Chapter 2
Elliptic problems on polyhedral
domains
We now proceed to study solution to elliptic boundary value problems on polygonal and polyhedral
domains. Our goal is to gain a thorough understanding of singularity structure and regularity of
Poisson’s problem in such domains. Much of what we learn about the Laplacian can be extended
in similar fashion to other uniformly elliptic boundary value problems with constant coefficients.
Writing down results for more complicated problems in a straightforward, concrete way is however
somewhat messy, and the intuition we gain from the Laplacian is sufficient for our purposes. We
first handle the two-dimensional case.
2.1
The Laplacian on polygonal domains
The material and references for Da88
this section are mainly a fewG85
(or even several) decades old.
See
Gr76
especially the books by Dauge [6, Chapter 4] and Grisvard [13]. The conference report [12] of
Grisvard also has a good overview.
2.1.1
Polygonal domains: Notation
Let ⌦ ⇢ R2 be a polygonal domain with boundary = @⌦. Denote by v1 , ..., vJ the vertices of
arranged so the numbering increases in counterclockwise fashion. The edges are denoted by 1 , .., J
where j = (vj 1 , vj ) with the obvious modification 1 = (vJ , v1 ). Let ⇢j (x) be the distance from
x 2 ⌦ to the vertex vj . Let !j be the interior opening angle at vj , that is, the angle from j+1 to j
measured in counterclockwise fashion. Also, let ✓j be the interior angle between the segment j+1
and (x, vj ), which again is measured counterclockwise. The pair (⇢j , ✓j ) acts as polar coordinates
fig2-1
centered at the vertex vj , with angles always measured on the interior of ⌦. See Figure 2.1.
2.1.2
Singularity structure: Basic intuition
We begin by giving some basic intuition concerning the behavior of solutions to
u = f in ⌦. We
begin with the Dirichlet boundary condition u = 0 on @⌦ and assume for now that f is smooth. Let
23
24
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
Figure 2.1: Notation for polygons.
then
uj (x) = ⇢j (x)⇡/!j sin
fig2-1
⇡✓j (x)
.
!j
Note that uj (x) = 0 on the edges j and j+1 which share the vertex vj . In a very rough sense, the
⇡✓ (x)
purpose of the term sin !j j is thus to ensure that the boundary condition is satisfied, and we shall
see that changing the boundary condition changes this term. The term ⇢j (x)⇡/!j , on the other hand,
is generally singular (we will talk about exceptional cases below, e.g., !j = ⇡2 ). The functions uj (x)
can be seen as the “least regular” portions of the solutions to
u = f with Dirichlet boundary
conditions. More precisely, let j (x) be a smooth cuto↵ function which is 1 in a neighborhood of vj
and 0 at any other vertex and on any edge not abutting vj . Then we may write
u(x) = u0 (x) +
J
X
↵j
j (x)uj (x),
(2.1)
j=1
where u0 is the “regular” part of the solution u in the sense that it is in general more regular than
u itself (we make this explicit later). The coefficients ↵j are sometimes called (or are related to)
“stress intensity factors” because in mechanics they may be related to stresses arising at crack tips
when !j = 2⇡.
To given an example, let !j = 3⇡
2 . This corresponds to the non convex vertex in the L-shaped
domain that we used for sample computations in Chapter 1. We then have
⇡/(3⇡/2)
uj (x) = ⇢j
sin
⇡✓j (x)
2✓j
2/3
= ⇢j sin
.
3⇡/2
3
sing1
2.1. THE LAPLACIAN ON POLYGONAL DOMAINS
25
2 3⇡
Note first that ✓j = 0 and ✓j = 3⇡
2 correspond to @⌦, and that sin 0 = 0 and sin 3 2 = sin ⇡ = 0.
Thus uj is 0 on the the portions of @⌦ lying near vj . Also, if we assume that vj is the origin, we
p
2/3
2/3
can rewrite ⇢j (in xy coordinates) as (x2 + y 2 )
= (x2 + y 2 )1/3 .
We now try to gain some intuition about the regularity of uj . Assume that vj is the origin and
that the coordinates (⇢j , ✓j ) correspond to the usual polar coordinates (r, ✓). The gradient in polar
coordinates is given by
1 @uj
@uj
ruj = e✓
+ er
,
r @✓
@r
where e✓ and er are unit vectors pointing along the given coordinate directions. Thus
r(r2/3 sin
2✓
) = e✓ r
3
12
3
cos
2✓
2
+ er r
3
3
1/3
sin
2✓
2
= r
3
3
1/3
(e✓ cos
2✓
2✓
+ er sin ).
3
3
We may more generally calculate that we lose one power of r for each derivative of uj , so that roughly
speaking Ds uj ⇠ r⇡/!j s (we ignore the trigonometric portion of uj here since it is smooth).
To continue the example, we now ask the following question: If !j = 3⇡
2 is the maximum interior
opening angle of ⌦, for what p can we expect that u 2 W p,2 (⌦)? Integrating in polar coordinates,
we have
Z
Z
Z diam⌦
2
p
⇡/!j 2 p
|D uj | dx ⇠
|r
| r dr 
rp(2/3 2)+1 dr = rp(2/3 2)+2 |C
0.
⌦
⌦
0
In order for this quantity to be finite, we must have p(2/3 2) + 2 > 0. Thus 4p/3 > 2, or
p,2
4p/3 < 2, or p < 3/2. Thus if !j = 3⇡
(⌦) for p < 3/2 (only). If 3⇡/2 is the
2 , we have uj 2 W
maximum interior opening angle, the singularities at the other vertices will be weaker, that is, ui
(i 6= j) will be at least as smooth as uj , and we may expect (or at least reasonably hope) that the
solution u to
u = f with f 2 Lp (⌦) will satisfy u 2 W p,2 (⌦) as long as 1 < p < 3/2. This is in
fact the case, as we state more precisely in the next subsection below.
