A uniformly stable Fortin operator for the
Taylor–Hood element
Kent-Andre Mardal, Joachim Schöberl &
Ragnar Winther
Numerische Mathematik
ISSN 0029-599X
Volume 123
Number 3
Numer. Math. (2013) 123:537-551
DOI 10.1007/s00211-012-0492-6
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Numer. Math. (2013) 123:537–551
DOI 10.1007/s00211-012-0492-6
Numerische
Mathematik
A uniformly stable Fortin operator for the Taylor–Hood
element
Kent-Andre Mardal · Joachim Schöberl ·
Ragnar Winther
Received: 2 November 2011 / Revised: 29 June 2012 / Published online: 9 September 2012
© Springer-Verlag 2012
Abstract We construct a new Fortin operator for the lowest order Taylor–Hood
element, which is uniformly stable both in L 2 and H 1 . The construction, which is
restricted to two space dimensions, is based on a tight connection between a subspace
of the Taylor–Hood velocity space and the lowest order Nedelec edge element. General
shape regular triangulations are allowed for the H 1 -stability, while some mesh restrictions are imposed to obtain the L 2 -stability. As a consequence of this construction, a
uniform inf–sup condition associated the corresponding discretizations of a parameter
dependent Stokes problem is obtained, and we are able to verify uniform bounds for
a family of preconditioners for such problems, without relying on any extra regularity
ensured by convexity of the domain.
K.-A. Mardal was supported by the Research Council of Norway through Grant 209951 and a Centre of
Excellence grant to the Centre for Biomedical Computing at Simula Research Laboratory. R. Winther was
supported by the Research Council of Norway through a Centre of Excellence grant to the Centre of
Mathematics for Applications.
K.-A. Mardal
Department of Informatics, Centre for Biomedical Computing at Simula Research Laboratory, Norway,
University of Oslo, Oslo, Norway
e-mail: kent-and@simula.no
URL: http://simula.no/people/kent-and
J. Schöberl
Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria
e-mail: joachim.schoeberl@tuwien.ac.at
URL: http://www.asc.tuwien.ac.at/∼schoeberl
R. Winther (B)
Department of Informatics, Centre of Mathematics for Applications, University of Oslo,
0316 Oslo, Norway
e-mail: ragnar.winther@cma.uio.no
URL: http://folk.uio.no/rwinther
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Mathematics Subject Classification (1991)
65N22, 65N30
1 Introduction
The stability of mixed finite element methods for the linear Stokes problem is
intimately connected to the existence of bounded Fortin operators, i.e., bounded interpolation operators which properly commute with the divergence operator. For some
discretizations the construction of locally defined Fortin operators is well known. However, for the celebrated Taylor–Hood element the direct construction of a commuting
interpolation operator is not obvious, and, as a consequence, most of the stability
proofs found in the literature typically use alternative approaches. The main contribution of the present paper is to construct a new Fortin operator for the lowest order
Taylor–Hood element in two space dimensions which, under proper conditions, is
uniformly bounded both in L 2 and H 1 . In fact, by utilizing a tight connection between
a subspace of the Taylor–Hood velocity space and the lowest order Nedelec edge
element, we perform a construction which resembles the corresponding well known
construction for the Mini element, cf. [1].
The discussions of this paper is motivated by the study of finite element discretizations of a singular perturbation problem related to the linear Stokes problem. More
precisely, let ⊂ Rn be a bounded Lipschitz domain and ∈ (0, 1] a real parameter.
We will consider singular perturbation problems of the form
(I − 2 )u − grad p = f
div u = g
u=0
in ,
in ,
on ∂,
(1.1)
where the unknowns u and p are a vector field and a scalar field, respectively. For
each fixed positive value of the perturbation parameter the problem behaves like the
Stokes system, but formally the system approaches the mixed formulation of the scalar
Laplace equation as this parameter tends to zero. In physical terms this means that we
are studying fluid flow in regimes ranging from linear Stokes flow to porous medium
flow. Another motivation for studying these systems is that they frequently arise as
subsystems in time stepping schemes for time dependent Stokes and Navier–Stokes
equations, cf. for example [2,6,17,19,20]. Therefore, the study of preconditioners for
these systems is of substantial interest.
