1) withf
2PH
s ;1().If for any vertex
v
the straight lineRe=s
;12 does not meet the pseudo-spectrum e Sv
],and if for any edge
E
the straight line Re=s
does not meet the pseudo-spectrum e SE
],there holds with
s
0 = minf min vvertex v+1
2
emin2edge eg :
(17)
a) If
s < s
0, thenu
2PH
s+1() (Theorem 6.1).b) If
s
0s < s
0+ 1, thenu
2H
+1() for all< s
0 and there exist coecients kv
2C and functions
kE 2V
s;Re
u
; Xvvertex
X
2 e Sv] ;
1
2< Re<s ;
1 2
KX
k=1
kv
U
k
v
; X
E edge
X
2 e SE]
0<Re<s KE
X
k=1
K
kE]U
kE 2H
s+1():
(18)
c) If
s
s
0+1,u
still admits an expansion similar to(18), but with new sorts of terms,the shadow terms:
kv
U
kp
v and
K
@
pzkE]U
kp
E
p
= 12::::
The degree of homogeneity of the functions
U
vkp andU
kp
E is
+p
and +p
respec-tively.
For a complete statement in the situation satisfying to i) and ii) of the assumption (14), but without excluding logarithmic singularities, see 16].
7.2 Vertex{ Edge interaction
We still assume (14), and only in order to simplify the notations, that the \multiplicities"
K
v andK
E are equal to 1, which allows to drop the exponentk
for the singularfunctions. We moreover assume that we are in situation b) of Theorem 7.1. In Figure 6, the edge
E
has the verticesv
1 andv
2as end points. The vertices
v
1 andv
2 correspondto spherical polygons
G
1 andG
2 respectively. The corners ofG
1 correspond to the edgesE
,E
11 andE
12 and the corners ofG
2 correspond to the edgesE
,E
21 andE
22.G1 G2
v1 E v2
E
E
E11
E12 E22
E21
The expansion (18) is obtained by rst splitting the solution near the vertices and second along the edges. Splitting only along the edges yields new edge coecients ~
E2V
s;Re0 (
E
). It is interesting to compare E and ~E.The vertex singular functions are
Uv(#
) and each Uv has singularities at thecorners of
G
(corresponding to the edges such thatE
3v
): these singularities are thealready introduced
U
E and there exist coecientsa
vE such that there holds
Uv ;
X
E3v X
2 e SE]
0<Re<s
a
vE
U
E 2H
s+1(G
):
By plugging this expansion into splitting (18) we obtain the following expansion of the edge coecients in a neighborhood of
v
:~
e ;X
2 e Sv] ;1=2<Re< s;1=2
a
vE
= e:
7.3 Curved edges or variable coecients
In presence of curved edges or variable coecients, the operator valued Mellin symbol
P
e
] and the singular functionsu
ke depend on
e
2
E
. Let us assume now that thereare only edges and no vertices (since the edges are curved, this is possible!). A diculty arises now, coming from possible changes in the multiplicity.
If we avoid this diculty by the assumption that for all
e
2E
the algebraic andgeometric multiplicities of the poles of the resolvent R
e
] do not change in the strip0
<
Res
and that they never coincide with integers (in other words, there exists away to dene the bases
u
ke with a
C
1 dependence on
e
2
E
), then an expansion likeu
; XE edge
X
2 e Se]
0<Re<s KE
X
k=1
K
kE]U
k e2
H
s+1()(19)
holds with coecients which belong to Sobolev spaces with variable exponents, seeMaz'ya,
Rossmann36] andNazarov, Plamenevskii39]: if
is a simple polekE2H
s;Re(
E
):
But the above assumption is not generic. Two basic situations escape to this as-sumption
i) For
e
0 2E
and 0 2Se
0], the resolvent 7;! Re
0] has a double pole and fore
2E
,e
6=e
0 there are two simple poles 1(e
) ! 0 and 2(e
) ! 0 ase
!e
0.This is the classical situation of a branching1. In any
e
2E
neare
0, there exist two 1The well known example is the rst singular function of the Stokes Dirichlet problem:e
0 corresponds
associated singularities: in
e
0 these arer
0U e00(
) andr
0
log
r
Ue00(
) +V e00(
)and for
e
6=e
0, these areu
1(e
) :=r
1Ue
1(
) andu
2(e
) :=r
2U e
2(
):
ii) The pole
is simple and depends smoothly one
2E
but for an isolated valuee
02E
, =02N2and interacts with polynomials. This is the case of the crossing2. Thereis only one singular function for each value of
e
but it has dierent expressions ife
=e
0r
0log
r
Ue00(
) +V e00(
)and if
e
6=e
0r
Ue:
In this situation
r
0U e00(
) is a polynomial in cartesian variables3 and is the limit of
r
Ue ase
!