To wrap up this section, we investigate the singularities uj a bit more. The Laplacian in polar
2
1 @2u
⇡/!j
coordinates can be written as u(r, ✓) = @@ru2 + 1r @u
sin ⇡✓
@r + r 2 @✓ 2 . Writing uj = r
! with respect
to generic polar coordinates, we compute
uj = r⇡/!j
2
(
⇡ ⇡
(
!j !j
1) +
⇡
!j
(⇡/!j )2 ) sin
⇡✓
= 0.
!j
Thus
each of the uj ’s is harmonic. Recall also that the singular functions added in the expansion
sing1
(2.1) are also multiplied by cuto↵ functions j , in order to ensure that boundary conditions are
satisfied. The functions uj j are however harmonic close to the vertices vj , which is important as
this is the region where uj is singular. In fact, (uj j ) is smooth with proper choice of j .
2.1.3
Singularity structure: Regularity results in W p,2
sing1
We first generalize the singularity expression (2.1) slightly and then give a regularity result in W p,2 .
Given 1 < p < 1, let p0 be the conjugate index, i.e., p1 + p10 = 1. For fixed 1  j  J, let
1`<
2!j
:= `j .
p0 ⇡
(2.2)
ellcond
26
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
We also generalize the structure of the singular functions uj above to include “smoother” singularities:
l⇡/!j
uj,` (x) = ⇢j
singtheorem
sin `⇡
✓j
.
!j
(2.3)
udef
Theorem 2.1.1 As above let j be a cuto↵ function whhich is smooth, 1 in a neighborhood of vj ,
2!
and 0 at all other vertices of @⌦. Assume that 1 < p < 1 be such that p0 ⇡j is non-integer for
1
1  j  J. Let u 2 H0 (⌦) be the weak solution to
u = f with f 2 Lp (⌦). Then there exist
coefficients ↵j,` such that
u = u0 +
J
X
X
↵j,`
j uj,` ,
(2.4)
j=1 1`<`j
where u 2 W p,2 (⌦).
This theorem confirms our rough calculation above that if 1 < p < 3/2 and max1jJ !j = 3⇡
2 ,
then the solution to
u = f possesses W p,2 regularity. In particular, 1 < p < 3/2 implies that
ellcond
2!
p0 > (1 2/3) 1 = 3. Thus in (2.2) we have `j = p0 ⇡j < 6⇡
6⇡ = 1, so that the condition becomes
sing2
empty. Thus the sum in (2.4) is empty, and u = u0 2 W p,2 (⌦).
Many a priori error results in the finite element literature assume a convex polyhedral domain. A
major reason for this is that elliptic boundary value problems on such domains possess H 2 regularity,
so standard duality arguments for proving L2 estimates may be used. We calculate more precisely
in two dimensions. Given maximum interior opening angle !j , we wish to find the maximum p
ellcond
2!
2!
for which p0 ⇡j < 1 so that (2.2) is empty. Then p0 > ⇡ j , which implies that p = (1 1/p0 ) 1 <
2!
⇡
(1 2!
) 1 = 2!j j ⇡ := p̄. For ⌦ convex, !j < ⇡, so p̄ is always greater than 2, decreases to 2
j
as !j " ⇡, and increases to 1 as !j # ⇡/2. Thus as stated, H 2 regularity always holds on convex
polygonal domains.
More generally, H 2 regularity holds on any convex domain in Rd (d 2) for a general class of
elliptic
boundary value problems satisfying reasonable assumptions. This important fact is proved
G85
in [13, Chapter 3].
2.1.4
Other boundary conditions
We now assume homogeneous Dirichlet boundary conditions u = 0 on j (j 2 D) and Neumann
@u
boundary conditions @~
n = 0 on j (j 2 N ). Here D [ N is a disjoint partition of the index set
{1  j  J}. As above, we may decompose solutions u to
u = f with the above boundary
conditions into regular and singular portions. Let S be the set of indices j for which the boundary
condition is the same for both edges abutting vj . That is, j, j + 1 2 D or j, j + 1 2 N . Let M
(“mixed”) be the set of indices for which the type of boundary condition changes at vj , that is,
j 2 D and j + 1 2 N or j + 1 2 D and j 2 N . Note that for j 2 M , opening angles !j = ⇡ are
allowed. We will see that even if @⌦ is smooth, the change from Dirichlet to Neumann boundary
conditions induces a singularity in u just as corners do.
sing2
2.1. THE LAPLACIAN ON POLYGONAL DOMAINS
27
ellcond
We generalize (2.2) as follows:
udef
and (2.3) as follows:
8
2!
>
1  ` < ⇡pj0 , j 2 S;
>
>
<
2!
1  ` < p0 ⇡j + 12 , j 2 M, !j 6= ⇡2 , 3⇡
2 ;
> no ` when j 2 M and !j = ⇡2 ;
>
>
: 1  ` < 3 + 1 , ` 6= 2 when j 2 M and ! =
j
p0
2
uj,` =
8 `⇡/!
✓
>
⇢j j sin `⇡ !jj , j, j + 1 2 D;
>
>
>
>
< ⇢`⇡/!j cos `⇡ ✓j , j, j + 1 2 N ;
j
!j
(`
>
⇢j
>
>
>
>
: ⇢(`
j
1
2 )⇡/!j
1
2 )⇡/!j
sin(`
sin(`
1 ⇡✓j
2 ) !j , j 2
1 ⇡(!j ✓j )
,
2)
!j
(2.5)
genellcond
(2.6)
genudef
3⇡
2
D, j + 1 2 N,
j 2 N, j + 1 2 D.
singtheorem
The results of Theorem 2.1.1 then hold essentially verbatim, but with the generalized restrictions
on ` and definitions of singular functions above substituted in.
The essential intuition we gain is that Neumann boundary conditions induce the same strengths of
singularities as do Dirichlet conditions, while switching conditions causes even stronger singularities.
To illustrate this, first consider the case of mixed boundary conditions on a line segment,genudef
that is,
we switch from Dirchlet to Neumann conditions at vj with !j = ⇡. The third line of (2.6) with
⇡/2!
