Chapter 4
Local and pointwise error estimates
In this chapter we make a brief excursion into the area of local and pointwise (maximum-norm)
error estimates for the finite element method. We shall emphasize a priori error estimates. In
keeping with the spirit of the previous chapters, we will pay careful attention to domain regularity
in our estimates. Another issue which arises when proving error estimates in norms other than the
global energy norm is mesh regularity. It is straightforward to prove optimal a priori error estimates
in the energy norm on shape-regular grids, and in particular on adaptively-generated grids. The
situation for other norms has proved to be much more difficult, and we will highlight this issue in
our discussion as well.
4.1
Local energy estimates
Local energy estimates are both interesting in their own right and useful as a tool for proving
maximum-norm error estimates. We begin with motivation from analysis, then establish an important technical tool (superapproximation), and finally proceed to prove local error estimates.
4.1.1
Caccioppoli inequalities
It is standard intuition that one cannot bound a stronger norm of a function by a weaker norm, but
this intuition turns out to be wrong (or at least incomplete) for harmonic functions.
Proposition 4.1.1 Let
u = 0 on ⌦, with u 6= 0 on @⌦. Let also D ⇢ ⌦ and d > 0 be given such
that Dd := {x 2 ⌦ : dist(x, D)  d} ⇢⇢ ⌦. Then
krukL2 (D) . d
1
kukL2 (Dd ) .
(4.1)
Proof. Let ! 2 C01 (Dd ) be a cuto↵ function satisfying ! ⌘ 1 on D and kDm !kL1 (Dd ) . d m ,
m 0. Then
Z
2
2
krukL2 (D)  k!rukL2 (⌦) =
! 2 ru · ru
⌦
Z
Z
(4.2)
=
ru · r(! 2 u) 2 (!ru) · (ur!)
⌦
⌦
77
cacc
78
But
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES
R
⌦
ru · r(! 2 u) = 0 because u is harmonic. Applying Hölder’s inequality, we thus obtain
k!ruk2L2 (⌦)  2k!rukL2 (⌦) kukL2 (Dd ) kr!kL2 (⌦) . d
1
k!rukL2 (⌦) kukL2 (dd ) .
Dividing through by k!rukL2 (⌦) completes the proof.
4.1.2
2
Superapproximation
Superapproximation is an essential
tool in local and pointwise error analysis for FEM. It was introNS74
duced in the classical DGS11
paper [41] in which local error estimates were first proved. We give a slightly
sharper version from [22] which will allow us to prove local error estimates on general shape regular
grids.
Lemma 4.1.2 Assume that T is a shape-regular simplex in Rd , and that ! 2 C 1 (T ) satisfies
kDm !kL1 (T ) . d m , m = 0, ..., r + 1 with d
hT := diam(T ). Let also Ih be the Lagrange
interpolant on T into Pr . Then for 2 Pr , there holds
Ih (! 2 )kH 1 (T ) .
k! 2
hT
hT
k!r kL2 (T ) + 2 k kL2 (T ) .
d
d
(4.3)
superapprox
(4.4)
eq2-3
Proof. We use Hölder’s inequality and standard approximation theory to find
n/2
k! 2
Ih (! 2 )kH 1 (T ) ChT k! 2
n/2+r
ChT
1 (T )
Ih (! 2 )kW1
r+1
|! 2 |W1
(T ) .
Noting that D↵ = 0 for all multiindices ↵ with |↵| = r + 1, recalling that
inverse estimates, we compute
n/2+r
hT
n/2+r
r (T ) . h
|! 2 |W1
T
.(
r+1
X
i=2
+
r+1
X
i=1 |↵|=i,| |=r+1 i
kD↵ ! 2 D
n/2+r
hT
X
↵
2
kD ! D
hT
n/2+r
k kL2 (T ) + hT
d2
X
 1, and employing
kL1 (T )
i (T ) )k kL (T )
hiT 1 |! 2 |W1
2
|↵|=1,| |=r
.