e
0. Then we set1(e
) =(e
) and2(e
)0, andu
1(e
) :=r
1Ue
1 and
u
2(e
) :=r
0U e
00
:
In both situations, if we try to write an expansion of the solution
u
with the original singular functionsu
1(e
) andu
2(e
) in the case of branching, andu
1(e
) in the case of crossing respectively, we obtain coecients which behave in general like (1;2);1 and
thus blow up as
e
!e
0. We obtain Stable Singularities by taking the sum and the divideddierence of the previous expressions
e
u
1(e
) =u
1(e
) +u
2(e
) andu
e2(e
) =u
1(e
);u
2(e
) 1(e
);2(e
):
These new singular functions depend smoothly on
e
2E
and yield asymptotic expansionslike (19) with coecients which do not blow up as
e
!e
0, seeCostabel, Dauge10, 11].We note that in case ii), we have to combine a singular function with a polynomial (which is a regular function). Here the correct concept is that of asymptotic expansion rather that of splitting into regular and singular parts.
8 Eects of singularities on nite element discretizations
The rst obvious eect of singularities is the limitation of the regularity of solutions, even if the data are very smooth. This fact hampers the convergence of Galerkin methods based on nite element discrete spaces for the discretization of problem (1).
2The well known example is the rst singular function of the Laplace Dirichlet problem:
e0 corresponds
to the opening != =2 ande6=e
0 to dierent openings. Then
Concerning the classical
h
-version of nite elements, if a regular and uniform mesh is used with continuous piecewise polynomials of degreed
, then the maximal convergence rate in energy normH
1() is, withs
0 the upper bound (17) for the regularity, see 6] for instanceh
minfd s 0g
where
h
is the characteristic length of the mesh. As may be seen from Table 1,s
0 can be very small.Concerning the
p
-version of nite elements or the spectral element method, where a mesh is xed and the degreeN
of the polynomial spaces is increased, from the regularityof the solution
u
of (1) in Sobolev spaces we can deduce that the convergence rate inenergy norm is
N
;s0. But this result is not optimal, indeed, for smooth enough data, the
convergence rate is doubled: it is
N
;2s 0see Dorr19] for the
p
-version and Bernardi, Maday 5] for the spectral method.Let us give the main arguments leading to this improved convergence rate in the framework of spectral methods. Let # be the model interval (;1
1). The bestapproxi-mation by polynomials of degree
N
inL
2(#) of a functionbehaves like
N
; not only if belongs toH
(), but also ifbelongs to the domain of the operator
A
=2, whereA
is the Sturm{Liouville operator on the interval #:A
() = ((1;2)
0) 0
2#
whose eigenvalues are
N
(N
+1) and associated eigenfunctions are the Legendre polynomi-als of degreeN
. The domain ofA
=2coincides with the weighted Sobolev spaceH
=2(#) where form
2N,H
m(#) is dened asH
m(#) =f2L
2(#)j (1;2)@
k2L
2(#)k
= 01:::m
gand by interpolation for non-integer
m
(compare with (16), there is no shift of the weight with the order of derivatives). In a rectangle or a parallelepiped 0, we have similarapproximation properties with weighted spaces obtained by tensorization from
H
=2(#). Then the key argument is the following one: for a singular functionu
of the form (11) with 0 2C there holds (withn
the dimension of the space)u
2H
s( 0) () Re
0> s
;n
2 ()
u
2H
s2s(0
)
:
The doubled convergence rate is basically a consequence of this and of the classical C ea inequality.