1/2
` = 1 then gives uj,` = ⇢j j sin(⇡ ✓j )/2 = ⇢j sin(⇡ ✓j )/2. The singularity ⇢1/2 is as strong
as is encountered for Dirichlet or Neumann conditions on a crack domain (!j = 2⇡). If we instead
consider a change in boundary conditions on an L-shaped domain at a vertex vj with !j = 3⇡/2, we
⇡/(2·3⇡/2)
1/3
have principle singularity ⇢j
= ⇢j . This is a nastier singularity that is encountered with
pure Dirichlet or Neumann conditions on any polygonal domain.
We make some final remarks. First, the above results are valid also for domains with cracks
1/2
(!j = 2⇡). Here
the leading singularity strength is ⇢j . In addition, when p and ⌦ are such that
genellcond
no ` satisfy
(2.5), then solutions to
u = f with f 2 Lp (⌦) additionally satisfy the regularity
G85
estimate [13, Theorem 4.3.2.4, Remark 4.3.2.5]
kukW p,2 (⌦) . kf kLp (⌦) .
Finally, W p,k regularity results for k > 2sing2
may be deduced in a similar way. That is, for given k, p
the singular function representation in (2.4) must be suitably expanded to take into account all
singularities with regularity less than W p,k (at least
this is the intuition for most values of !j ) in
G85
which case a similar representation holds (see e.g. [13, Theorem 5.1.1.4]).
2.1.5
Fractional Sobolev regularity
Da88
We now give fractional Sobolev regularity results for the Dirichlet problem; cf. [6, Theorem 14.6].
hsreg
Theorem 2.1.2 Assume that the maximum opening angle in the polygon ⌦ is given by !j  2⇡,
!j 6= ⇡. Then is an isomorphism from H s+1 (⌦) \ H01 (⌦) onto H s 1 (⌦) if and only if 0 < s < !⇡j .
28
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
Da88
Although notDa88
directly stated in [6, Theorem 14.6], essentially the same result holds for the case
of plane cracks [6, Theorem 14.10]. That is, if 0 < s < 1/2 and maxj !j = 2⇡, then f 2 H s 1
implies that u 2 H s+1 (⌦).
hsreg
Dauge_web
The results of Theorem 2.1.2 hold for Neumann boundary conditions also [8, Slide 12]. In the
⇡
case of mixed boundary conditions, the condition s < !⇡j is replaced by s < 2!
. Thus in the extreme
j
case !j = 2⇡ with mixed boundary conditions at vj , the solution to
u = f will in general only
lie in H 1/4 ✏ (any ✏ > 0).
We briefly explore these results. Returning to the L-shaped domain with maxj !j = 3⇡/2, we
have !⇡j = 2/3. Thus if f 2 L2 (⌦), we have u 2 H 1+s (⌦) for any 0 < s < 2/3. Similarly, for
the crack domain u 2 H 1+s (⌦) for any 0 < s < 1/2. Returning to our computational examples
in Chapter 1, we observe that the finite element convergence rates on quasi-uniform meshes match
exactly these regularity results. In particular, it can be proved that if Vh is a space of Lagrange
polynomials defined on a mesh of diameter h, then
inf ku
2Vh
kH 1 (⌦) . hs |u|H 1+s (⌦) .
We observed convergence rates close to O(h2/3 ) for the L-shaped domain and O(h1/2 ) for the crack
domain. These match quite precisely the regularity results given above.
We finally consider a square domain with maxj !j = ⇡/2. Then if f 2 H s 1 (⌦), 0 < s < 2,
we have u 2 H s+1 (⌦). Thus for f smooth, u 2 H k for any k < 3. udef
There does
however remain a
sing2
slight disconnect
between
the
singular
function
expansion
given
by
(
2.3)
and
(
2.4)
on the one hand
hsreg
and Theorem 2.1.2. In particular, for !j = ⇡/2 and ` = 1 we obtain uj,` = ⇢2j sin 2✓j , which has
infinite smoothness.
(In standard polar coordinates, r2 = x2 + y 2 is a polynomial.) On there other
hsreg
hand, Theorem 2.1.2 indicates limited regularity. The reason is that the correct singular
function
Dauge_web
expansion in exceptional
cases
where
⇡/!
is
an
integer
also
includes
a
logarithmic
term
[8,
Slide
24].
j
udef
We thus modify (2.3) as follows: Assume that !`⇡j 2 N and !j < 2⇡. Then in the case of Dirichlet
udef
boundary conditions, we modify (2.3) as follows:
`⇡/!j
uj,` = ⇢j
(ln ⇢j sin
`⇡✓j
`⇡✓j
`⇡
+ ✓j cos
),
2 N.
!j
!j
!j
(2.7)
A similar modification holds for Neumann boundary conditions (I don’t immediately have a reference
for mixed boundary conditions). It is easy to compute that if !j = 2⇡, then u1,` ⇠ ⇢2j log ⇢j is not
hsreg
in H 3 , but is in H s for any s < 3. This corresponds precisely with Theorem 2.1.2.
The case !j = 2⇡ (the case of a plane crack) is something of an exception to this exception.
There for even ` the singular functions are simply omitted, so that the leading singularities are r1/2 ,
r3/2 , r5/2 , etc.
2.2
The Laplacian on polyhedral domains
A good (and relatively readable) overview of singularity
structureDa88
for the Laplacian on polyhedral
Dauge_web
domains
can
be
found
in
slides
by
Monique
Dauge
[8].
The
book
[6]
also has useful information, as
Da92
MR10
does [7]. We also refer at times to the monograph [14] of Maz’ya and Roßmann.
logudef
2.2. THE LAPLACIAN ON POLYHEDRAL DOMAINS
2.2.1
29
Polyhedral domains: Notation
A polyhedral domain ⌦ ⇢ R3 has boundary = @⌦ which may be broken into a set V of vertices, E
of edges (consisting of line segments), and F of polygonal faces. The set F of faces will play little
or no roll below. As in the case of polygonal domains, we index the vertices by j (1  j  J). As
necessary we index the edges e 2 E as eij , where vi , vj 2 eij are the boundary vertices of e. However,
for both edges and vertices such indexing is less meaningful here than in the 2D case and we shall
omit it when doing so causes no confusion.
