Domain embedding preconditioners
for mixed systems1
Torgeir Rusten
SINTEF, P. O. Box 124 Blindern, N-0314 Oslo, Norway.
Panayot S. Vassilevski
Center of Informatics and Computer Technology, Bulgarian Academy of Sciences, \Acad.
G. Bontchev" street, Block 25 A, 51113 Soa, Bulgaria
Ragnar Winther
Department of Informatics, University of Oslo, P. O. Box 1080 Blindern, N-0316 Oslo,
Norway.
In this paper we study block diagonal preconditioners for mixed systems derived from the Dirichlet
problems for second order elliptic equations. The main purpose is to discuss how an embedding
of the original computational domain into a simpler extended domain can be utilized in this case.
We show that a family of uniform preconditioners for the corresponding problem on the extended,
or ctitious, domain leads directly to uniform preconditioners for the original problem. This is in
contrast to the situation for the standard nite element method, where the domain embedding
approach for the Dirichlet problem is less obvious.
KEY WORDS second order elliptic problems, Dirichlet boundary conditions, mixed
nite elements, preconditioning, domain embedding, auxiliary space method, nonconforming nite elements
1
This work was partially supported by the Research Council of Norway (NFR), programs
100998/420, 107547/410 and STP.29643. The work of the third author was also partially supported by the Bulgarian Ministry for Education, Science and Technology under grant I{95 # 504
2
Rusten, Vassilevski, and Winther
1. Introduction
We consider mixed nite element approximations for second order elliptic equations,
with Dirichlet boundary conditions, of the form
; div(a grad p) = f in p = g on @ :
(1.1)
Here R2 is a bounded polygonal domain and @ is the boundary. The coecient
matrix a is assumed to be bounded and uniformly positive denite on . The
main purpose of this paper is to discuss how to construct preconditioners for the
corresponding discrete linear systems where an embedding of the domain into an
extended, or ctitious, more regular domain e is utilized.
If the boundary value problem (1.1) is discretized by a standard conforming nite
element method a symmetric and positive denite system of the form
Ah ph = fh
is obtained. Here, h > 0 is a small parameter indicating the mesh size. However,
since the condition number of the coecient operator Ah increases with decreasing values of h, the behavior of iterative methods depends on the construction of
eective preconditioners Bh .
In a domain embedding approach we utilize an embedding of the original domain
into an extended domain e to construct a proper preconditioner Bh . This is in
contrast to domain decomposition methods where the domain is split into a number of subdomains. In practical applications the geometry of e will be simpler, or
more regular, than the geometry of . Hence, it is reasonable to assume that corresponding preconditioners on the domain e , for example constructed by multilevel
methods, is more easily obtained. In certain applications, it can even be possible
to apply a fast solver on the extended domain. For early papers discussing domain
embedding as a tool for solving discrete systems we refer to Astrakhantsev 4] and
Buzbee, Dorr, George and Golub 10]. For some more recent developments, cf. for
example Proskurowski and Vassilevski 23] and 24].
For problems of the form (1.1), but with the Dirichlet boundary conditions replaced by natural boundary conditions, application of domain embedding is rather
straightforward. In this case we can apply preconditioners Bh of the form
Bh = Rh Beh Eh (1.2)
where Beh is a corresponding preconditioner with respect to the domain e , Eh
is essentially the extension by zero operator and Rh is the operator dened by
restricting functions to . Under proper conditions on Beh the preconditioner Bh
will be a uniform preconditioner for Ah , i.e., the spectral condition number of
Bh Ah is independent of the mesh parameter h (cf. for example Astrakhantsev 4]
or Marchuk, Kuznetsov and Matsokin 19]).
For Dirichlet (or essential) boundary conditions the situation is not as straightforward. One way to remove the essential boundary condition is to use the Lagrange
multiplier method for the Dirichlet problem introduced by Babuska 5]. The resulting saddle point system can then be preconditioned by a block diagonal operator,
Domain embedding preconditioners
3
where one of the blocks will correspond to a preconditioner for the Neumann problem. Hence, this part of the operator can be constructed by domain embedding as
indicated above. The second block of the preconditioner is a boundary operator
dened on the interface between and + e n . For a general description
and various applications of Lagrange multipliers and domain embedding for the
Dirichlet problem we refer to Dinh, Glowinski, He, Kwock, Pan and Periaux 13]
and Rossi 26]. An alternative approach, utilizing an approximate harmonic extension, is suggested by Nepomnyaschikh 20] and studied in detail in 21], 22], while
Finogenov and Kuznetsov 14] study a two{stage method for the Dirichlet problem.
The latter approach utilizes inexact but suciently accurate solutions of problems
on the extended domain e . The problem of inexact solutions on the extended domain was also treated in Borgers and Widlund 6]. Another version of the domain
embedding method in the framework
of 20], based on recently proposed bounded
1
2
extension operators exploiting H {stable wavelet{like hierarchical decomposition
of conforming nite element spaces, was presented in Vassilevski 28].
In the present paper we shall consider the discrete systems obtained from the
mixed nite element method for the problem (1.1). This discretization procedure
will lead to an indenite saddle point system. Also, the Dirichlet boundary conditions are natural boundary conditions for the mixed weak formulation of the
problem (1.1). Therefore, as we shall see below, domain embedding preconditioners
for these systems can be constructed rather naturally of the form (1.2).
In x2 we will give a brief review of the mixed nite element method for elliptic
boundary value problems of the form (1.1), while preconditioning of saddle point
problems is discussed in x3. In particular, we will describe the tight connection between preconditioners for the indenite mixed system and preconditioners for the
positive denite operator derived from the inner product in the space H (div). Domain embedding preconditioners for this positive denite operator will be discussed
in x4. A key tool in the analysis is the construction of an extension operator which
is bounded in H (div). In x5 we show how the H (div) preconditioner proposed in
x4 can be combined with the auxiliary space technique (cf. Xu 32]), to construct
preconditioners for the systems obtained from the nonconforming Crouzeix{Raviart
method by viewing this method as a nonconforming mixed method. Finally, some
numerical experiments are presented in x6.