X
hT
d
(4.5)
kL1 (T )
|↵|=1,| |=r
kD↵ ! 2 D
kL1 (T ) .
We next consider the terms kD↵ ! 2 D kL1 (T ) . Since |↵| = 1, we have D↵ ! 2 = 2!D↵ !. Let
R
hT
1 (T ) .
!
ˆ = |T1 | T ! dx so that k! !
ˆ kL1 (T ) . hT |!|W1
d . Employing inverse estimates, we thus
eq2-4
4.1. LOCAL ENERGY ESTIMATES
79
have
X
n/2+r 1
hT
|↵|=1,| |=r
.d
1 n/2+r
hT
kD↵ ! 2 D
X
| |=r
.d
1 n/2+r
hT
X
| |=r
hT
k kL2 (T ) +
d2
hT
.( 2 k kL2 (T ) +
d
.(
kL1 (T )
k!D
kL1 (T )
(k(!
!
ˆ )D
kL1 (T ) + kˆ
!D
kL1 (T ) )
(4.6)
eq2-5
(4.7)
eq2-6
hT
|ˆ
! |H 1 (T ) )
d
hT
hT
|(ˆ
! !) |H 1 (T ) +
|! |H 1 (T ) ).
d
d
Using an inverse inequality, we find that
hT
|(ˆ
!
d
eq2-6
hT
1 (T ) k kL (T ) + kˆ
(|!|W1
! !kL1 (T ) | |H 1 (T ) )
2
d
hT 1
hT
C
( k kL2 (T ) +
| |H 1 (T ) )
d d
d
hT
C 2 k kL2 (T ) .
d
!) |H 1 (T ) 
eq2-5
eq2-4
eq2-3
Inserting (4.7) into (4.6) and the result into (4.5) and (4.4) completes the proof.
4.1.3
2
Local energy estimates
NS74
We now prove a local error estimate. Such estimates were first proved
in [41] and improved to allow
DGS11
general shape-regular
(as
opposed
to
quasi-uniform)
meshes
in
[22].
As
an aside, the local energy
DGS11
estimate in [22] is the only error estimate we are aware of for finite element methods in a norm other
than a global energy norm that is valid on arbitrary shape-regular grids.
Theorem 4.1.3 Assume that Th is a shape-regular simplicial decomposition of a domain ⌦ ⇢ Rd ,
and that Sh is a Lagrange finite element space of degree r on Th . Let also D ⇢ ⌦ be given, and given
d > 0 let Dd := {x 2 ⌦ : dist(x, D) < d}. Assume that KhT  d for each T 2 Th with T \ Dd 6= ;
and K sufficiently large. Finally, let u 2 H 1 (⌦), and assume that uh 2 Sh satisfies
Z
Z
rurvh =
ruh rvh , vh 2 Sh , vh = 0 on ⌦ \ Dd .
Dd
⌦
Then
kr(u
uh )kL2 (D) . inf
2Sh
✓
kr(u
loc_est
1
)kL2 (Dd ) + ku
d
kL2 (Dd )
◆
1
+ ku
d
uh kL2 (Dd ) .
(4.8)
We discuss the bound (4.8). The first two terms kr(u
)kL2 (Dd ) + d1 ku
kL2 (Dd ) are local
almost-best-approximation terms; we could not expect much better of the local error behavior. The
last term d1 ku uh kL2 (Dd ) is a “pollution” term which expresses influence of the global solution
loc_est
80
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES
quality on the local error behavior. Note carefully two things. First, the pollution term is measured
in a weaker norm. If u is smooth, we can expect this term to decrease at a faster rate than the
energy error as the mesh is refined. Second, the pollution term is penalized by a factor of d 1 . Thus
if we try to cut the extended (cuto↵) domain Dd too close to D and thus measure the approximation
error over a smaller domain, we will be penalized for it in the pollution term.