Thus, augmenting the degree is a priori more ecient than increasing the number of
subdomains in the mesh. However, within the
h
-version, a renement of the mesh usingspecial gradings can restore optimal convergence rates, seeRaugel 44] for polygons and
Apel, Nicaise 2] for polyhedra. With a geometrical grading and an augmentation of
9 Eects of singularities: the Maxwell bug
Classical time harmonic Maxwell equations are given by
curl
E
;i! H
= 0 andcurl
H
+i! "E
=J
in:
(20)
Here
E
is the electric part andH
the magnetic part of the electromagnetic eld, whereas"
and are the permittivity and the permeability of the medium. The right hand sideJ
is the current density. The exterior boundary conditions on
@
are those of the perfect conductor (n
denotes the unit outer normal on@
):E
n
= 0 andH
n
= 0 on@
:
(21)
When is polyhedral, or if the medium is heterogeneous with polyhedral homoge-neous subdomains j where
"
and are constant, the electromagnetic eld has dierent kinds of singularities which are thoroughly described in 12, 15]. The main dierence with what we described in the previous sections is the variational space: Equations (20) and (21) can be transformed in elliptic boundary value problems associated with the form(
uv
) 7;!Z
curl
u
curl
v
+ divu
divv
(22)
involving
E
orH
alone, but the variational space for the electric partE
is dened asX
N() =fu
2L
2()3 jcurl
u
2L
2()3 div"u
2L
2()E
n
j@= 0gand the corresponding space
X
T() for the magnetic part is dened similarly.The remarkable feature is that
X
N() orX
T() are not contained inH
1()3 in general. Even if"
and are constant (what we assume from now on), we have for any polyhedron with plane facesX
N() =H
N() () is convexwhere
H
N() :=X
N()\H
1()3. If is not convex the singularities inX
N() are thegradients of the singular functions of the Dirichlet Laplace operator on . Moreover, the
closure in
X
N() of smooth vector elds with zero tangential component isH
N(). Theform (22) is coercive on any of the spaces
X
N() andH
N() and yields dierent solutions in general if the variational problem is posed inX
N() orH
N(), see Costabel 7, 8] and 15].10 Conclusions
Singularities are not there only to give trouble to mathematicians, to their theories and numerical methods. They are a part of the correct description of phenomena. For ex-ample, the singularities along a crack front, especially in continuum mechanics, are very important. First they are strong, behaving in
r
1=2 for displacements, whence inr
;1=2for stresses. Second, the associated coecients
c
v are called stress intensity factors andintervene in dierent fracture criteria, see Nazarov, Polyakova40] and the references
therein.
Indeed, even if the domain has not real sharp corners, but only regions where the curvature of its boundary is very large, see in the middle of Figure 7, the behavior of its solutions are almost singular. In the example of Figure 7, the limit domain with corner is represented in (
a
). The solutions of coercive elliptic boundary value problems on can be expanded with respect to the curvature parameter (for example, the curvature radius"
) as"
! 0 and involves solutions of problems on the domain (a
) but also corner layerterms whose proles are solutions of problems on the unbounded domain (
b
), seeMaz'ya,Nazarov, Plamenevskii 33].
(a) (b)
Ω
Figure 7: Curved corner
Thus, it can be important for theoretical or practical reasons, to know precisely the exponents
in the spectra S associated with 2D corners, edges or 3D vertices, and alsothe angular shape U(
) or U(#
) of the singular functions. So let us end this short survey paper by a few references about the computation of singular functions.For plane problems (associated with polygons or edges), concerning the Laplace op-erator and the Dirichlet or Neumann conditions the exponents are
k=!
, for integerk >
0, with!
the opening of the sector ;. Concerning Lam e and Stokes operators, the spectrum coincides with the set of the zeros of special analytic functions, the root discriminantfunctionssee Lozi 32] andOrlt, Sandig 43] for Stokes, Grisvard 21] andNicaise,
Sandig42] for isotropic elasticity. The determination of the root discriminant functions
can be extended to any elliptic system in 2D, see Costabel, Dauge 9] and its
imple-mentation for general elasticity transmission problems inCostabel, Dauge, Lafranche
Szabo, Yosibash47, 48].
Very few works are devoted to the computation of three dimensional vertex
sin-gularities. Concerning the Laplace operator let us quote Stephan, Whiteman 46] and
Beagles, Whiteman4] using nite element approach on the solid angle
G
, orSchmitz,Volk, Wendland 45] using boundary element method on its boundary
@G
.As a nal conclusion, let us say that many regions are left to future mathematical or numerical explorations. The interaction of singularities with small (or large) parameters constitute one of these regions, endowed with very rich and various landscapes...
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Institut Math ematique de Rennes, UMR-CNRS 6625,