We adopt, throughout this paper the following notation: functions and spaces in
boldface will denote vector elds and vector valued function spaces, respectively.
Similarly, operators in boldface (such as grad or curl) will have actions in vector
valued function spaces.
2. Preliminaries
The inner product in L2 (
) will be denoted ( ) and the same notation will be used
to denote the corresponding inner product for vector valued functions. The Sobolev
space of functions with derivatives of order less than or equal to m in L2 (
) will
be denoted H m(
), and the associated norm by jj jjm . Furthermore, H (div )
is the space consisting of square integrable 2{vectorelds with square integrable
4
Rusten, Vassilevski, and Winther
divergence. The inner product in H (div ) is given by
(u v) = (u v) + (div u div v)
(2.1)
and we let
jjv jjdiv = (v v)1=2 :
For domains K R2 , dierent from , we will in general use an extra subscript to
denote norms with respect to the domain K , for example jjjj0K and jjjjdivK will
be the norms in L2 (K ) and H (div K ), respectively.
The mixed weak formulation of the problem (1.1) is given by:
Find (u p) 2 H (div ) L2 (
) such that
(a;1 u v) + (p div v) = G(v ) for all v 2 H (div )
(2.2)
(div u q) = F (q) for all q 2 L2 (
):
Here u is the auxiliary variable a grad p. The functionals F and G are given by
Z
Z
F (q) = ; fq dx and G(v ) = g v n ds
@
where s denotes the arc length along @ and n is the exterior unit normal vector on
@ . A key observation for the discussion in this paper is the well known fact that
the Dirichlet boundary conditions are natural boundary conditions in the mixed
weak formulation.
The system (2.2) can alternatively be written in operator form as
A up = G
F where the coecient operator, A, is given by
;1
a
I
;
grad
A=
:
div
0
(2.3)
The operator A can be seen to be an isomorphism mapping H (div ) L2 (
) into
it's L2{dual H (div ) L2 (
). A corresponding mapping property for the discrete coecient operator will be the main tool in deriving the structure of eective
preconditioners for the corresponding discrete systems.
The mixed nite element method is derived from the weak formulation (2.2).
Let f(Vh Wh )gh2(01] be pairs of nite element spaces, derived from a family of
triangulations fTh gh2(01], such that Vh H (div ) and Wh L2 (
) where
h denotes the characteristic mesh size. The discrete weak formulation is to nd
(uh ph ) 2 Vh Wh such that
(a;1 uh v) + (ph div v) = G(v ) for all v 2 Vh (2.4)
(div uh q) = F (q) for all q 2 Wh :
This system can similarly be written in the operator form
Ah uphh = GFhh 5
Domain embedding preconditioners
where the operator Ah : Vh Wh 7! Vh Wh has the block structure
h B h :
Ah = A
Bh 0
(2.5)
Here the operators Ah : Vh 7! Vh and Bh : Vh 7! Wh are dened implicitly by
the system (2.4), i.e.,
(Ah u v) = (a;1 u v) for all u v 2 Vh and
(Bh u q) = (div u q) for all u 2 Vh q 2 Wh :
Furthermore, B h : Wh 7! Vh is the L2 {adjoint of Bh .
The family of spaces f(Vh Wh )g are assumed to satisfy a Babuska{Brezzi condition of the form
v) jjqjj0 for all q 2 Wh
sup (qjjvdiv
(2.6)
jjdiv
v2Vh
where > 0 is independent of h. In addition, we assume that
div(Vh ) Wh :
(2.7)
It is well known that the two conditions (2.6) and (2.7) imply the stability of the
mixed nite element method for second order elliptic problems of the form (1.1).
More precisely, let Xh = Vh Wh with norm
jj(v q)jj2Xh = jjvjj2div + jjqjj20 and let Xh be equal to Xh as a set, but with the corresponding dual norm, i.e.,
jj(v q)jj2Xh = jjvjj2 + jjqjj20 where
jjvjj = sup jj(zvjj z ) :
div
z2Vh
The conditions (2.6) and (2.7) imply that the operator norms
jjAh jjL(Xh Xh ) and jjA;h 1 jjL(Xh Xh ) are bounded uniformly in h:
(2.8)
This qualitative description of the system (2.4) is the basic information needed in
order to construct the structure of possible preconditioners for the system.
3. Preconditioning mixed systems
Consider the discrete system (2.4) with coecient operator Ah given by (2.5). Our
aim is to solve this system by an iterative method. However, since Ah arises as a
6
Rusten, Vassilevski, and Winther
discretization of the unbounded operator (2.3), a preconditioning of the system is
necessary in order to obtain an eective iteration.
The coecient operator Ah of the discrete mixed system is L2{symmetric, but
indenite. For a short review of various iterative methods proposed for saddle point
problems we refer to Section 7 of 3] and references given there. In the numerical
experiments, presented in Section 6., we will use the minimum residual method.
For discussions on the minimum residual method for saddle point problems we
refer to 17], 27], 31] and Chapter 9 of the text 16]. The derivation given below of
the desired structure of the preconditioner for the saddle point operator is closely
related to similar discussions given in 2] and 3], see also 29].
A preconditioner for the system (2.4) is a positive denite operator Bh : Xh 7!
Xh. The basic properties required for an eective preconditioner Bh is that the
action of Bh can be eciently computed and that the spectral condition number of
the operator BhAh , (Bh Ah ), is bounded uniformly in h. Here (BhAh ) is given by
jj (BhAh ) = sup
inf jj
where the supremum and inmum is taken over the spectrum of Bh Ah . We observe
that BhAh is symmetric with respect to the innerproduct (Bh;1 ). Furthermore,
this operator is the coecient operator of the preconditioned system
u
h
BhAh ph = Bh GFhh :
If the condition number (Bh Ah ) is bounded, uniformly in h, then an iterative
method like the minimum residual method, or another method with similar properties, will converge with a rate independent of h. More precisely, the number of
iterations needed to achieve a given factor of reduction of the error in a proper
norm is bounded independently of h.