The proof we
give is short and quite elementary given the superapproximation estimate above.
NS74, DGS11
That given in [41, 22],
on the other hand, is slightly more roundabout in that it first proves an
cacc
estimate similar
to
(
4.1)
for “discrete harmonic” functions and then obtains the estimate below.
loc_est
Proof (4.8). cacc
Let 2 Sh , and let ! be a cuto↵ function which is 1 on D and 0 outside of Dd/2 as
in the proof of (4.1). Then
)k2L2 (D)  k!r(uh
Z
=
r(uh
Z⌦
.
r(uh
kr(uh
)k2L2 (Dd )
2
)r[! (uh
)]
2
Z
!r(uh
⌦
)r!]
(4.9)
1
)] + k!r(uh
d
)r[! 2 (uh
)[(uh
⌦
)kL2 (Dd ) kuh
kL2 (Dd )
:= I + II
The assumption that ! is 0 outside of Dd/2 and that d Kh with K sufficiently large ensures that
Ih (! 2 (uh
)) is 0 outside of Dd . We may thus apply Galerkin orthogonality to obtain
Z
Z
2
I=
r(uh u)r[! (uh
)] +
r(u
)r[! 2 (uh
)]
⌦
⌦
Z
.
r(uh u)r[! 2 (uh
) Ih (! 2 (uh
))]
(4.10)
⌦
1
+ kr(u
)kL2 (Dd ) ( kuh
kL2 (Dd ) + k!r(uh
)kL2 (Dd ) )
d
:= Ia + Ib .
superapprox
We finally employ the superapproximation estimate (4.3), an inverse estimate, hT /d . 1, and
Young’s inequality to find for ✏ > 0
X
Ia 
kr(u uh )kL2 (T ) hT (d 1 k!r(uh
)kL2 (T ) + d 2 kuh
kL2 (T ) )
T \Dd 6=;
X
.
(kr(u
)kL2 (T ) + kr(
uh )kL2 (T ) )hT (d
kr(u
)kL2 (T ) (k!r(uh
)kL2 (T ) + d
T \Dd 6=;
X
.
T \Dd 6=;
+k
.✏
1
uh kL2 (T ) (d
r(u
1
)k2L2 (Dd )
k!r(uh
+d
2
(ku
)kL2 (T ) + d
uh k2L2 (Dd )
2
1
kuh
+ ku
1
k!r(uh
kuh
)kL2 (T ) + d
2
kuh
kL2 (T ) )
kL2 (T ) )
kL2 (T ) )
2
kL2 (Dd ) ) +
✏k!r(uh
)k2L2 (Dd ) .
(4.11)
Similarly employing Young’s inequality on the terms Ib and II yields
k!r(uh
)k2L2 (Dd ) . ✏
1
r(u
+ ✏k!r(uh
)k2L2 (Dd ) + d
)k2L2 (Dd ) .
2
(ku
uh k2L2 (Dd ) + ku
k2L2 (Dd ) )
(4.12)
4.2. L1 ERROR ESTIMATES ON SMOOTH DOMAINS
81
Taking ✏ small enough to reabsorb the last term completes the proof.
2
4.2
L1 error estimates on smooth domains
We
next prove sharp pointwise error estimates on smooth domains, following the work of Schatz
S98
[44]. From these we can easily recover standard maximum-norm error estimates.
4.2.1
Statement of results
We assume that ⌦ ⇢ Rd is given with @⌦ smooth, and that u 2 H 1 (⌦) satisfies
Z
Z
L(u, v) :=
Arurv + b · ruv + cuv =
f v =: (f, v), all v 2 H 1 (⌦)
⌦
⌦
for a coercive and continuous bilinear form L with smooth coefficients A, b, c. Here natural boundary
conditions Aru · ~n = 0 are enforced implicitly.