Some preliminary discussions are presented in Sect. 2, while the construction and
analysis of the new Fortin operator will be carried out in Sect. 3. Finally, in Sect. 4 we
will relate our results to the properties of a family of preconditioners for the coefficient
operators of the discrete systems obtained from the Taylor–Hood method applied to
(1.1). We will show that these preconditioners behave uniformly well with respect to
the model parameter , as well as the discretization parameter. For the present model
similar results have been derived before, but only by utilizing extra regularity ensured
by convexity of the domain.
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2 Preliminaries
If X is a Hilbert space, then · X denotes its norm. We will use H m = H m ()
to denote the Sobolev space of functions on with m derivatives in L 2 = L 2 ().
The corresponding spaces for vector fields are denoted H m (; Rn ) and L 2 (; Rn ).
Furthermore, ·, · is used to denote the inner products in both L 2 () and L 2 (; Rn ).
In general, we will use H0m to denote the closure in H m of the space of smooth functions
with compact support in , while L 20 is the space of L 2 functions with mean value
zero. We will use L(X, Y ) to denote the space of bounded linear operators mapping
elements of X to Y , and if Y = X we simply write L(X ) instead of L(X, X ). If X and
Y are Hilbert spaces, both continuously contained in some larger Hilbert space, then
the intersection X ∩ Y and the sum X + Y are both Hilbert spaces with norms given
by
x2X ∩Y = x2X + x2Y and
z2X +Y =
inf
x∈X,y∈Y
z=x+y
x2X + y2Y .
The system (1.1) admits the following weak formulation:
Find (u, p) ∈ H01 (; Rn ) × L 20 () such that
u, v + 2 Du, Dv + p, div v = f, v, v ∈ H01 (; Rn ),
div u, q = g, q, q ∈ L 20 ()
(2.1)
for given data f and g. Here Dv denotes the gradient of the vector field v. With the
choice of H01 (; Rn ) × L 20 () as a solution space the bounds on the solution will
blow up as tends to zero. To obtain a proper uniform bound on the solution operator
we are forced to introduce dependent spaces and norms. A possible choice in the
present setting is to take
(u, p) ∈ (L 2 ∩ H01 )(; Rn ) × ((H 1 ∩ L 20 ) + −1 L 20 )(),
with more explicit norms given by
u2L 2 ∩ H 1 = u2L 2 + 2 Du2L 2 and p2H 1 + −1 L 2
= grad(− 2 + I )−1 p, grad p.
Here, a natural boundary condition is assumed for the inverse of the elliptic operator.
Uniform bounds for the solution in these norms can be derived from the Brezzi conditions [4,5]. In fact, the only nontrivial condition needed to obtain such bounds is the
uniform inf–sup condition
div v, q
≥ αq H 1 + −1 L 2 , q ∈ L 20 (),
v
2
1
1
n
L ∩ H
v∈H (;R )
sup
(2.2)
0
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where the positive constant α is independent of ∈ (0, 1]. This condition is
closely tied to the existence of a proper right inverse of the divergence operator in
L(L 20 (), H01 (; Rn )). If there exist such a right inverse, which can be extended to
an operator in L(H0−1 (), L 2 (; Rn )), then the uniform condition (2.2) will follow.
Here, H0−1 ⊃ L 20 corresponds to the dual space of H 1 ∩ L 20 , obtained by extending
the L 2 inner product to a duality pairing. In other words, the uniform condition (2.2)
will follow from the existence of an operator S with the properties
S ∈ L(L 20 , H01 ) ∩ L(H0−1 , L 2 ), and div S = I,
(2.3)
cf. [15] for more details. A proper operator which satisfy these conditions is the
Bogovskiĭ operator, see for example [10, Section III.3], or [8,11]. On a domain ⊂
Rn , which is star shaped with respect to an open ball B, this operator is explicitly
given as an integral operator of the form
Sg(x) =
z
g(y)K (x − y, y) dy, where K (z, y) = n
|z|
∞ z
r n−1 dr,
θ y +r
|z|
|z|
where θ ∈ C0∞ (B) is positive, and with integral equal to one. This operator is a right
inverse of the divergence operator, and it has exactly the desired mapping properties
given by (2.3), cf. [8,11]. Furthermore, the definition of S can be extended to general
bounded Lipschitz domains, by using the fact that such domains can be written as
a finite union of star shaped domains. The constructed operator will again satisfy
the properties given by (2.3). We refer to [10, Section III.3], [11, Section 2], and [8,
Section 4.3] for more details.