As a consequence of (2.8) the desired bound on the condition number (Bh Ah )
will hold if
jjBh jjL(Xh Xh ) and jjBh;1jjL(Xh Xh ) are bounded uniformly in h: (3.1)
Hence, we conclude that a uniform preconditioner Bh for the mixed system (2.4)
is a positive denite operator Bh : Xh 7! Xh which satises the bounds (3.1) and
which is easy to evaluate.
Let h : Vh 7! Vh be dened by (cf. (2.1))
(u v) = (h u v) for all u v 2 Vh :
It is straightforward to see that h is L2 {symmetric and positive denite. The
operator h should be thought of as an approximation of the operator I ; grad div
in the sense that the equation h uh = f h approximates problems of the form
u ; grad div u = f (3.2)
with the natural boundary condition
div u = 0 on @ :
(3.3)
Domain embedding preconditioners
7
Note that the operator I ; grad div is not an elliptic operator. In fact, when
restricted to gradient elds this operator acts like a second order elliptic operator,
while it coincides with the identity operator on curl elds.
The signicance of the operator h is that the bounds (2.8) state that the operator Ah has the same mapping properties as the diagonal operator
h 0 :
0 I
Furthermore, the simplest choice of an operator which satises the condition (3.1)
is a block diagonal operator of the form
Bh = 0h I0 (3.4)
where h is a uniform preconditioner for h . The operator h should therefore be
spectrally equivalent to ;h 1 , i.e., there are positive constants c1 and c2 independent
of h such that
c1 (v v) (h h v v) c2 (v v):
(3.5)
Hence, the construction of a preconditioner Bh for the coecient operator Ah of
(2.4) has been reduced to the construction of a preconditioner for the positive
denite operator h .
The construction of eective preconditioners for the h has been discussed by
Cai, Goldstein and Pasciak 11], Vassilevski and Wang 30] and Arnold, Falk and
Winther 3]. In particular, it is established in 3] that, if Vh is a Raviart{Thomas
space, a standard multigrid V-cycle operator, utilizing a proper smoothing operator,
will in fact be a uniform preconditioner for h . In the analysis below we will discuss
the possibility of using a preconditioner constructed with respect to an extended
domain to construct a uniform preconditioner for the operator h .
4. Domain embedding
If the geometry of the domain is, in some sense, irregular then for computational
reasons it is frequently desirable to embed into a more regular extended domain
e . The purpose of this section is to discuss the use of such an extended domain
to construct the preconditioner h for h . We emphasize that even if we are
approximating a Dirichlet problem of the form (1.1), the boundary condition given
by (3.3) is a natural boundary condition for the problem (3.2). Therefore, the
domain embedding approach is well suited for the operator h .
In the rst subsection, x4.1, we present the setting of domain embedding and
prove some auxiliary results, the main one being an extension result of normal
trace data into the interior of the neighboring subdomain + . Note, that unlike
the domain embedding approach studied, e.g., in Nepomnyaschikh 20], we do not
need this extension mapping explicitly in the algorithm implementing the actions
of the preconditioner. In subsection x4.2 we give the construction of the domain
embedding preconditioner and study its spectral equivalence properties.
8
Rusten, Vassilevski, and Winther
+
;
Figure 1. The domain e = ; + .
4.1. An extension result for H (div){functions
Let e be a xed extended domain, i.e., e . The part of @ which is contained
in the interior of e will be denoted ;. For simplicity, we assume that ; is connected.
Furthermore, we let + = e n (
;). Hence,
e = ; + cf. Figure 1. The inner products in L2(
e ) and H (div e ) will be denoted ( )e
and e ( ), respectively.
We will use H01 (;) to denote the space
H01 (;) = fj; : 2 H 1 (@ ) 0 on @ n ;g
and H01=2 (;) will be the interpolation space half way between L2 (;) and H01 (;)
(This space is frequently referred to as H01=02 (;), cf. 18].) If K @ e ; we let
h iK denote the1 inner product of L2 (K ) and j jmK the norm in H m(K ). The
dual space of H02 (;) with respect to the inner product h i; of L2(;) is denoted
H ;1=2 (;).
As above let n = (n1 n2 ) denote the outward unit normal on @ . It is well known
(cf. e.g., the text 15]) that the trace operator
v ;! (v n)j;
Domain embedding preconditioners
9
is a bounded map from H (div ) onto H ;1=2 (;). More precisely, there exists a
positive constant c, only depending on and ;, such that
jv nj;1=2; cjjvjjdiv for all v 2 H (div ):
(4.1)
We will assume that Vh can be embedded into a larger nite element space,
Vh(
e ) H (div e ). Hence, the space Vh is obtained by restricting elements
of Vh (
e ) to . We recall that if v 2 H (div e ) then the normal component of v
is required to be continuous over the interface ; in the sense that
(v n)j;+ = (v n)j;; i.e., the normal components taken from the outside or the inside of are the same.
The subspace of functions in Vh (
e ) with support in the closure of + will be
denoted V0h , i.e.,
V0h = fv 2 Vh(
e ) : v 0 on g:
We observe that if v 2 V0h then (v n)j; = 0. The orthogonal complement of V0h
in Vh (
e ) with respect to the inner product e ( ) will be denoted V0?h . Hence,
V0?h consists of all functions in Vh(
e ) which are \discrete {harmonic" on +.
We shall throughout this section assume that the spaces fVh (
e )g are Raviart{
Thomas spaces constructed from a quasiuniform triangulation fTeh g of e . Hence,
for a nonnegative integer r,
Vh (
e ) = fv 2 H (div e) : v 2 (Pr (T ))2 + (x y)Pr (T ) for all T 2 Tehg:
Here Pr (T ) denotes the space of polynomials of degree at most r on T . The associated spaces Wh and Wh (
e ) consist of discontinuous piecewise polynomials of
degree r with respect to the triangulations Th and Teh , respectively. It is well known
that the stability condition (2.6) holds for the pairs of spaces f(Vh Wh )g (cf. 25]).