2.2.2
Singularity structure: Basic intuition
As in the 2D case we wish to write solutions to
u = f in ⌦, u = 0 on @⌦ as a sum of singular
and regular portions. Here however there are two types of singularities, those occurring at the edges
and those occurring at the vertices. That is, we write:
X
X
u = u0 +
↵ v v uv +
↵ e e ue ,
(2.8)
v2V
3dexpansion
e2E
where as in the 2D case u0 is “more regular” than u, e and v are “cuto↵” functions that are 1 in
neighborhoods of e and v respectively, and uv and ue are singular functions. The vertex coefficient
↵v 2 R. The edge coefficient ↵e varies along e. In the case of Dirichlet boundary conditions it is 0
(and in fact
“flat” in the sense that derivatives of sufficiently high order are 0) at the ends (vertices)
Dauge_web
of e (cf. [8, Slide 32]). It must also satisfy certain regularity conditions in order to make any
statement about the regularity of u, but we shall not make precise statements about requirements
on ↵e .
We first describe the edge singularities ue , since in contrast to the vertex singularities uv their
basic form may be described quite concretely. Given a point x 2 e 2 E, let (⇢e , ✓e , z) be cylindrical
coordinates with z-axis along e. Let also !e be the interior edge opening of e. (We may obtain this
by intersecting ⌦ near e with a plane in the (⇢e , ✓e ) direction and measuring the opening angle of
the resulting 2D wedge.) Define also
⇡
.
e =
!e
(This is the same quantity that appeared repeatedly in our 2D definitions; we now give it a shorthand
definition for convenience.) Then
ue (⇢e , ✓e , z) = ⇢e e sin
e ✓e .
uedgedef
(2.9)
Generally the same heuristics apply to the expansion (2.9) as apply to the corresponding expressions
for polygonal domains. That is, the term ⇢e e is the singular portion, and is the same for both
Dirichlet and Neumann boundary conditions but becomes stronger for mixed boundary conditions.
The purpose of the term sin e ✓e is to enforce the boundary conditions and naturally takes on a
di↵erent form for di↵erent boundary conditions. Also, in exceptional cases where e is an integer,
r e is not generally singular and so logarithmic terms must be added to the expansion. Finally,
the fractional Sobolev regularity of the singular functions expansions is quite easy to calculate by
integrating in polar coordinates: ue 2 H s+1 (⌦) for s < e , except possibly in exceptional cases.
Thus larger edge opening angles induce lower regularity.
There is however one major exception to this easy comparison between 2D vertex singularities
and 3D edge singularities, and it has quite important implications for finite element convergence
uedgedef
30
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
rates. 2D vertex singularities can always be resolved with optimal convergence rate by shape-regular
(possibly adaptive) finite element meshes, while there is a limit to the convergence rates that can be
obtained when approximating 3D edge singularities using shape-regular meshes. Thus anisotropic
meshes are sometimes necessary.
Description of the vertex singularities requires a bit more abstraction. Given v 2 V, we begin by
defining coordinates (⇢v , ✓v ). Here ⇢v is the distance to v and ✓v is a (two-dimensional) coordinate
system on the unit sphere S 2 centered at v. We also recall the Laplace-Beltrami operator (surface
Laplacian). Assume momentarily that S is a smooth, orientable n 1-dimensional surface embedded
in Rn . Let also ⌫ be the unit normal to S. Given a function u defined on Rn , then its tangential
derivative when restricted to S is given by rS u = ru (ru · ⌫)⌫ = (I ⌫ ⌦ ⌫)ru. Denoting by
rS = (I ⌫ ⌦ ⌫)r the tangential gradient operator, the surface Laplacian is given by S = rS · rS ;
here rS · is the surface divergence.
We next briefly review the case of a two-dimensional vertex singularity uv = ⇢v v sin v ✓v with
uv = 0. Generalizing the
v = ⇡/!v . Recall that we calculated using polar coordinates that
Laplacian in polar coordinates to arbitrary space dimension n using spherical coordinates, we obtain
u=
1
@ 2 u n 1 @u
+
+ 2
2
@r
r @r
r
Sn
1
u.
(2.10)
spher_laplac
When n = 2, the surface Laplacian is calculated on Sv = S 1 \ {0 < ✓v < !v }. An essential
observation for understanding the 3D case is that sin v ✓v is a Dirichlet eigenfunction of Sv with
2
eigenvalue µv =
v.
We now return to the case of a 3D vertex v 2 V and seek singular functions uv which are singular,
harmonic, and satisfy Dirichlet boundary conditions near v. Following the discussion above for the
2D case, we first define the “spherical cap” Sv as the solid angle of the tangent cone Kv . Unpacking
this definition, Kv may be thought of as taking the intersection of ⌦ with a small ball about v not
touching any other vertices and extending the result infinitely tangentially to @⌦. Sv = Kv \ S 2
is then the intersection of Kv with the unit sphere. Alternatively, in order to obtain Sv we may
intersect ⌦ with a sphere centered at v having radius small enough so that no there vertices are
enclosed in it, then scaling the result up to a spherical segment of radius 1.
v
In analogy to the 2D case we now define a model singular function uv = ⇢spher_laplac
v '(✓v ) and try to
determine v 2 R and a function ' on Sv so that u is harmonic near v. From (2.10) we have
@ 2 u n 1 @u
1
+
+ 2 Sn 1 u
@r2
r @r
r
= v ( v 1)⇢v 2 uv + 2 v ⇢v 2 uv + ⇢v v
uv =
= 0.
2
Sv '
(2.11)
eq1
We now briefly recall some facts about eigenfunctions and eigenvalues of the Laplacian. First, the
eigenvalue problem
S u = µu, u = 0 on @S has an infinite set of eigenvalues 0 < µ1 < µ2  µ3 ....
Second, with Dirichlet boundary conditions the base eigenvalue µ1 is monotone with respect to the
domain:
S ( S 0 ! µ1,S > µ1,S 0 .