Remark 2.1 An alternative right inverse of the divergence operator, mapping L 2 ()
boundedly into H01 (; Rn ), can be constructed from the Stokes problem, i.e., problem (1.1) with = 1. However, this operator can, in general, not be extended to an
operator in L(H0−1 , L 2 ), unless the solution operator for the Stokes problem admits
extra regularity. Therefore, this right inverse will, in general, not lead to the uniform
inf–sup condition (2.2). The construction of preconditioners for the coefficient operators of the discrete systems derived from (2.1) has been frequently studied, cf. for
example [2,6,17–20]. Several of these studies have shown that a family of preconditioners, described in Sect. 4 below, behave uniformly well, both with respect to the
perturbation parameter and the discretization parameter h. However, all the theoretical verifications of this result relies on additional properties of the domain which
lead to extra regularity of the solution of Stokes problem. In fact, it seems that the
non-optimal results of these studies partly can be traced to the use of the wrong right
inverse in the continuous case. We will return to the discussion of preconditioners
for discretizations of (2.1) in Sect. 4 below. Finally, we mention that an alternative
derivation of the uniform inf–sup condition (2.2) for non-convex domains, based on
domain decomposition, is presented in the recent manuscript [13].
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We will consider finite element discretizations of the problem (1.1) of the form:
Find (u h , ph ) ∈ Vh × Q h such that
u h , v + 2 Du h , Dv + ph , div v = f, v, v ∈ Vh ,
div u h , q = g, q, q ∈ Q h .
(2.4)
Here Vh and Q h are finite element spaces such that Vh × Q h ⊂ H01 (; Rn ) × L 20 (),
and h is the discretization parameter. For the Taylor–Hood element, and the Mini
element, the pressure space Q h is in fact a subspace of H 1 , and this will be assumed
throughout our discussions below. The proper discrete uniform inf–sup condition,
which will guarantee uniform stability of the method (2.4), is of the form
sup
v∈Vh
div v, q
≥ αq H 1 + −1 L 2 , q ∈ Q h ,
v L 2 ∩ H 1
(2.5)
where the positive constant α is independent of both and h. The technique we will
use to establish the discrete inf–sup condition (2.5) is in principle rather standard. We
will just rely on the corresponding continuous condition (2.2) and bounded interpolation operators into the spaces Vh and Q h which commute properly with the divergence
operator. In the present setting, this means that we need to construct interpolation operators Πh : L 2 (; Rn ) → Vh which are uniformly bounded, with respect to and h,
in the norm of L 2 ∩ H01 . However, this is equivalent to the requirement that
Πh L(L 2 ) and Πh L(H 1 ) are bounded independently of h.
(2.6)
Furthermore, the operators Πh must satisfy the commuting relation
div Πh v, q = div v, q v ∈ H01 (; Rn ), q ∈ Q h .
(2.7)
Operators Πh , satisfying (2.6) and (2.7) will be referred to as a uniformly bounded
Fortin operators. If such operators are constructed, the discrete uniform inf–sup condition (2.5) will follow from the corresponding continuous condition (2.2). In the
case of the Mini element the construction of uniformly bounded Fortin operators, in
any space dimension, is straightforward. In fact, the design of this element, originally
presented in [1], was highly motivated by the desire to construct a suitable Fortin
operator (cf. also [5, Chapter VI] and [17]). However, for the Taylor–Hood element
the direct construction of a bounded Fortin operator is not obvious. In fact, most of the
stability proofs found in the literature for this discretization typically use alternative
approaches, cf. for example [5, Section VI.6], [12, Chapter II.4.2] and the discussion given in the introduction of [9]. An exception is [9], where a local interpolation
operator satisfying (2.7) is constructed. However, this operator is not defined in L 2 .
Below we propose a new construction of an interpolation operator, satisfying (2.6)
and (2.7), by utilizing a technique for the Taylor–Hood method which is similar to the
standard construction for the Mini element. As for the construction carried out in [9],
this analysis is restricted to two space dimensions.
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3 A Fortin operator for the Taylor–Hood element
This section is devoted to the construction of the new local Fortin operator Πh for
the lowest order Taylor–Hood element. We will assume that the domain ⊂ R2 is
a polyhedral domain which is triangulated by a family of shape regular, triangular
meshes {Th } indexed by decreasing values of the mesh parameter h = maxT ∈Th h T .