Our FEM is defined as follows. We assume that Th is a quasi-uniform mesh (NOT shape-regular;
we have significantly restricted our choice of meshes!) of mesh width h and corresponding Lagrange
finite element space Sh of degree r. The finite element solution is the unique uh 2 Sh satisfying
L(uh , vh ) = (f, vh ) for all vh 2 Sh . Because @⌦ is smooth, this means that some “triangles” have a
curved face, so it is not obvious that standard approximation properties, etc., automatically hold.
We will however assume that all needed properties (approximation estimates, inverse estimates,
superapproximation) are valid even on boundary elements. We fix a point x0 2 ⌦ at which the error
is to be measured and finally define a weight
x0 ,h (x)
Note that 0 <
x0 ,h
=
|x
h
.
x0 | + h
 1 with equality holding at x0 , and that
(4.13)
⇡ h when |x
sigma_def
x0 | ⇡ 1.
Theorem 4.2.1 Let the assumptions on the solution u, finite element approximation uh , and finite
element space Sh given above hold, and let 0  s  r 1. Then for x0 2 ⌦
|(u
1
uh )(x0 )| . h(ln )
h
s,r
1
inf [k
2Sh
s
x0 ,h (u
)kL1 (⌦) + k
s
x0 ,h r(u
)kL1 (⌦) ]
(4.14)
pointwise
(4.15)
maxnorm
Thus
ku
1
uh kL1 (⌦) . h(ln )
h
maxnorm
1r
inf ku
2Sh
1
r
1 (⌦) . h (ln
kW 1
)
h
1r
r+1
|u|W1
(⌦) .
The estimate (4.15) is contained in a number
of previous papers going back to the 1970’s under
pointwise
various assumptions on the regularity of @⌦. (4.14) is a “finer” estimate in that it measures the error
at a point instead of in the maximum norm and expresses a more local dependence of the error at x0
on the properties of u. More
precisely, when r > 1 (piecewise quadratic and higher-degree elements),
pointwise
we may take s > 0 in (4.14). In this case the approximation error at points x with |x x0 | ⇡ 1 is
multiplied by xs 0 ,h (x0 ) ⇡ hs , and we have | x0 ,h (x)s r(u
)(x)|  hr+s . On the other hand, at x0
we have x0 ,h (x0 ) = 1 and thus here xs 0 ,h (x0 )|r(u
)(x0 )| ⇡ hr in general. That is, the weight
de-emphasizes the influence of the approximability of u at points removed from the target point x0 .
82
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES
4.2.2
proof
pointwise
The proof of (4.14) is rather involved; for the sake of time we shall omit some details.
Step 1: Error representation by a regularized Green’s function. Given x0 2 ⌦, there is a smooth
regularized -function x0 supported in the element T 2 Th containing x0 such that
(vh ,
x0
) = vh (x0 ), vh 2 Sh ,
(4.16)
delta
(4.17)
deltaest
(4.18)
delta_rep
and
x0
k
kW p,m (⌦) . h
d(1 1/p) m
.
2 Sh ,
Thus for
|(u
uh )(x0 )|  |(u
)(x0 )| + |(
 |(u
)(x0 )| + |(
. ku
x0
uh ,
u,
kL1 (Tx0 ) + |(u
x0
)|
)| + |(u
uh ,
x0
x0
uh ,
)|
)|.
The first term above is bounded by Ck x0 ,h (u
)kL1 (⌦) , so we are left to bound the second.
Step 2: Dual representation by a regularized Green’s function. Let g 2 H 1 (⌦) satisfy L(v, g) =
(v, x0 ) for all v 2 H 1 (⌦). Let also gh 2 Sh satisfy L(vh , gh ) = L(v, g), vh 2 Sh . Using Galerkin
orthogonality yields
(u
uh ,
x0
) = L(u
 (k
⇥
uh , g)
s
x0 ,h (u
(k xs 0 ,h (g
= L(u
,g
gh )
s
)kL1 (⌦) + k x0 ,h r(u
gh )kL1 (⌦) + k x0s,h r(g
)kL1 (⌦) )
(4.19)
error_rep
gh )kL1 (⌦) ).