(2.12)
Assume
now that ' is the smallest Dirichlet eigenfunction of Sv with eigenvalue µv . The relationeq1
ship (2.11) then reduces to
[ v ( v 1) + 2 v µv ]⇢v v 2 ' = 0,
mono_eig
2.2. THE LAPLACIAN ON POLYHEDRAL DOMAINS
31
or
2
v
+
v
µv = 0.
Solving this equation for the positive root yields
v
=
1+
p
1 + 4µv
.
2
(2.13)
lambda3d
Combining this with the 2D result, we then have that the strength of a vertex singularity in Rn
(n = 2, 3) is given by ⇢v v , where v is the positive root of v ( v + n 2) = µv and µv is the smallest
Dirichlet eigenvalue of the surface Laplacian on the spherical cap Sv .
Recall also that in 2D, the regularity of the singularity decreases (or, the nastiness
increases)
mono_eig
as the interior opening angle increases. The eigenvalue monotonicity relationship (2.12) allows us
to make a similar statement in the case of Dirichlet boundary conditions: The strength of a vertex
singularity increases as the solid angle of the interior opening increases. Of course, in contrast to
the 2D case, it is not always possible to make such a direct geometrical comparison. In addition,
such a monotonicity result does not hold for Neumann eigenvalues.
Generally speaking values of µv (and thus of v ) are not known analytically. They may
however
DG12
be approximated using a surface finite element method. We givefig2-4
such an example from [9]. Recall
from Chapter
1 the two-brick domain, pictured again in Figure 2.2 with the vertex v denoted. In
fig2-3
Figure 2.3 is the spherical cap Sv (or more accurately a triangulated approximation thereof). In
order to characterizer uv , we solved the Dirichlet eigenvalue problem on Sv using an adaptive surface
finite element method and found that
p
1 + 1 + 4µv
µv ⇡ 3.90722, v =
⇡ 1.5389.
2
fig2-3
The mesh and 'v are pictured in Figure 2.3. Note that obtaining usable information about ' is
relatively difficult from an implementation standpoint. In particular, if one wanted to use a singular
function finite element method by incorporating uv = ⇢v v 'v into an FEM as a basis function,fig2-3
it
would be necessary to transfer information from the surface FEM calculation pictured in Figure 2.3
into a 3D finite element code.
We conclude this subsection by discussing the regularity of the singular function uv ⇡ ⇢1.5389
'v .
v
As in 2D, it is not difficult to calculate that we (heuristically) lose one power of ⇢v for each derivative
we take. To estimate the maximum s for which uv 2 H s , we then calculate:
Z
Z 1
v s)+3
(Ds uv )2 dx ⇠
(⇢v v s )2 ⇢2v d⇢v ⇠ ⇢2(
.
v
⌦
0
This integral is finite when 2( v s) + 3 > 0, or s < v + 3/2. Because v > 1.5, we have the
uv 2 H 3 (and in fact uv 2 H s fig2-4
for some values of s > 3).
The vertex v1 (see Figure 2.2) is a rarity in that v1 = 5/3 can be computed analytically. The
resulting singularity uv1 ⇠ ⇢5/3 is weaker (more regular) than that for v, so v is the vertex which
most strongly limits the regularityfig2-4
of u.
In contrast, note from Figure 2.2 that the maximum edge opening angle in ⌦ is 3⇡
2 (at e1 and
e2 ). As in the 2D case, the corresponding edge singularities will lie in H s only for s < 5/3. Thus
in this case (and in general), the edge singularities are nastier than the vertex singularities in the
sense that they tend to be the limiting factor in determining the regularity of u.
32
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
v1
e2
e1
v
Figure 2.2: Non-Lipschitz Two-brick domain ⌦
fig2-4
Figure 2.3: Spherical cap with adaptive mesh and 'v color-coded.
fig2-3
2.2. THE LAPLACIAN ON POLYHEDRAL DOMAINS
2.2.3
33
Fractional Sobolev regularity
Dauge_web
We next state fractional Sobolev regularity results [8]. We generalize our notation slightly in order
@u
to also handle Neumann boundary conditions @~
n = 0 on @⌦. Given v 2 V , we let Sv be the spherical
“cap” at v as above. Let µv,D be the first Dirichlet eigenvalue on Sv and µv,N be the first positive
Neumann eigenvalue on Sv . We also recall that !e is the edge opening angle at the edge e.
Theorem 2.2.1 Consider the problem
u = f in ⌦ with u = 0 on @⌦. Given v 2 V, let
the positive root of 2 + = µv,D . Assume that s > 1/2 satisfies
⇢
8v 2 V,
8e 2 E,
s 12 < min{
s < !⇡e .
Then f 2 H s 1 (⌦) ) u 2 H s+1 (⌦).
If we instead impose the Neumann boundary condition
positive root of 2 + = µv,N . Assume that
⇢
Then f 2 H s
1
8v 2 V,
8e 2 E,
s 12 < min{
s < !⇡e .
v,D , 2},
@u
@~
n
= 0 on @⌦, we first let
v,N , 1},
v,D
be
(2.14)
v,N
be the
(2.15)
(⌦) ) u 2 H s+1 (⌦).
As discussed in the previous subsection, analytical
values of v,D and v,N are relatively hard
p
5 1
to come by. However, if ⌦ is convex, then v,N
> 12 and hence s > 1. In addition, v,D = 1
2
if Sv is a half-sphere. By monotonicity of the Dirichlet eigenvalues, we have v,D > 1 for convex
vertices v, so again we always have s > 1 (and over regularity of at least H 2 ) for convex domains.
In addition, as noted above the edges tend to be the limiting factor in determining regularity, and
there the singularity strengths may be computed quite precisely.
We now give a more general (and more precise with regards to the form of the singular functions)
statement concerning singular functions expansions. Given v 2 V , we define (abstracted) spaces of
model functions:
X
S0 (Kv ) = { : =
⇢v (logq ⇢v )'q (✓v ), 'q 2 H01 (Sv )},
q 0,f inite
S (Ke ) = { :
=
X
q 0,f inite
⇢e (logq ⇢e )'q (✓e ), 'q 2 H01 (0, !e )}.