Here h T is the diameter of the triangle T . We recall that the mesh is shape regular if
there exists a positive constant γ0 such that for all values of the mesh parameter h
h 2T ≤ γ0 |T |, T ∈ Th .
Here |T | denotes the area of T . For the lowest order Taylor–Hood element the velocity
space, Vh , consists of continuous piecewise quadratic vector fields, while Q h is the
standard space of continuous piecewise linear scalar fields with respect to the triangulation Th . For technical reasons we will assume throughout the analysis below that
any T ∈ Th has at most one edge in ∂. Such an assumption is frequently made for
convenience when the Taylor–Hood element is analyzed, cf. for example [5, Proposition 6.1], since most approaches requires a special construction near the boundary. On
the other hand, this assumption will not hold for many simple triangulations. Therefore, at the end of this section we will outline how to refine the analysis such that this
assumption can be removed.
We note that if we have established the discrete inf–sup condition for a pair of
spaces (Vh− , Q h ), where Vh− ⊂ Vh , then this condition will also hold for the pair
(Vh , Q h ). This observation will be utilized here. We let Vh− ⊂ Vh ⊂ H01 (; R2 ) be
the space of piecewise quadratic vector fields which has the property that on each edge
of the mesh the normal components of elements in Vh− are linear. Each function v in
the space Vh− can be determined from its values at each interior vertex of the mesh, and
of the mean value of the tangential component along each interior edge. In fact, the
space Vh− can be decomposed as Vh− = Vh1 ⊕ Vhb , where the space Vh1 is the space of
continuous piecewise linear vector fields, while the space Vhb is spanned by quadratic
“edge bubbles.” To define this space of bubbles we let
1 (Th ) = i1 (Th ) ∪ ∂1 (Th )
be the set of the edges of the mesh Th , where i1 (Th ) is the set of interior edges and
∂1 (Th ) is the set of edges on the boundary of . Furthermore, if T ∈ Th then 1 (T ) is
the set of edges of T , and i1 (T ) = 1 (T )∩i1 (T ). For each e ∈ 1 (Th ) we let e be
the associated macroelement consisting of the union of the two triangles T ∈ Th with
e ∈ 1 (T ). The scalar function be is the unique continuous and piecewise quadratic
function on e which vanish on the boundary of e , and with e be ds = |e|, where
|e| denotes the length of e. The space Vhb is defined as
Vhb = span{be te | e ∈ i1 (Th ) },
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where te is a tangent vector along e with length |e|. Alternatively, if xi and x j are
the vertices corresponding to the endpoints of e then the vector field ψe = be te is
determined up to a sign as ψe = 6λi λ j (x j − xi ), where {λi } are the piecewise linear
functions corresponding to the barycentric coordinates, i.e., λi (xk ) = δi,k for all
vertices xk . In particular,
ψe · (x j − xi ) ds = 6
e
λi λ j ds|e|2 = |e|3 .
e
The construction of the desired interpolation operator Πh utilizes a standard Clement
operator Rh mapping into the space Vh1 , cf. [7]. The operator Rh : L 2 (; R2 ) → Vh1
is local, it preserves constants, and it is stable in L 2 and H01 . More precisely, we have
for any T ∈ Th that
k− j
(I − Rh )v H j (T ) ≤ ch T v H k (T ) ,
0 ≤ j ≤ k ≤ 1,
(3.1)
where the constant c is independent of h and v. Here T denote the macroelement
consisting of T and all elements T ∈ Th such that T ∩ T = ∅. The interpolation
operator Πh will be of the form
Πh = Πhb (I − Rh ) + Rh ,
where Πhb : L 2 (; R2 ) → Vhb will be constructed below. A key part of the construction
is that the operator Πhb , mapping into the bubble space Vhb , satisfies an analog of (2.7),
i.e.,
(3.2)
div Πhb v, q = div v, q, q ∈ Q h .
If this identity holds then we will also have that
div (I − Πh )v, q = div (I − Πhb )(I − Rh )v, q = 0
for all q ∈ Q h , which is (2.7). Hence, it remains to construct the operator Πhb , satisfying
(3.2), and such that the corresponding operator Πh satisfies (2.6).