We thus must show that
k
s
x0 ,h (g
gh )kL1 (⌦) + k
gh )kL1 (⌦) . h
s
x0 ,h r(g
s,r
1
.
Step 3: Dyadic decomposition and local energy bounds. We first decompose ⌦ into dyadic annuli
about x0 . Let M be a positive constant which we shall eventually take to be sufficiently large, and
let dj := 2j M h, j = 0, 1, ..., J with J the smallest integer so that dJ > diam(⌦). Let
⌦j := {x 2 ⌦ : dj
⌦0 := BM h (x0 ),
Then ⌦ = [Jj=0 ⌦j . Note also that
⌦0j = ⌦j
1
x0 ,h (x)
1
 |x
⇠ hdj 1 for x 2 ⌦j , j
[ ⌦j [ ⌦j+1 , ⌦00j = ⌦0j
1
loc_est
s
x0 ,h (g

.
gh )kL1 (⌦) + k
J
X
j=0
J
X
j=0
d/2
s
x0 ,h r(g
(h/dj )s dj kg
(h/dj )
gh )kL1 (⌦) . h
s,r
1.
)kL2 (⌦j ) + dj 1 k(g
dyad_def
(4.21)
dyadic
1. We also define
2 Sh
1
gh kH 1 (⌦j )
s d/2
dj (kr(g
(4.20)
[ ⌦0j [ ⌦0j+1 , etc.
Using a local estimate such as (4.8), we then compute for any
k
x0 | < dj }, j
)kL2 (⌦j ) + dj 1 kg
gh kL2 (⌦0j ) ).
4.2. L1 ERROR ESTIMATES ON SMOOTH DOMAINS
83
We next use the Green’s function (not the regularized variety), which satisfies L(Gx0 , v) =
v(x0 ) = (Gx0 , Lv) for any x0 2 ⌦, where L(w, v) = (w, Lv), w 2 H 1 (⌦). It is known that if all
problem data are smooth,
Dx↵0 Dx Gx0 (x) . |x
Thus for y 2 ⌦0j (j
d |↵| | |
x0 | 2
, |↵| + | |
1.
green_decay
(4.23)
green_approx
2) and |↵| = r + 1,
D↵ g(y) = Dy↵ (Gy ,
x0
) = (Dy↵ Gy ,
 kDy Gy kL1 (Tx0 ) k
x0
x0
)
2 d (r+1)
kL1 (Tx0 ) . dj
.
Applying approximation properties and recalling that h/dj  1, we thus have for j
(h/dj )
s d/2
dj (kr(g
. (h/dj )
. (h/dj )
. (h/dj )
)kL2 (⌦j ) + dj 1 k(g
)kL1 (⌦j ) )
s d r
r+1
dj h |g|W1
(⌦0j )
(4.24)
s d r 2 d r 1
dj h dj
1 s
= h(h/2j M h)r
1 s
. h2
j(r 1 s)
For j = 0, 1, we have dj ' h. Using standard H 2 regularity kgkH 2 (⌦) . k
have
s d/2
dj (kr(g
2
)kL2 (⌦j ) )
)kL1 (⌦j ) + dj 1 k(g
s d
dj (kr(g
= h(h/dj )r
(h/dj )
(4.22)
x0
.
kL2 (⌦) . h
)kL2 (⌦j ) ) . hd/2 hkgkH 2 (⌦) . h.
