RESIDUAL-FREE BUBBLES FOR ADVECTION-DIFFUSION
PROBLEMS: THE GENERAL ERROR ANALYSIS
F. BREZZI, D. MARINI
Dipartimento di Matematica and I.A.N.-C.N.R.
Via Abbiategrasso 215, 27100 Pavia (Italy)
E. S
ULI
University of Oxford, Computing Laboratory,
Wolfson Building, Parks Road, Oxford OX1 3QD (UK)
We develop the general
a priori
error analysis of residual-free bubble nite element
approximations to non-self-adjoint elliptic problems of the form (
"A
+
C
)
u
=
f
sub ject to
homogeneous Dirichlet b oundary condition, where
A
is a symmetric second-order elliptic
operator,
C
is a skew-symmetric rst-order dierential op erator, and
"
is a positive
parameter. Optimal-order error b ounds are derived in various norms, using piecewise
polynomial nite elements of degree
k
1.
1. Intro duction
Although the pap er deals with a slightly more general case, let us consider here,
for simplicity, the following mo del problem: nd
u
2
H
1
0
() such that
"
u
+
u
x
=
f
in
;
(1.1)
where is a bounded p olygonal domain in the plane,
f
is given in
L
2
(), and
" >
0
is very small compared with the diameter of , so that (1.1) is advection-dominated.
Let
fT
h
g
h
be a sequence of partitions of into triangles
T
, let
k
1 b e an integer,
and consider the nite element space
W
h
W
k
h
(
T
h
;
) =
f
v
2
H
1
0
() :
v
j
T
2
P
k
j
T
for each
T
in
T
h
g
:
(1.2)
Here
h
is a positive discretisation parameter which measures the granularity of the
partition
T
h
, and
P
k
j
T
denotes the space of polynomials of degree
k
on
T
. The
usual Galerkin nite element approximation to (1.1) is then
8
<
:
nd
u
G
h
2
W
h
such that
R
(
"
r
u
G
h
r
v
+
u
G
x
v
) d
x
=
R
f v
d
x
8
v
2
W
h
:
(1.3)
1
It is well known that, whenever
"=h <<
1, this method is unstable, which manifests
itself in large maximum-principle-violating oscillations in the numerical solution.
Among several possible remedies for this undesirable feature of the usual Galerkin
approximation, the SUPG metho d ([7], [15]) has attracted considerable attention
over the last decade, primarily b ecause of its attractive combination of structural
simplicity, generality and the quality of the resulting numerical solution. For prob-
lem (1.1) the SUPG method reads
8
>
>
>
>
<
>
>
>
>
:
nd
u
S
h
2
W
h
such that
R
(
"
r
u
S
h
r
v
+
u
S
h;x
v
) d
x
S
(
u
S
h
; v
) =
R
f v
d
x
8
v
2
W
h
;
where
S
(
u
S
h
; v
) =
P
T
T
R
T
(
"
u
S
h
+
u
S
h;x
f
)(
"
v
v
x
) d
x;
(1.4)
and
T
is a parameter which needs to b e chosen suitably. Reasonable rules of thumb
for the choice of
T
can b e found, for instance, in [9] and the references therein; the
corresponding error analysis (for mo del problems like (1.1)) is given in [14].
In recent years, the SUPG metho d has been frequently viewed in a more general
context (see, e.g., [1], [2] and the references therein), and appropriate choices for
the value of
T
(or, more generally, for a suitable form of the stabilizing term to b e
added to (1.3)) found a dierent, and phylosophically more app ealing, justication.
For the particular case of piecewise linear elements, for instance, it was shown
that SUPG can b e also derived by the so-called
residual-free bubble
approach (RFB
from now on; see [6], [10]) as well as by the
local Green's function
approach ([12]).
The connections b etween these two approaches were claried in [2]: both strategies
lead precisely to (1.4), with a very sp ecic value for
T
which can be, therefore,
considered as optimal, at least from the theoretical point of view.
Since for
k
= 1 the SUPG method and the RFB approach (or its equivalent
local Green's function counterpart) yield the same scheme, the results of [14] can
be used for the error analysis. However, as it was shown in [4], an analysis based
on the residual-free bubble framework can arrive at the same results by means of a
completely dierent procedure, casting a new light on the basic underlying features
of the new metho dology.