As we saw above, we often may take q = 0 only and 'q as an eigenfunction of the Laplace-Beltrami
operator on Sv (or on (0, !e )), but not always. To define the set of possible exponents at v, we
first let ( D
2 R for
v ) be the spectrum of the Dirichlet Laplacian on Sv . Then ⇤v is the set of
which there is a non-polynomial 2 S0 (Kv ) with
polynomial. To gain some intuition, note that
non-polynomial harmonic functions always satisfy this condition. This definition thus generalizes
our approach of seeking harmonic singular functions. The more general form becomes important in
exceptional cases where v , or corresponding to a higher eigenvalue of the Dirichlet Laplacian on
Sv , is an integer.
34
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
Theorem 2.2.2 Let s > 1/2 and f 2 H s 1 (⌦), and assume that u = f with u = 0 on @⌦.
Assume also that s 1/2 2
/ ⇤v for all v 2 V, and e 6= s for all e 2 E. Then
X
X
sing
sing
u = ureg +
+
,
v uv
e ue
v2V
where u
reg
2H
s+1
(⌦),
using
=
v
X
X
e2E
↵v,
,q v
,q
, ↵v,
,q
2⇤v , 1/2< <s 1/2 q,f inite
and
using
=
e
X
X
↵e,
,p e
,2p
,
2 R,
e
,2p
`2N:0< =`⇡/!e <s p2N,p>0, +2ps
Above ↵e,
2.2.4
,p
v
,q
2S
2 S0 (Sv ),
+2p
(Ke ).
is a coefficient function whose regularity we do not precisely describe.
W p,k regularity
In this subsection we present results concerning Da92
W p,k regularity
of solutions, with main focus on the
MR10
cases k = 1, 2. We mainly present results from [7]; cf. [14].
We shall consider Dirichlet, Neumann, and mixed boundary conditions. Recalling that F is the
set of all faces in @⌦, we recall that we have (homogeneous) Dirichlet boundary conditions when
@u
u = 0 on all F 2 F and Neumann boundary conditions when @~
n = 0 on all F 2 F. In the case of
mixed boundary conditions, in analogy to the 2D case we admit edge opening angles !e = ⇡. We
divide = @⌦ into a Dirichlet portion D on which u = 0 and a Neumann portion N on which
@u
@~
n = 0. Here D and N both are (closures of) unions of faces in F. We also partition E into a
set ED of edges for which Dirichlet conditions are prescribed on both incident faces, Neumann edges
EN , and mixed edges EM , and similarly for the vertices VD , VN , and VM . We then define
⇢ ⇡
e 2 ED or e 2 EN ,
!e ,
(2.16)
e =
⇡
,
e
2 EM .
2!e
To define a similar quantity on vertices, we as above let µv be the smallest positive eigenvalue of
the Laplace-Beltrami operator on the spherical cap Sv , but now with boundary conditions for the
eigenvalue problem given by the relevant corresponding conditions (Dirichlet, Neumann, or mixed)
on . We then let
r
1
1
+ µv + , 2}.
(2.17)
v = min{
2
4
Let q be the conjugate exponent to p. Then W p,
q,1
WD
(⌦)
p, 1
1
e_cond
v_cond
(⌦) is the dual space of
= {v 2 W q,1 (⌦) : v = 0 on
D },
q,1
WD
(⌦)
that is, W
(⌦) consists of bounded linear functionals f :
For technical reasons, we additionally suppose below that
p > 2, p 6= 3, k = 1,
6
p
, p 6= 2, k = 0,
5
p > 1, p 2
/ {2, 3}, k 1.
! R¿
(2.18)
p_cond
2.2. THE LAPLACIAN ON POLYHEDRAL DOMAINS
preg3d
35
Theorem 2.2.3 Let k 2 { 1, 0, 1, ...}. Let u 2 H 1 (⌦) weakly solve
u = f with f 2 W p,k (⌦) and
Dirichlet,
Neumann, or mixed boundary conditions described as above. Assume also that p satisfies
p_cond
(2.18), and that
e 2 E,
k+2
2/p <
e,
k+2
3/p <
v , v 2 V.
(2.19)
k_cond
(2.20)
preg
Then u 2 W p,k+2 (⌦), and
kukW p,k+2 (⌦)  Ckf kW p,k (⌦) .
e_cond v_cond
k_cond
We
now work to unpack the relationship between the conditions (2.16), (2.17), and (2.19) above;
Da92
cf. [7, Section 4]. First assume that ⌦ is convex.
Assume Dirichlet boundary
conditions, and to
e_cond
k_cond
begin with assume k = 1. e > 1 in (2.16), and the first line of (2.19) is always satisfied
if
v_cond
1 + 2 2/p < 1, that is, 2/p < 0. This condition is satisfied for p < 1. Now consider (2.17).
In the Dirichlet case 1 = 1 when Sv is a half-sphere,
and monotonicity of the eigenproblem yields
k_cond
1
for
⌦
convex.
The
second
line
of
(
2.19)
is
thus
satisfied for p < 1, and we then find that
v
reg
(1.1)
is
satisfied
for
2
<
p
<
1,
k
=
1.
A
duality
argument
can be used to
show that in fact
preg
preg
(2.20) is satisfied for 1 < p < 1. For k = 0 we similarly may compute that (2.20) is satisfied for
6
5  p < p0 for some p0 > 2 depending on ⌦. For k = 1 and ⌦ convex, we may compute that there
is always some p > 1 for which f 2 W p,1 (⌦) implies u 2 W p,3 (⌦).
For convex ⌦ and Neumann conditions,
we have the same conditions on e as for Dirichlet
p
5 1
conditions. It is also known that v
his more restrictive than in the Dirichlet case.
2 , which
p
preg
Thus (2.20) holds for k = 1 when p < f rac63
5
⇡ 7.854, and when k = 0 for some p > 2. For
preg
the mixed problem, it can be shown that (2.20) holds for k = 1, 2 < p < 3 and k = 0, p < 4/3.
On general (nonconvex)
polyhedral domains ⌦, we consider only the Dirichlet and Neumann
preg
problems.