)kL2 (⌦j ) + dj 1 k(g
d/2
, we thus
(4.25)
dyadic
Inserting the last two inequalities into (4.21) yields
k
s
x0 ,h (g
gh )kL1 (⌦) + k
.h+h
J
X
2
gh )kL1 (⌦) . h
s
x0 ,h r(g
j(r 1 s)
j=2
+
J
X
d/2 1
(h/dj )s dj
j=0
1
. h(1 + (ln )
h
s,r
1
)+
J
X
d/2 1
(h/dj )s dj
j=0
s,r
kg
kg
1
gh kL2 (⌦0j )
(4.26)
eq4-26
gh kL2 (⌦0j ) .
PJ
s > 0, a geometric sum yields j=2 2 j(r 1 s) . 1,
PJ
j(r 1 s)
= j=2 1 ⇡ J ⇡ ln h1 .
PJ
d/2 1
Step 4: Second duality argument. We now approach the term j=0 (h/dj )s dj
kg gh kL2 (⌦0j ) .
This is the heart of the proof. We first clean up our notation a little by noting
Here we have used the fact that when r
PJ
and when r 1 s = 0 we have j=2 2
J
X
(h/dj )
j=0
s d/2 1
dj
kg
1
gh kL2 (⌦0j ) .
J
X
(h/dj )
j=0
s d/2 1
dj
kg
gh kL2 (⌦j ) .
(4.27)
For any j, we have
kg
gh kL2 (⌦j ) =
sup
vj 2C01 (⌦j ),kwj kL2 (⌦) =1
(g
gh , vj ).
(4.28)
eq4-27
84
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES
Given such a vj , we let zj solve L(zj , w) = (vj , w), w 2 H 1 (⌦). Then for
chosen vj and associated zj ,
kg
gh kL2 (⌦j ) . L(zj , g
 kzj
gh ) = L(zj
,g
kW 1,1 (⌦\⌦00j ) kg
We first compute for properly chosen
kzj
2 Sh and properly
gh )
gh kW 1,1 (⌦\⌦00j ) + kzj
kH 1 (⌦00j ) kg
gh kH 1 (⌦00j ) .
(4.29)
eq4-29
2 Sh that
kH 1 (⌦00j ) . hkzj kH 2 (⌦) . hkvj kL2 (⌦) . h.
(4.30)
r+1
kW 1,1 (⌦\⌦00j ) . hr |zj |W1
(⌦\⌦0 ) .
(4.31)
Also,
kzj
j
We compute for y 2 ⌦ \ ⌦0j and |↵| = r + 1 that
d/2
D↵ zj (y) = Dy↵ (vj , Gy ) = (vj , Dy↵ Gy )  kvj kL2 (⌦j ) dj kDy↵ Gy kL1 (⌦j )
d/2 2 d |↵|
 dj dj
since |y
2 d/2 r 1
= dj
1 r d/2
= dj
(4.32)
,
dj for any y 2 ⌦ \ ⌦0j , x̃ 2 ⌦j . Finally, we may compute that
x̃|
gh kL2 (⌦) . h inf kg
kg
2Sh
kH 1 (⌦) . h2 kgkH 2 (⌦) . h2
eq4-29
d/2
.
(4.33)
eq4-27
Inserting the previous four inequalities into (4.29) and the result into (4.27) yields
J
X
(h/dj )
j=0
.h
s d/2 1
dj
kg
d/2 1
.h+
kg
J
X
gh kL2 (⌦0j )
gh k
L2 (⌦00
0)
+
J
X
j=0
s d/2 1
dj
(hkg
gh kH 1 (⌦j ) + (h/dj )r
s
x0 ,h (g

gh )kL1 (⌦) + k
J
X
j=0
s
kg
kg
gh kW11 (⌦) )
gh kW 1,1 (⌦) .