Let also Zh(
e ) be the usual space of continuous piecewise polynomials of degree
at most r + 1, i.e.,
Zh (
e ) = fz 2 H 1 (
e ) : z jT 2 Pr+1 (T ) for all T 2 Teh g
and let curl be the dierential operator
@ curl = ; @y@ :
@x
It is easy to check that
curl(Zh(
e )) Vh(
e ):
In fact, curl(Zh (
e )) consists of piecewise polynomials of degree at most r, and
curl(Zh(
e )) is exactly the subspace of divergence free vector elds in Vh(
e ) (cf.
9]).
Finally, we let Sh (;) be the trace space given by
Sh (;) = f : = (v n)j; for v 2 Vh g:
10
Rusten, Vassilevski, and Winther
Hence, Sh (;) is a space of discontinuous piecewise polynomials on ;. Similarly, we
let
Sh (@ + ) = f : = (v n)j@ + for v 2 Vh (
e )g
such that Sh (;) Sh (@ + ). In the analysis below we shall utilize the following
extension result for boundary functions.
Lemma 4.1. There exists a bounded extension operator E@ : H ;1=2(;) 7! H ;1=2(@ +)
such that
Z
(E@ g)(s) ds = 0 for all g 2 H ;1=2 (;):
@ +
Proof Let 1 > 0 be the length of ; and 2 > 0 the length of @ + n ; such that
functions on @ + can be identied with (periodic) functions on (;
1 2 ). For each
g 2 H ;1=2 (;) = H ;1=2 ((;
1 0)) dene E@ g as the odd extension of g given by
g(s)
for s 2 (;
1 0)
E@ g = ; 1 g(; 1 s) for
(4.2)
s 2 (0 2 ):
2
2
It is straightforward to check that the L2{dual operator E@ is given by
(E@ )(s) = (s) ; (; 2 s)
1
for s 2 (;
1 0). Furthermore, the operator E@ is easily seen to satisfy
E 2 L(L2 (@ + ) L2 (;)) \ L(H 1 (@ + ) H 1 (;)):
@
0
By interpolation we therefore can conclude that
E@ 2 L(H 1=2 (@ + ) H01=2 (;))
and by duality this implies
E@ 2 L(H ;1=2 (;) H ;1=2 (@ + )):
Finally, the mean value property for E@ g follows from the fact that E@ 0 for
any constant function .
Let Qh : L2 (@ + ) 7! Sh(@ + ) be the L2 {projection. Since the space Sh (@ + )
in general will contain piecewise constants the operator Qh admits an estimate of
the form
j(I ; Qh )j0@ + ch jj@ + for 0 1
(4.3)
where c is independent of h. We now have the following discrete analog of Lemma 4.1..
Lemma 4.2. For each g 2 Sh(;) there exists an extension g~ 2 Sh(@ +) such that
Z
g~(s) ds = 0 and jg~j;1=2@ + cjgj;1=2; @ +
where the constant c is independent of g and h.
Domain embedding preconditioners
11
Proof From Lemma 4.1. it seems that the obvious choice for g~ is E@ g. However,
in general E@ g 2= Sh (@ + ) for g 2 Sh (;). But we observe that the denition
of E@ implies that E@ g is a piecewise polynomial with respect to a partition of
@ + implicitly dened by (4.2). Therefore, the function E@ g will satisfy an inverse
estimate of the form
jE@ gj0@ + ch;1=2 jE@ gj;1=2@ + ch;1=2 jgj;1=2; (4.4)
where the nal estimate follows from Lemma 4.1..
Dene instead g~ = QhE@ g. Since Sh (@ + ) consists of discontinuous functions
the operator Qh is local and therefore
g = E@ gj; = g~j; :
Furthermore, since the constant functions are elements of Sh (@ + ), the mean value
property for g~ is inherited from the mean value property of E@ g. Finally, if 2
H 1=2 (@ + ) we have
hg~ i@ + = hE@ g i@ + + hE@ g (Qh ; I )i@ +
jE@ gj;1=2@ + jj1=2@ + + jE@ gj0@ + j(I ; Qh )j0@ +
cjgj;1=2; jj1=2@ + where we have used the estimates (4.3) and (4.4) in the nal step. Hence, it follows
from the denition of jg~j;1=2@ + that
jg~j;1=2@ + =
hg~ i@ +
cjgj;1=2;:
2H 1=2 (@ + ) jj1=2@ +
sup
Let
Zh(
+ ) = fz j+ : z 2 Zh (
e )g:
Lemma 4.3. (Discrete divergence free extension) Let g 2 Sh(;). There
exists a v 2 curl (Zh (
+ )) such that
(v n)j; = g and jjvjjdiv+ = jjvjj0+ cjgj;1=2; where c is independent of g and h.
Proof For a given g 2 Sh(;) let g~ 2 Sh (@ + ) be as in Lemma 4.2.. Since g~ has
mean value zero, there exists a unique 2 H 1=2 (@ + ) such that
t = g~ and
Z
ds = 0:
@ +
Here and in what follows, t will denote the tangential derivative of , where the
unit tangent vector t is dened such that t = (n2 ;n1) on ;. Furthermore,
jj1=2@ + cjg~j;1=2@ + cjgj;1=2;:
(4.5)
12
Rusten, Vassilevski, and Winther
Also, observe that corresponds to a trace on @ + of a function in Zh (
e ).
Let be the unique element in Zh (
+ ) such that = on @ + and
Z
grad grad z dx = 0
+
for all z 2 Zh (
+ ) such that z j@ + = 0. Hence, is a \discrete harmonic" function
on + and therefore (cf. e.g. 7])
jjjj1+ cjj1=2@ + = cjj1=2@ + :
Dene now v = curl . Then
(4.6)
(v n)j; = (t )j; = (t )j; = g:
Furthermore, since
jjvjjdiv+ = jjv jj0+ jjjj1+ the desired estimate follows from (4.5) and (4.6).