In contrast with the case of
k
= 1, for
k >
1 the RFB approach produces a
stabilizing term similar but not identical to that of (1.4). As a matter of fact, for
k >
1, (1.4) may b e obtained by suitable
virtual
bubbles (see [1]) { an approach
which do es not follow from the RFB methodology.
The ob jective of this pap er is then to perform the error analysis of the residual-
free bubble metho d for the case of
k >
1. This can b e seen, in a sense, as an
extension of [4], although the techniques of error analysis presented here are quite
dierent and, to the best of our knowledge, completely new in this context. They
are based on sharp interpolation results in certain Besov spaces of dierential order
1
=
2 which do not coincide with the usual Hilbertian Sob olev spaces
H
1
=
2
or
H
1
=
2
00
.
The outline of the pap er is as follows. In Section 2 we present our mo del problem,
and we recall the basic features of the RFB metho d applied to it. In Section 3 we
2
recall the denitions of the Besov spaces mentioned ab ove and we prove some simple
properties of these which will be used in the subsequent analysis. Finally, the error
analysis is presented in Section 4.
Numerical experiments to compare the relative p erformances of SUPG and RFB
for values of
k >
1 would be very interesting but are beyond the scope of this pap er
and are not discussed here.
2. Statement of the problem
Suppose that is a b ounded p olyhedral domain in
R
n
and let
L
be a second-order
linear dierential op erator of the form
L
=
"A
+
C;
(2.1)
where
A
and
C
are dened, for
w
,say, in
H
1
(), by
Aw
n
X
i;j
=1
@
@ x
j
a
ij
(
x
)
@ w
@ x
i
; C w
n
X
i
=1
c
i
(
x
)
@ w
@ x
i
:
We assume that, for almost every
x
in , the
n
n
matrix (
a
ij
(
x
)) is symmetric
and positive denite, with smallest eigenvalue
>
0 and largest eigenvalue
1,
independent of
x
. In a sense, we are
normalizing
the op erator
A
in the pro duct
"A
.
To the operator
A
we assign the bilinear form
a
(
w; v
) =
Z
n
X
i;j
=1
a
ij
(
x
)
@ w
@ x
i
@ v
@ x
j
d
x; w; v
2
H
1
()
:
(2.2)
With the ab ove assumptions
A
is a symmetric operator from
H
1
0
() into
H
1
()
verifying
a
(
w; v
) =
h
Aw; v
i
=
h
w; Av
i 8
w; v
2
V ;
(2.3)
where, from now on,
V
=
H
1
0
()
; V
0
=
H
1
()
;
equipped with resp ective norms
k k
H
1
()
and
k k
H
1
()
, and
h
;
i
denotes the
duality pairing between
V
and
V
0
. Moreover, the bilinear form
a
(
;
) is
V
-elliptic
and
normalised to
1, that is,
j
v
j
2
H
1
()
a
(
v ; v
)
8
v
2
V ;
(2.4)
a
(
v ; w
)
j
v
j
H
1
()
j
w
j
H
1
()
8
v ; w
2
V ;
(2.5)
where
j j
H
1
()
is the seminorm of
V
=
H
1
0
().
Similarly, we assume that, for almost every
x
in , the
n
-component vector
(
c
i
(
x
)) has Euclidean norm
, indep endent of
x
, and we introduce the bilinear
form
c
(
;
), dened by
c
(
w; v
) =
Z
n
X
i
=1
c
i
(
x
)
@ w
@ x
i
v
d
x; w
2
H
1
()
; v
2
L
2
()
:
(2.6)
3
As a consequence, we have
c
(
w; v
) = (
C w ; v
)
8
w
2
H
1
()
;
8
v
2
L
2
()
;
(2.7)
where (
;
) signies the inner product in
L
2
(). Our hypotheses imply that
j
c
(
w; v
)
j
j
w
j
H
1
()
k
v
k
L
2
()
8
w
2
H
1
()
;
8
v
2
L
2
()
;
(2.8)
where
k k
L
2
()
is the norm of
L
2
(). We make the additional assumption that the
bilinear form
c
(
;
) is skew-symmetric on
V
; namely,
c
(
w; v
) =
c
(
v ; w
) = (
C w ; v
) =
(
w; C v
)
8
w; v
2
V :
(2.9)
This can be ensured by requiring that the vector eld
c
= (
c
1
; : : : ; c
n
) is divergence-
free on in the sense of distributions.