(4.34)
Step 5: Double kickback. Recall
now that h/dj 
eq4-26
while recalling the results of (4.26) to find that
k
1 r d/2
gh kH 1 (⌦00j ) + hr dj
j=0
s+1 d/2
dj kg
(h/dj )
(h/dj )
s
x0 ,h r(g
d/2
(h/dj )s dj kg
1
. h(1 + (ln )
h
s,r
1
1
M,
j
eq4-34
dyadic
0. We now insert (4.34) into (4.21)
gh )kL1 (⌦)
gh kH 1 (⌦j )
)+
J
1 X
(h/dj )
M j=0
s d/2
dj kg
gh kH 1 (⌦j ) +
J
X
j=0
(h/dj )r
s
kg
gh kW 1,1 (⌦) .
(4.35)
eq4-34
4.3. W 1,1 ESTIMATES ON POLYHEDRAL DOMAINS
85
Taking M large enough to kick back the terms involving kg
k
s
x0 ,h (g
gh )kL1 (⌦) + k
1
. h(1 + (ln )
h
s,r
s
x0 ,h r(g
1
)+
J
X
gh kH 1 (⌦) above yields
gh )kL1 (⌦)
(h/dj )r
j=0
s
kg
gh kW 1,1 (⌦) .
(4.36)
Pj
PJ
PJ
r s
j
r s
1. In
But
=
= M s r j=0 2 j(r s) . M 1 , since r s
j=0 (h/dj )
J=0 (h/2 M h)
1
s
s
addition, because x0 ,h 1 we have kg gh kW 1,1 (⌦)  k x0 ,h (g gh )kL1 (⌦) + k x0 ,h r(g gh )kL1 (⌦) .
We may thus take M large enough to reabsorb the term involving kg gh kW 1,1 (⌦) above, which
completes the proof.
2
4.3
W 1,1 estimates on polyhedral domains
In this section we briefly discuss issues that arise when trying to prove pointwise error estimates on
polygonal or polyhedral domains. In addition, we shall also discuss attempts to prove such error
estimates on solution-adapted meshes.
4.3.1
Statement of results
GLRS09
RS82, BS08
We state there a theorem from [38]; cf. [43, 15] for other results (including the 2D case).
Theorem 4.3.1 Assume that u 2 H01 (⌦) weakly satisfies
u = f on a convex polyhedral domain
⌦ ⇢ R3 with f sufficiently smooth. Let also uh 2 Sh be its finite element approximation, with
Sh ⇢ H10 (⌦) a Lagrange finite element space on a quasi-uniform mesh. Then
kuh kW 1,1 (⌦) . kukW 1,1 (⌦) ,
(4.37)
winfstab
1 (⌦) . inf ku
u h kW 1
(4.38)
winfapprox
and consequently
ku
2Sh
1 (⌦) .
kW 1
The proof of the above theorem is very similar to the one given in the previous subsection for
bounding |(u uh )(x0 )|. One first represents the error by using regularized Green’s and functions.
Let x0 be a discrete function as above, and let ⌫ be a unit vector in R3 . Then for 2 Sh ,
r · ⌫ = (r ,
x0
⌫) =
( ,r
x0
· ⌫.
(4.39)
Our regularized Green’s function g 2 H01 (⌦) then solves
g = r x0 · ⌫. We may thus represent
the error and proceed much as in the previous subsection, using dyadic decompositions, local energy
estimates, a double kickback argument, etc. The main
di↵erence comes in the regularity of the
green_decay
Green’s function. Because @⌦ is no longer smooth, (4.22) giving the decay of the Green’s function
near the singularity only holds for very restricted ↵, . In fact, if we consider only
integer orders
GLRS09
we must have |↵|  1 and | |  1, which is not sufficient. The main advance in [38] was to prove
86
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES
Hölder estimates for Green’s functions on convex polyhedral domains. More precisely, the estimates
needed are of the form
|@xi @⇠j G(x, ⇠) @yi @⇠j G(y, ⇠)|
. |x
|x y|
GLRS09
⇠|
3
+ |y
⇠|
3
.
(4.40)
greenest
greenest
It is shown in [38] that (4.40) holds on any convex polyhedral domain (under appropriate assumptions) for 0 <
< 1 depending on the maximum interior opening angle of ⌦. Recall that ru
itself is not generally bounded on nonconvex domains; in particular there will in this case be edge
⇡/!