Lemma 4.4. There is a constant c, independent of h, such that
jjujjdiv+ cju nj;1=2; for all u 2 V0?h :
Proof Let u 2 V0?h and let g = (u n)j; 2 Sh (;). Since u 2 V0?h it follows
that this function minimizes jjvjjdiv+ over all elements in Vh (
e ) which satisfy
the condition (v n)j; = g. Therefore, it is enough to establish the existence of a
v 2 Vh(
+ ) = Vh(
e )j+ such that (v n)j; = g and
jjv jjdiv+ cjgj;1=2;:
(4.7)
However, the existence of such a function v follows from Lemma 4.3.. We observe
that, together with the trace inequality (4.1), this lemma implies an inequality of
the form
jjvjjdiv+ cjjvjjdiv for all v 2 V0?h :
Hence, since jjvjj2dive = jjvjj2div = jjvjj2div+ we obtain
jjvjjdive jjv jjdiv for all v 2 V0?h (4.8)
where the positive constant is independent of h. Also, for each v 2 Vh there
exists a unique extension v~ 2 V0?h such that v~ = v on . In fact, v~j+ is dened
by
e (~v z) = 0 for all z 2 V0h (4.9)
(~v n)j;+ = (v n)j;; :
Hence, if v~ 2 V0?h is dened from v 2 Vh by (4.9) then (4.8) implies that
jjv~ jjdive jjvjjdiv :
(4.10)
Domain embedding preconditioners
13
4.2. The domain embedding preconditioner
In this subsection we present the construction of the domain embedding preconditioner for the operator of main interest h and analyze its properties.
Let Rh : Vh (
e ) 7! Vh be the operator dened by restricting elements of Vh (
e )
to and let E h : Vh 7! Vh (
e ) be the adjoint operator, i.e.
(E h u v)e = (u Rh v ) for all u 2 Vh v 2 Vh (
e ):
Alternatively, E h u can be characterized as the L2 {projection into Vh (
e ) of u0 ,
where u0 2 L2 (
e ) denotes the extension of u by zero outside e .
The analog of the operator h with respect to the extended space Vh (
e ) will
be denoted eh , i.e., eh : Vh (
e ) 7! Vh (
e ) is dened by
(eh u v)e = e (u v) for all u v 2 Vh (
e ):
We now dene Th : Vh 7! Vh (
e ) by
Th = ;eh1 E hh:
The operator Th preserves the H (div){inner product in the sense that
e (Th u v) = (u Rh v ) for all u 2 Vh v 2 Vh (
e ):
(4.11)
This follows since
e (Th u v) = (eh Th u v)e = (E h h u v)e
= (h u Rh v ) = (u Rh v):
Furthermore, if u 2 Vh it follows from (4.11) and Cauchy{Schwarz inequality that
e (Th u Th u) = (u Rh Th u) (u u)1=2 (Rh Th u Rh Th u)1=2
(u u)1=2 e (Th u Th u)1=2
or
e (Th u Th u) (u u) for all u 2 Vh :
(4.12)
The following result shows that ;eh1 can be used to construct a uniform preconditioner for h .
Lemma 4.5. The operator Rh;eh1 Eh is spectrally equivalent to ;h 1 in sense that
;2 (v v) ((Rh ;eh1 E h )h v v) (v v) for all v 2 Vh :
The constant is from estimate (4.10).
14
Rusten, Vassilevski, and Winther
Proof This result is just a special case of \the ctitious space lemma" given in
Nepomnyaschikh 22]. However, for completeness we outline the proof in the present
setting. The right inequality above follows since
((Rh ;eh1 E h )h v v) = (Rh Th v v) = e (Th v Th v) (v v)
where we have used (4.11) and (4.12).
The left inequality follows from the identity
(v v) = (v Rh v~ ) = e (Th v v~)
where v~ 2 V0?h is the extension of v dened by (4.9). Therefore, by (4.10), (4.11)
and Cauchy{Schwarz inequality
(v v) e (Th v Th v )1=2 e (~v v~ )1=2
e (Th v Th v)1=2 (v v)1=2
= ((Rh ;eh1 E h )h v v)1=2 (v v)1=2 :
This completes the proof.
The lemma above shows that ;eh1 can be used to construct a uniform preconditioner for h . However, it is clear that it suces to use a preconditioner eh
instead of eh .
Assume that eh : Vh (
e ) 7! Vh (
e ) is L2 {symmetric and spectrally equivalent
to ;eh1 , i.e.
c1 ((;eh1 v v) (eh v v) c2 ((;eh1 v v) for all v 2 Vh (
e )
(4.13)
where the positive constants c1 and c2 are independent of h. Dene an operator
h : Vh 7! Vh by
h = RhehE h:
(4.14)
Observe that h is L2 {symmetric since
(h u v) = (eh E h u E h v)e = (u h v) for all u v 2 Vh :
The following theorem is the main result of this section.
Theorem 4.1. Assume that the L2{symmetric operator eh : Vh (
e ) 7! Vh(
e )
satises (4.13) and let h : Vh 7! Vh be dened by (4.14). Then the two quadratic
forms
(v v) and (h h v v)
are equivalent on Vh , uniformly in h.
Domain embedding preconditioners
15
Proof We observe that for any v 2 Vh
(h h v v) = (eh E h h v E h h v ):
Furthermore, by (4.13) this quadratic form is equivalent, uniformly in h, to the
quadratic form
(;eh1 E h h v E h h v) = ((Rh ;eh1 E h )h v v)
and by Lemma 4.5. this form is equivalent to (v v).
Hence we have demonstrated that a preconditioner h , constructed from the
extended domain e as indicated by (4.13) and (4.14), is spectrally equivalent to
;h 1 . Therefore, it follows from the discussion given in Section 3 above that a
preconditioner Bh for the mixed system (2.4), derived from the Dirichlet problem
(1.1), is naturally constructed by the domain embedding approach.