For
f
given in
L
2
(), say, we consider the boundary value problem
Lu
=
f
in
;
u
= 0 on
@
:
(2.10)
Let
L
(
;
) be the bilinear form on
V
V
associated with the op erator
L
, namely,
L
(
w; v
) =
"a
(
w; v
) +
c
(
w; v
)
8
w; v
2
V :
(2.11)
We consider the variational form of (2.10):
nd
u
2
V
such that
L
(
u; v
) = (
f ; v
)
8
v
2
V :
(2.12)
Applying (2.11), (2.4) and (2.9), it is easy to check that
"
j
v
j
2
H
1
()
L
(
v ; v
)
8
v
2
V :
(2.13)
By virtue of (2.13) and the Lax-Milgram lemma, (2.12) has a unique solution in
V
.
Next we formulate the RFB-approximation of (2.12). Suppose that we are given
a shap e-regular family of partitions
fT
h
g
h
of into op en
n
-simplices
T
(referred
to as
elements
), and an integer
k
1. We recall that
fT
h
g
h
is said to be a shape-
regular family if there exists a xed positive constant
such that, for each
T
h
and
each
T
2 T
h
,
h
T
T
;
(2.14)
where
h
T
denotes the diameter of the
n
-simplex
T
(i.e. its longest edge), and
T
is
the diameter of the largest ball inscribed in
T
. We set
V
h
V
k
h
(
T
h
;
) =
f
v
2
V
:
v
j
e
2
P
k
j
e
for each (
n
1)-dimensional
face
e
of any element
T
in the partition
T
h
g
:
(2.15)
Here
P
k
j
e
denotes the set of all p olynomials in (
n
1) variables of degree
k
on
the face (or edge for
n
= 2)
e
. The discrete counterpart of (2.12) is then
4
nd
u
h
2
V
h
such that
L
(
u
h
; v
h
) = (
f ; v
h
)
8
v
h
2
V
h
:
(2.16)
Notice that
V
h
is
not
the usual nite element space of continuous piecewise
polynomial functions (that would b e the space
W
h
dened in (1.2)), but can b e
thought of as b eing obtained by supplementing
W
h
by the space of
al l
functions in
H
1
0
(
T
), for
al l
T
in
T
h
. More precisely,
V
h
=
W
h
+
B
h
;
(2.17)
where
B
h
=
M
T
2T
h
H
1
0
(
T
)
:
(2.18)
In particular,
V
h
is not nite-dimensional. In the following discussion we shall
show that problem (2.16) is equivalent to a nite-dimensional one. However, work-
ing on formulation (2.16) makes the analysis simpler. For instance we can imme-
diately p oint out that, for every
T
2 T
h
, and for every
'
2
C
1
0
(
T
), it is possible
to construct
v
h
2
V
h
by selecting
v
h
=
'
in
T
, and
v
h
identically zero outside
T
.
Consequently, from (2.16), we conclude easily that
Lu
h
=
f
in each
T
in
T
h
;
(2.19)
which is the property that justies the name
residual-free
.
In the remaining part of this Section we shall analyse (2.16) from the point of
view of the possible computational techniques. In doing so, we shall also clarify its
relationships with the SUPG-method.
We start the analysis with the following considerations, typical of the RFB-
approach (see, e.g., [2]). The solution
u
h
of (2.19) is a p olynomial of degree
k
on
each
e
of
@ T
(see (2.15)). Let
p
k
be a p olynomial of degree
k
in
T
having the
same b oundary value (on
@ T
) as
u
h
. Such a p olynomial is not unique for
k > n
,
but this is not essential. Then,
u
h
=
p
k
+
u
b
;
(2.20)
where
u
b
2
H
1
0
(
T
) and, using (2.19), solves
L
T
u
b
=
Lp
k
+
f
in each
T
in
T
h
:
(2.21)
where
L
T
:
H
1
0
(
T
)
!
H
1
(
T
) denotes the restriction of the op erator
L
to
T
;
namely,
L
T
w
=
Lw
for all
w
2
H
1
0
(
T
),
T
2 T
h
. Notice that
L
T
is injective, so that
(2.21) can b e written as
u
b
=
L
1
T
(
Lp
k
+
f
) in each
T
in
T
h
:
(2.22)
Assume now that
f
is a piecewise polynomial of degree
(
k
1), and
A
and
C
have
piecewise constant coecients. Then, in each
T
, the right-hand side of (2.21) is a
5