⇡/! 1
singularities of the form ⇢e with ! > ⇡ and thus ru ⇠ ⇢e
has negative exponent and blows
up near the edge e. Thus the restriction that ⌦ is convex is quite natural here.
As an interesting
historicalBS08
note, stability of the FEM in W 1,1 on 2D convex polygonal domains
RS82
was proved in [43]. The text [15] (and earlier versions) contains a 3D version of the result as well,
but with assumptions that restrict the maximum interior
edge opening angle to be strictly less than
GNS05
2⇡
.
This
restriction
is
discussed
more
explicitly
in
[35],
where similar bounds are proved for the
3
Stokes problem. Instead of using Green’s function Hø”lder bounds as above, they employed W p,2
regularity results which were more readily available in the literature at the time. With regard to the
restriction that the edge opening angles be less than 2⇡
3 , the authors stated that the restriction is
natural because of the relationship between the angle condition and W p,2 regularity needed to obtain
pointwise bounds on both the continuous and discrete solution. Such W p,2 estimates do not hold
for
sufficiently large p on arbitrary convex polyhedral domains, while the Green’s function estimates
greenest
(4.40) do. The takeaway lesson is that developing multiple ways to understand PDE regularity is
essential. Here both W p,2 and Hölder regularity come into play and give meaningful information,
but gaining a sharp understanding of pointwise behavior relies on the latter type of estimates.
4.3.2
Mesh regularity
Recall Céa’s Lemma
ku
uh kH 1 (⌦) . inf ku
2Sh
kH 1 (⌦) ,
which holds whenever the associated bilinear form is continuous and coercive. This stability estimate
holds for any mesh, and in particular on the shape-regular butwinfstab
not quasiuniform meshes generated
1,1
by adaptive
refinement
procedures.
The
W
stability
result
(
4.37)
and almost-best-approximation
winfapprox
GLRS09
result (4.38) look similar, but their proof in [38] assumes quasiuniform meshes. It is natural to ask
whether such estimates also hold on shape-regular (adaptive) grids. Several papers have tried to
answer this question, but it remains at least partially
open.
Er94
We briefly describe the approach of Eriksson [28]. To understand his assumption, assume that T
1
is a shape regular grid We construct a mesh size function hT 2 W1
(⌦) as follows. For each T 2 T ,
let hT = diam(T ) as usual. For each vertex z in T , we then let hT (z) be the average of the meshes
sizes hT over all elements T sharing the vertex z. Finally, hT is taken to be continuous and piecewise
linear, so specifying it at vertices determines it uniquely. The shape regularity of T guarantees that
hT (x) ' hT whenever x 2 T 2 T , and in addition that
|rhT | . 1
(4.41)
with constant depending on the shape regularity of ⌦. These facts are easy consequences of the observation that for shape-regular grids, neighboring elements (sharing a face) always have comparable
diameters, and the number of elements sharing a given vertex is always uniformly bounded.
mesh_function
4.3. W 1,1 ESTIMATES ON POLYHEDRAL DOMAINS
87
mesh_function
Eriksson’s idea was to assume instead of (4.41) that
|rhT | . µ, µ sufficiently small.
(4.42)
This condition is significantly more restrictive than shape regularity, but mild_grading
still allows for meaningful
mesh grading. For example, “smooth” mesh gradings which both satisfy (4.42) and which optimally
resolve corner singularities may be constructed.
Eriksson proved L1 estimates on 2D polygonal
mild_grading DLSW11
domains assuming
a
condition
much
like
(
4.42),
while
[24] contains a proof of the W 1,1 stability
winfstab
estimate (4.37) on convex polygonal and polyhedral domains.
mild_grading
88
CHAPTER 4. LOCAL AND POINTWISE ERROR ESTIMATES