For
k
=
1,
(
2.20)
holds for 3/2 ✏ < p < 3 + ✏ for some ✏ > 0 depending on ⌦. For
preg
k = 0, (2.20) holds at least for 6/5  p < 4/3.
2.2.5
Regularity in weighted Sobolev spaces
We now describe a further tool that has been used to understand finite element approximations
on polyhedral domains, weighted (Babuška-Kondrat’ev) spaces. Given a polyhedral domain ⌦, let
⇢(x) = max(maxv2V ⇢v (x), maxe2E ⇢e (x)) be the distance to the singular set (edges and vertices)
on @⌦. Our considerations apply also to polygonal domains in two space dimensions, where the
definition of ⇢(x) is naturally defined as the minimum distance from x to any vertex in @⌦. Given
an integer k > 0 and a 2 R, we define the weighted Sobolev space
Kak (⌦) = {u : ⇢|↵|
a
D↵ u 2 L2 (⌦), |↵|  k}.
(2.21)
We comment on the role of a in the definition of Kam (⌦). If a < k, then weight_def
the highest- (k th-)
order derivatives of u are weighted by a positive power of ⇢ in the definition (2.21). These positive
powers of ⇢ will have the e↵ect of counteracting or smoothing out singularities, so that we may have
u 2 Kak (⌦) even if u 2
/ H k (⌦).
We
now
consider
the strong-form problem: Find u such that
u = f in ⌦, u = 0 on @⌦. From
BNZ05
[4], we have the following theorem.
weight_def
36
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
Theorem 2.2.4 Let ⌦ ⇢ Rn , n = 2, 3, be a polyhedral domain and let k > 0 be an integer. Then
k+1
there exists a0 > 0 such that the above boundary value problem has a unique solution u 2 Ka+1
(⌦)
k 1
for any f 2 Ka 1 (⌦) whenever a < a0 . This solution depends continuously on f in the sense that
kukKk+1 (⌦)  Ckf kKk
a+1
a
1
1 (⌦)
.
Finally, the solution is the variational solution when k = a = 0.
We next make some brief comments about this theorem. It is a well-posedness result in the
sense that it ensures a unique solution to our boundary value problem under certain conditions,
along with continuous dependence on data. Secondly, we note the relationship between the gains
in the smoothness index k and the weight index a when solving our boundary value problem. As
we expect from our previous discussion of elliptic regularity, the above theorem indicates that we
gain two orders of smoothness in passing from f to u, from order k 1 to order k + 1. It may at
first glance seem a little strange that the weight index a also changes (from a 1 for f to a + 1 for
u), but the exponent of ⇢ is dimensionally consistent in the following sense. f is a combination of
second derivates of u, and in the definition of kf kKk 1 (⌦) the highest derivative of f is multiplied by
a
1
the weight ⇢k 1 (a 1) = ⇢k a . Correspondingly, the highest-order (k + 1-st) derivative of u in the
definition of kukKk+1 (⌦) is multiplied by the weight ⇢k+1 (a+1) = ⇢k a . Thus the k + 1-st derivatives
a+1
of u are multiplied by the same power of ⇢ as are the k 1-st derivates of f in our definition, as
they should be.
2.3
Other elliptic model problems
We describe briefly how the above results for the Laplace operator extend to other basic stationary
second-order models; we consider in particular the case of Stokes’ equations and Maxwell’s equations.
2.3.1
Stokes’ equation
MR10
A good reference for this section is the monograph [14] of Maz’ya and Roßmann.
As above, let ⌦ ⇢ R3 be a polyhedral domain. Consider the incompressible, stationary Stokes
problem: Find u : ⌦ ! R3 such that
u + rp = f in ⌦,
r · u = 0 in ⌦,
(2.22)
u = 0 on @⌦.
Here u is the (fluid) velocity and p is the pressure, and the divergence-free condition u = 0 is the
incompressibility condition.
The basic heuristic for the Stokes problem is that the edges and vertices in @⌦ will cause the
same strength of singularities in u as arise in solutions to Poisson’s problem on ⌦. The pressure p
will have one order of regularity less than u. ForMR10
example, if we wish to measure regularity in Lp
spaces, we may summarize our results as follows [14, p. 519]:
2.3. OTHER ELLIPTIC MODEL PROBLEMS
37
1. If ⌦ is convex and f sufficiently regular, then for some ✏ > 0:
(u, p) 2 W p,1 (⌦) ⇥ Lp (⌦), 1 < p < 1,
(u, p) 2 W 2+✏,2 (⌦) ⇥ W 2+✏,1 (⌦),
(u, p) 2 C 1,✏ (⌦) ⇥ C 0,✏ (⌦).
2. If ⌦ is an arbitrary Lipschitz polyhedron, then for some ✏ > 0:
(u, p) 2 W 1,3+✏ (⌦) ⇥ L3+✏ (⌦),
(u, p) 2 W 4/3+✏,2 (⌦) ⇥ W 4/3+✏,1 (⌦),
u 2 C 0,✏ (⌦).
preg3d
As per the discussion following Theorem 2.2.3, these results mirror exactly those for Poisson’s
problem.
2.3.2
Maxwell’s equations
Dauge_web
CD00
In this section we again follow [8]; cf. [5]. As a precursor, we briefly review the de Rham complex:
r
r⇥
r·
H 1 (⌦) ! H(curl; ⌦) ! H(div; ⌦) ! L2 (⌦).
Here r⇥ and r· are the curl and divergence operators, respectively. We may also consider the de
Rham complex with boundary conditions:
r
r⇥
r·
H01 (⌦) ! H0 (curl; ⌦) ! H0 (div; ⌦) ! L2 (⌦).