5. The nonconforming Crouzeix{Raviart method
The purpose of this section is to adopt the preconditioning strategy discussed above
for the mixed nite element method to an alternative discretization procedure which
is not directly based on the mixed weak formulation (2.2). More precisely, we shall
analyze the nonconforming Crouzeix{Raviart method (cf. 12]). We will show that
this system can be interpreted as a nonconforming mixed system. Furthermore,
the obtained saddle point problem will be preconditioned by utilizing a preconditioner of the form studied in x 4 above, on the lowest order Raviart{Thomas space,
combined with the the auxiliary space method (cf. Xu 32]). Hence, since we have established above that the preconditioners studied in x 4 can be naturally constructed
by domain embedding, this shows, implicitly, that also the Crouzeix{Raviart system
can be preconditioned by domain embedding.
Hence, the idea is to reformulate the positive denite Crouzeix{Raviart system as
a saddle point problem, precondition the saddle point problem and then solve the
preconditioned system by an iterative method for saddle point problems. This is in
contrast to the more common approach where saddle point problems are reduced
to positive denite problems in order to apply standard iterative techniques like
the conjugate gradient method.
5.1. The nonconforming method as a mixed system
As above, let fTh gh2(01] be a quasiuniform family of triangulations of . Furthermore, Eh is the set of edges in Th and for each E 2 Eh, xE is the midpoint of the
edge. Throughout this section we let
Wh0 = fw : w 2 P1 (T ) 8T 2 Th w is continuous at xE 8E 2 Eh g:
Here the continuity requirement at the boundary should be interpreted such that
w(xE ) = 0 for all boundary edges E .
We consider approximations of the Dirichlet problem (1.1) with homogeneous
boundary conditions. The discrete solutions are determined by:
16
Rusten, Vassilevski, and Winther
Find ph 2 Wh0 such that
XZ
T 2Th T
(a grad ph ) (grad qh ) dx = (f q) for all q 2 Wh0 :
(5.1)
For simplicity we will assume throughout this section that the coecient matrix a is piecewise constant with respect to the triangulation Th for all h 2 (0 1].
(Otherwise, just replace a by it's average on each triangle in the discussion below.)
Let Vh0 denote the space of discontinuous piecewise constant 2{vectorelds with
respect to the triangulation Th . Dene an operator gradh : Wh0 7! Vh0 by taking
the gradient elementwise, i.e., (gradh q)jT = grad(qjT ). Then the equation (5.1)
can be equivalently written as
(a gradh ph gradh q) = (f q) for all q 2 Wh0 :
We also dene a discrete divergence operator divh : Vh0 7! Wh0 by duality, i.e.,
(divh v q) = ;(v gradh q) for all q 2 Wh0 :
A more explicit characterization of the operator divh can be derived from the fact
that div(vjT ) = 0 for any v 2 Vh0 . For each E 2 Eh let n be chosen unit normal
vector. If v 2 Vh0 then v n]E will denote the jump of v in this direction. Then we
have
(divh v q) =
X
E 2E0h
jE j q(xE ) v n]E for all q 2 Wh0 (5.2)
where E0h denotes the set of interior edges and jE j is the length of E .
If ph 2 Wh0 is the solution of (5.1) we let the discrete #ux uh 2 Vh0 be given by
Z
a;1 uh v dx =
Z
gradh ph v dx for all v 2 Vh0 :
The two unknowns uh and ph can now be determined as the unique solution
(uh ph ) 2 Vh0 Wh0 of the saddle point problem
(a;1 uh v) ; (gradh ph v) = 0 for all v 2 Vh0 (divh uh q) = ;(f q) for all q 2 Wh0 :
(5.3)
We note that, under the assumption that the coecient matrix a is piecewise constant, the system (5.3) is exactly equivalent to the original system (5.1).
We observe that the system (5.3) can be given the operator form:
Ah uphh = F0h where the coecient operator, Ah : Xh 7! Xh , is given by
;1
Ah = a I ; gradh :
divh
0
Domain embedding preconditioners
17
Here Xh = Vh0 Wh0 .
If v 2 Vh0 we dene a mesh dependent \H (div){norm" by
jjvjj2divh = jjvjj20 + jj divh vjj20 :
Furthermore, in analogy with the notation above for the standard mixed method,
we let
h (u v) = (u v) + (divh u divh v)
be the corresponding inner product.
Lemma 5.1. For each q 2 Wh0 there is a v 2 Vh0 such that
divh v = q and jjvjjdivh cjjqjj0 where the constant c is independent of q and h.
Proof We rst recall that the functions in w 2 Wh0 satisfy a discrete Poincar$e
inequality of the form
jjwjj0 cjj gradh wjj0 (5.4)
where c is independent of w and h. A proof of this fact can for example be found in
1] (cf. Lemma 5.3 of 1]). For a given q 2 Wh0 let 2 Wh0 be uniquely determined
by
(gradh gradh w) = ;(q w) for all w 2 Wh0 (5.5)
and let v = gradh 2 Vh0 . By construction divh v = q. Furthermore, (5.4) and
(5.5) imply that
jjvjj20 jjqjj0 jjjj0 cjjqjj0 jjvjj0 and hence the desired bound on jjvjjdivh is established.
It is a direct consequence of Lemma 5.1. that a Babuska{Brezzi condition of the
form
inf sup (divh v q) > 0
q2Wh0 v2Vh0 jjv jjdivh jjq jj0
holds, where is independent of h. Therefore, if we let
jj(v q)jj2Xh = jjvjj2divh + jjqjj20 and, if the dual norm jj(v q))jjXh on Xh is dened by L2 {duality, we immediately
obtain that the operator norms
jjAh jjL(Xh Xh ) and jjA;h 1 jjL(Xh Xh ) are bounded uniformly in h: (5.6)
Hence, the properties of the operator Ah corresponds to similar properties for the
coecient operator of the standard mixed method studied above (cf. (2.8)). By
18
Rusten, Vassilevski, and Winther
arguing exactly as we did above we therefore conclude that if Bh : Xh 7! Xh is a
positive denite operator such that
jjBh jjL(Xh Xh ) and jjBh;1jjL(Xh Xh ) are bounded uniformly in h (5.7)
then the condition number of the operator BhAh is bounded uniformly in h.