We will need to separate the triangles which have an edge on the boundary of from the interior triangles. With this purpose we define
Th∂ = {T ∈ Th | T ∩ ∂ ∈ ∂1 (Th ) } and Thi = Th \ Th∂ .
In order to define the operator Πhb we introduce Z h as the lowest order Nedelec space
with respect to the mesh Th . Hence, if z ∈ Z h then on any T ∈ Th , z is a linear vector
field such that z(x) · x is also linear. Furthermore, for each e ∈ 1 (Th ) the tangential
component of z is continuous. As a consequence, Z h ⊂ H (curl; ) where the operator
curl denotes the two dimensional analog of the curl-operator given by
curl z = curl(z 1 , z 2 ) = ∂x2 z 1 − ∂x1 z 2 .
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It is well known that the proper degrees of freedom for the space Z h is the mean
value of the tangential components of v, v · t, with respect to each edge in 1 (Th ).
Furthermore, we let
Z h0 =
⎧
⎨
⎩
z ∈ Zh |
z · t ds = 0, T ∈ Th∂
∂T
⎫
⎬
⎭
.
(3.3)
Alternatively, the elements of Z h0 are those vector fields in Z h with the property that
curl z|T = 0 if T ∈ Th∂ , i.e., z is a constant vector field on T for T ∈ Th∂ . It is a
key observation that the mesh assumption given above, that any T ∈ Th intersects
∂ in at most one edge, implies that the spaces Vhb and Z h0 have the same dimension.
Furthermore, we note that grad q ∈ Z h0 for any q ∈ Q h .
For each e ∈ 1 (Th ) let φe ∈ Z h be the basis function corresponding to the Whitney
form, i.e., φe satisfies
(φe · te ) ds = |e|, and
(φe · te ) ds = 0, e = e,
e
e
where, as above, te is a tangent vector of length e. Hence, if e = (xi , x j ) then the
vector field φe can be expressed in barycentric coordinates as
φe = λi grad λ j − λ j grad λi .
Any z ∈ Z h0 can be written uniquely on the form
z=
a e φe +
ce φe ,
e∈∂1 (Th )
e∈i1 (Th )
where the coefficients ae corresponding to interior edges can be chosen arbitrarily, but
where the coefficients ce for each boundary edge should be chosen such that curl z = 0
on the associated triangle in Th∂ . We note that there is a natural mapping h between
the spaces Vhb and Z h0 given by h (ψe ) = φe for all interior edges, or alternatively,
e
h (v) · te ds = |e|−2
v · te ds, e ∈ i1 (Th ).
(3.4)
e
Below, we will use Vhb (T ) and Z h0 (T ) to denote the restriction of the spaces Vhb and
Z h0 to a single element T , and T will denote the corresponding restriction of the map
h .
We will define the operator Πhb : L 2 (; Rn ) → Vhb by,
Πhb u, z = u, z
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∀z ∈ Z h0 .
(3.5)
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To show that this operator is well defined, and behaves properly, the following general
formula for integration of products of barycentric coordinates over a triangle T will
be useful (cf. for example [14, Section 2.13]),
2α!
|T |,
(2 + |α|)!
λα1 1 λα2 2 λα3 3 d x =
T
where α! = α1 !α2 !α3 ! and |α| =
i
(3.6)
αi .
Lemma 3.1 Let T ∈ Thi with edges e1 , e2 , e3 . For any v ∈ Vhb (T ) we have
1 2
|a| |T | ≤
5
v · T (v) d x ≤
3 2
|a| |T |,
5
T
where T = h |T , v =
i
ai ψei , and |a|2 =
i
ai2 .
Proof A direct computation gives
v · T (v) d x =
3 3
ψei · φe j d x
ai a j
i=1 j=1
T
=
3 3
T
bei φe j · tei d x = a T Ma,
ai a j
i=1 j=1
T
where the 3 × 3 matrix M is given by
M = {Mi, j }i,3 j=1 =
⎧
⎨
⎩
T
ψi · φe j d x
⎫3
⎬
⎧
⎨
⎭
⎩
i, j=1
bei φe j · tei d x
⎫3
⎬
,
⎭
T
i, j=1
The desired result will follow from the diagonal dominance of this matrix.