(2.23)
deRhambd
Here the relevant condition on H(curl) functions is u ⇥ ~n = 0 on @⌦, and on H(div) functions
u · ~n = 0 on @⌦. We also recall the Helmholtz (Hodge) decomposition for H(curl) functions:
H0 (curl; ⌦) = rH01 (⌦) H1 Z? . This decomposition is orthogonal with respect to the L2 inner
produce. H1 is the set of harmonic fields, i.e., vector fields which are divergence- and curl-free (and
which satisfy the given boundary condition), and Z? is the orthogonal complement of rH01 H1 in
H0 (curl). H1 is a finite-dimensional space and is trivial if ⌦ is simply connected. We recall as well
the relationships curl r = 0, div curl = 0.
We are interested in understanding regularity of (for example) the time-harmonic Maxwell’s
equation
curl curl u
! 2 u = f in ⌦,
u ⇥ ~n = 0 on @⌦.
(2.24)
Here u is a vector field representing the electric field; we will be more precise about regularity
momentarily. Also, r⇥ = curl is the curl operator, and u ⇥~n = 0 is an essential boundary condition
(i.e., it must be prescribed in the natural variational formulation). Typically we also assume that
r · f = 0. The Hodge decomposition implies that if r · f = 0 and ! 6= 0, then we must have r · u = 0.
If ! = 0 it is standard to enforce the gauge constraint r · u = 0.
maxwell
38
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS
maxwell
We only concern ourselves with regularity of solutions to (2.24) (not with solvability). For this
purpose we may instead study the regularized problem with ! = 0:
u = curl curl u
r div u = f in ⌦,
u ⇥ ~n = 0 on @⌦,
(2.25)
vec_laplace
div u = 0 on @⌦.
The operator curl curl r div is referred to as the vector Laplacian, and can equivalently be obtained
by taking the Laplacian of the input vector componentwise. Let
X = {u 2 H0 (curl; ⌦) \ H(div; ⌦)}.
vec_laplace
The variational formulation of (2.25) is: Find u 2 X such that
Z
Z
curl u · curl v + div u div v =
f · v, v 2 X.
⌦
(2.26)
var_vecl
⌦
(Here we no longer assume div f = 0.)
We now make a few side notes about the space X. Let
D( ) = { 2 H01 (⌦) :
2 L2 (⌦)}.
deRhambd
The de Rham complex with boundary conditions (2.23) implies that if
2 D( ), then rpsi 2
H0 (curl; ⌦). In addition, r 2 H(div; ⌦), since
= div r 2 L2 (⌦). That is, 2 D( ) implies
that r 2 X, and so D( ) ⇢ X. This implies that if D( 6⇢ H 2 (⌦), then X 6⇢ H 1 (⌦). We know
from above that D( ) ⇢ H 2 (⌦) if and only if ⌦ is convex. Thus if ⌦ is not convex, X 6⇢ H 1 (⌦), and
in fact it can be shown that X is even a closed subspace of H01 (⌦). Thus it is not always possible
to approximate v 2 Xvar_vecl
by a sequence {vn } ⇢ H01 (⌦). This causes some difficulties when attempting
to numerically solve (2.26) on nonconvex polyhedral domains. In this case, use of conforming finite
element methods requires a discrete space Vh ⇢ X, i.e., Vh must both be in H(curl) and H(div).
For typical finite element spaces of this form (i.e., piecewise polynomial spaces), this implies that in
fact Vh ⇢ H 1 (⌦). But, X 6⇢ H 1 (⌦), and in fact H 1 is a closed
subspace of X. It thus in essence
var_vecl
impossible to build a conforming finite element method for (2.26) in the usual way, and in fact on
a nonconvex
polygonal domain attempting to do so leads to a method that converges to the wrong
AFW10
solution [3]. Mixed methods should be used instead.
We return to our original goal of understanding the regularity of u on polyhedral domains. We
return to our strategy of looking for harmonic singular functions near edges and corners. We first
consider a model edge problem:
u = 0 in Ke ,
u ⇥ ~n = 0 on @Ke ,
(2.27)
vec_laplace_edge
div u = 0 on @Ke .
Writing x = (y, z) near e with z coordinates along e and y coordinates orthogonal to e, we further
decompose the above as u(x) = (v(x), w(x)) with v normal to e and w tangent to e.
yv
= f in Ke ,
v ⇥ ~n = 0 on @Ke ,
div v = 0 on @Ke .
(2.28)
vec_laplace_e2
2.3. OTHER ELLIPTIC MODEL PROBLEMS
39
and
yw
= 0 in Ke ,
w = 0 on @Ke .
(2.29)
We also consider a model corner problem:
u = 0 in Kv ,
u ⇥ ~n = 0 on @Kv ,
(2.30)
div u = 0 on @Kv .
In the above expressions we shall look for v, w, u in the forms r '(✓), with:
1. Re >
n/2 (L2 fields) and Re < 2
n/2 (L2 right hand side)
2. curl u = 0 and rot v = 0, or Re > 1
n/2 (L2 curls)
3. div u = 0 and div v = 0, or Re > 2
n/2 (H 1 divergence)
4. For w, Re > 1
n/2 (L2 vector curl)
⇡/!
We take w = ⇢e e sin ⇡✓e /!e to be a Laplace-Dirichlet singularity on Ke , and v to be the
gradient of a Laplace-Dirichlet singularity on Ke , which we can easily compute has an exponent of
1. u can take on one of two forms. It may either be the gradient r⇢v v '(✓v ) of a
e = ⇡/!e
Dirichlet-Laplace singularity, or it may satisfy curl u = r with a Laplace-Neumann singularity.
The smallest exponent of ⇢v is then v 1. We are thus left with the generally (but not absolutely
always) applicable heuristic that: “Regularity of Maxwell = Regularity of Dirichlet Laplacian minus
one.” More formally, we have the following.
Theorem 2.3.1 Let f 2 L2 (⌦)3 . Let s 2 (0, 2]. If
s
3/2 <
s
v
1 < ⇡/!e
1, v 2 V,
1, e 2 E,
then u 2 H s (⌦).
Note that this theorem can easily lead to regularity H s (⌦) with s < 1. For example, if !e = 3⇡/2,
then the condition s 1 < ⇡/!z 1 reduces to s < 2/3.
vec_laplace_co
40
CHAPTER 2. ELLIPTIC PROBLEMS ON POLYHEDRAL DOMAINS