Let 0h : Vh0 7! Vh0 be dened by
(0h u v) = h (u v) for all u v 2 Vh0 :
This operator is L2 {symmetric and positive denite. Furthermore, assume that
0h : Vh0 7! Vh0 is a uniform preconditioner for 0h, and that Ih : Wh0 7! Wh0 is
spectrally equivalent to the identity. If Bh : Xh 7! Xh is the block diagonal operator
0
Bh = 0h I0h
then this operator satises the mapping property (5.7). Hence, the construction of a
preconditioner Bh is essentially reduced to the problem of constructing an eective
preconditioner 0h for 0h . Such a preconditioner will be constructed below by the
auxiliary space method. The operator Ih is introduced as a replacement for the
identity operator, in order to avoid the inversion of a \mass{matrix".
5.2. The auxiliary space technique
Let the spaces Vh0 and Wh0 be as above, i.e. Vh0 is the space of discontinuous
piecewise constant vectors and Wh0 is the nonconforming Crouzeix{Raviart space.
Furthermore, Vh H (div ) will denote the corresponding lowest order Raviart{
Thomas space as described in x4 above, i.e. the parameter r = 0, and Wh the
corresponding space for the mixed method consisting of piecewise constants.
In the auxiliary space method the main tool for constructing a preconditioner
0h : Vh0 7! Vh0 for the operator 0h is a corresponding preconditioner h for the
operator h dened on Vh . Here we recall that h : Vh 7! Vh is dened by
(u v) = (h u v) for all u v 2 Vh where ( ) is the H (div){inner product. The preconditioner h : Vh 7! Vh is
assumed to be a uniform preconditioner for h , i.e. the bilinear forms
(v v)
and
(h h v v)
are equivalent on Vh , uniformly in h. Furthermore, we assume that h v can be
eectively evaluated from a given inner product representation of v, i.e. from the
data (v j ), where fj g is a nodal basis for Vh .
Let h : Vh 7! Vh0 be the L2{projection. Note that this operator is local, since
0
Vh is a space of discontinous
functions. Furthermore, let h : Vh0 7! Vh be the
2
L {dual operator, i.e. h is the L2 {projection onto Vh .
We shall consider a preconditioner 0h : Vh0 7! Vh0 of the form
0h = h2I + hhh
(5.8)
Domain embedding preconditioners
19
where is a positve constant independent of h. Let us rst remark that this operator
is computationally feasible. This just follows from the assumption on h together
with the fact that h is local. In order to use the theory developed in 32] we need
to verify that the projection h is stable and accurate in the sense that
h (h v h v ) c(v v) for all v 2 Vh (5.9)
and
jj(I ; h )v jj0 chjj div vjj0 for all v 2 Vh (5.10)
for a suitable constant c independent of h. Furthermore, we need to construct an
operator P h : Vh0 7! Vh such that
(5.11)
(P h w P h w) ch(w w) for all w 2 Vh0 and
jj(I ; P h )wjj0 chjj divh wjj0 for all w 2 Vh0 (5.12)
where again c is independent of h. The operator P h is only needed for the analysis.
In the present setting we dene P h : Vh0 7! Vh by averaging the normal components on each edge, i.e.
(P h w n)E = 12 ((w n)E; + (w n)E+ )
for all E 2 Eh :
This will uniquely determine P h w 2 Vh .
Lemma 5.2. The operators h and P h dened above satisfy the properties (5.9){
(5.12).
We will delay the verication of these properties. However, the following theorem
now follows more or less directly from 32].
Theorem 5.1. Let 0h0 : Vh0 7! Vh0 be dened by (5.8). If 0 h is a uniform preconditioner for h then h is a uniform preconditioner for h .
Proof For completeness we outline a proof in the present setting. We need to show
that the bilinear forms
h(w w) and h (0h 0h w w)
are uniformly equivalent on Vh0 . We have
h (w w) = ((I ; P h )w + (I ; h )P h w 0h w) + (P h w h 0h w):
The rst inner product on the right hand side is estimated by
(jj(I ; P h )wjj0 + jj(I ; h )P h wjj0 )jj0h wjj0 chh (w w)1=2 jj0h wjj0 where we have used the properties (5.10){(5.12). By Cauchy{Schwarz inequality,
(5.11) and the assumption on h we also have
(P h w h 0h w) (P h w P h w)1=2 (;h 1 h 0h w h 0h w)1=2
ch (w w)1=2 h (h h h 0h w w)1=2
20
Rusten, Vassilevski, and Winther
However these bounds imply that
h(w w) ch (0h 0h w w)
for all w 2 Vh0 :
From (5.2) it follows that the spectral radius of 0h is O(h;2 ). Therefore,
h(h2 0h w w) = h2 jj0h wjj20 ch(w w):
(5.13)
This is the rst part of the desired lower bound for h (w w). In order to complete
the argument note that (5.9) and Cauchy{Schwarz inequality imply
h((h ;h 1 h )0h w w) h ((h ;h 1 h )0h w (h ;h 1 h )0h w)1=2 h (w w)1=2
c(;h 1 h 0h w ;h 1 h 0h w)1=2 h (w w)1=2
= ch ((h ;h 1 h )0h w w)1=2 h (w w)1=2 :
Together with (5.13) and the assumptions on h this implies the desired lower
bound
h(0h 0h w w) ch (w w)
for all w 2 Vh0 and this completes the proof of the theorem.
5.3. Proof of Lemma 5.2.
In order to complete the analysis of the auxiliary space preconditioner (5.8) we have
to establish the properties (5.9){(5.12) for the operators h and P h .
We rst note that (5.9) holds with c = 1. The follows from the identity
(div v q) = ;(v gradh q) = (divh (h v) q)
for all v 2 Vh q 2 Wh0 : (5.14)
To see this identity note that for v 2 Vh , (v n)jE is constant on the edges. This
implies that
Z
E
(v n)q] ds = jE j(v n)(xE )q](xE ) = 0
for all v 2 Vh q 2 Wh0 where xE is the midpoint of E . This leads directly to (5.14).
Property (5.10) is straightforward. Since the L2 {projection onto Vh0 is local, we
have (letting v = (v1 v2 )),
jj(I ; h )vjj0 ch(
However, if v 2 Vh then
X
T 2Th
jj grad v1 jj20T + jj grad v2 jj20T )1=2 :
jj grad v1 jj20T + jj grad v2 jj20T = 21 jj div vjj20T :
Therefore, (5.10) is established.
Domain embedding preconditioners
21
We next verify (5.12). Let Vh1 be the discontinuous Raviart{Thomas space, i.e.
Vh1 has the same degrees of freedom as Vh0 on each triangle, but the continuity
requirements have been removed. Hence, Vh Vh0 Vh1 . If z 2 Vh1 then the forms
X;
jjz jj20 and h2
(z n)2E; + (z n)2E+
(5.15)
E 2Eh
are uniformly equivalent with respect to h. From the deniton of the operator P h
we therefore obtain
jj(I ; P h )wjj ch
2
0
2
X
E 2Eh
!
w n]E ch2 jj divh wjj20 2
which is (5.12).
Finally, we show that (5.11). It follows directly from (5.15) that P h is uniformly
bounded in L2 . In order to bound k div P h wk0 for w 2 Vh0 we note that it follows
from a standard inverse inequality that
k div P h wk20 =
X
T 2Th
ch;2
k div(P h w ; w)k20 T
X
T 2Th
k(P h w ; w)k20 T ch;2 kP h w ; wk20 :
Here, we have used that w is a constant vector on each triangle. However, together
with (5.12) this implies the desired estimate (5.11).
6. Numerical experiments
In this section we shall report on some numerical experiments using the various
preconditioners discussed above. The extended domain, e , will always be taken
to be the unit square. The domain is either equal to e , i.e. there is no eect
of domain embedding, or is equal to the L{shaped domain obtained from e by
removing the upper right 1=2 1=2 square. The triangulation of e is constructed
by dividing the unit square into h h sized squares, and then dividing each square
into two triangles by using the negative sloped diagonal. In all the examples below
Vh and Vh(
e ) will be the lowest order Raviart{Thomas spaces.
Example 6..1
We rst consider preconditioners for the operator h : Vh 7! Vh . Hence, we consider approximations of the boundary value problem (3.2){(3.3). In the experiment
we have taken f = (1 1)T . We investigate the behavior of the preconditioner h
dened by (4.14), i.e.
h = RhehE h:
The operator eh is a multigrid V{cycle operator with an additive smoother of
the form described in 3], where the scaling factor, , is taken to be 1=2.
22
Rusten, Vassilevski, and Winther
Domain: Unit square L{shaped domain
1=h
(h h )
(h h )
32
2.40
5.36
64
2.44
5.66
128
2.46
5.90
256
2.46
6.09
Table 1. Condition numbers for the preconditioned H (div){operator.
Domain:
Unit square
L{shaped domain
1=h
(Bh Ah ) MINRES (BhAh ) MINRES
32
2.33
14
5.31
21
64
2.35
14
5.62
21
128
2.34
14
5.86
22
256
2.34
14
6.05
22
Table 2. Condition numbers and the number of iterations for the mixed operator.
The condition numbers (h h ) are estimated from the conjugate gradient iterations. The results are given in Table 1. Of course, the rst column of these results
just conrms the theory developed in 3], while the second column seem to agree
with the result of Theorem 4.1., i.e. the condition numbers (h h ) appears to be
bounded, uniformly in h.
Example 6..2
In the next example with consider the mixed method for the problem (1.1), with
the coecient a equals the identity, f 1 and g 0. The discrete system (2.4) is
preconditioned by an operator Bh of the form (3.4), i.e.
Bh = 0h I0 : Vh Wh 7! Vh Wh :
Here Wh is the space of piecewise constants and the H (div){preconditioner h
is chosen exactly as in the previous example above. In Table 6. we peresent the
result of this experiment. In addition to the estimates for the condition numbers,
(BhAh ), we also report the number of iterations required by the minimum residual
method to reduce the residual of the preconditioned system by a factor 10;5 in the
norm induced by the inner product (Bh;1 ). As expected, the results appear to be
bounded, independently of h.
Domain embedding preconditioners
23
Domain:
Unit square
L{shaped domain
1=h
(Bh Ah ) MINRES (BhAh ) MINRES
32
2.79
25
5.30
32
64
2.79
24
5.61
34
128
2.79
24
5.86
34
256
2.78
22
5.85
34
Table 3. Contidion numbers and the number of iterations for the nonconforming
operator.
Example 6..3
Finally, we consider the nonconforming method studied in Section 5., i.e. the system (5.1). This system is formulated as a saddle point problem, cf. (5.3), and is
preconditioned by a block diagonal operator
0
Bh = 0h I0h : Vh0 Wh0 7! Vh0 Wh0 :
We recall that Vh0 is the space of piecewise constant vectors, while Wh0 is the
piecewise linear Crouzeix{Raviart space. Here the preconditioner 0h : Vh0 7! Vh0
is the auxiliary space preconditioner given by (5.8), i.e.
0h = h2I + hhh:
In the experiments we have chosen = 0:01 and h : Vh 7! Vh exactly as in
the two previous examples. The operator Ih on Wh0 is obtained from the identity
operator by replacing exact integration by the simplest numerical integration rule
based on the values at the midpoint of each edge. In the same way as the above
the iterations are terminated when the proper residual is reduced by a factor of
10;5. The results are given in Table 6.. Since the condition numbers appear to be
bounded, independently of h, this conrms, indirectly, the conclusion of Theorem
5.1..
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