Let x j be the vertex opposite to e j , and let λ j be the corresponding barycentric
coordinate on T . Then the first diagonal element M1,1 of the matrix M is given by
M1,1 = 6
λ2 λ3 (λ2 gradλ3 − λ3 gradλ2 )(x3 − x2 ) d x
T
(λ22 λ3 + λ2 λ23 ) d x = 2|T |/5,
=6
T
where we have used formula (3.6) in the final step. Actually, from this formula we
derive that all the diagonal elements are given by Mi,i = 2|T |/5, and similar calculations for the off-diagonal elements gives |Mi, j | = |T |/10. In addition, the matrix
M is symmetric. The matrix M is therefore strictly diagonally dominant, and by the
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Gershgorin circle theorem all eigenvalues are bounded below by |T |/5. By combining
this with the fact that M is symmetric we conclude that a T Ma ≥ |a|2 |T |/5, and this
is the desired bound.
The next lemma is a variant of the result above for T ∈ Th∂ .
Lemma 3.2 Let T ∈ Th∂ with interior edges e1 , e2 , and where e3 is the edge on the
boundary. For any v ∈ Vhb (T ) we have
v · T (v) d x =
1 2
|a| |T |,
2
T
where v = a1 ψe1 + a2 ψe2 and |a|2 = a12 + a22 .
Proof Let v = a1 ψe1 + a2 ψe2 = 6a1 λ2 λ3 (x3 − x2 ) + 6a2 λ1 λ3 (x3 − x1 ). It is a key
observation that in this case T (v) is simply given as (v) = −a1 gradλ2 −a2 gradλ1 .
Since gradλi · tei ≡ 0 we therefore obtain from (3.6) that
v · T (v) d x = 6a12
T
λ2 λ3 gradλ2 · (x2 −x3 ) d x+6a22
T
λ2 λ3 d x + 6a22
= 6a12
T
λ1 λ3 gradλ1 · (x1 −x3 ) d x
T
1
λ1 λ3 d x = |a|2 |T |.
2
T
This completes the proof.
b
Let v =
e∈i1 (Th ) ae ψe ∈ Vh such that z = h (v) =
e∈i1 (Th ) ae φe +
0 . It follows from scaling and shape regularity that the two norms
c
φ
∈
Z
e∈∂ (Th ) e e
h
1
⎞1/2
⎛
z L 2 ()
⎜
and ⎝
⎟
ae2 ⎠
,
(3.7)
T ∈Th e∈i (T )
1
are equivalent, uniformly with respect to h. Correspondingly, the two norms
⎞1/2
⎛
⎟
⎜
v L 2 () and ⎝
|T |2
ae2 ⎠
T ∈Th
,
(3.8)
e∈i1 (T )
are uniformly equivalent. In fact, these equivalences hold locally for each T ∈ Th .
It will be convenient to introduce the weighted L 2 -norms given by
⎛
v1,h = ⎝
T ∈Th
123
⎞1/2
2
⎠
h −2
T v L 2 (T )
⎛
and z−1,h = ⎝
T ∈Th
⎞1/2
h 2T z2L 2 (T ) ⎠
.
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In fact, for functions v ∈ Vhb the norm v1,h is equivalent, uniformly in h, to the
v = ordinary H 1 -norm. Furthermore, if e ae ψe then the two norms v1,h and
φh (v)−1,h are both equivalent to ( T |T | e ae2 )1/2 . This observation leads to the
following inf–sup property.
Lemma 3.3 There is a positive constant c0 , independent of h, such that for each
v ∈ Vhb
sup
z∈Z h0
v, z
≥ c0 v1,h .
z−1,h
Proof Let v = e∈i (Th ) ae ψe ∈ Vhb be given. We simply choose the corresponding
1
z = h (v). As a consequence of the norm equivalences (3.7) and (3.8), combined
with Lemmas 3.1 and 3.2, we obtain
v, h (v) =
h (v) · v d x ≥
T ∈Th T
1 |T |
ae2 ≥ c0 h (v)−1,h v1,h ,
5
i
T ∈Th
e∈1 (T )
where c0 > 0 is independent of h. This completes the proof.
The inf–sup condition just derived implies that the operator Πhb satisfies the stability
estimate
(3.9)
Πhb v1,h ≤ c0−1 v1,h , v ∈ L 2 ().
If the family {Th } is quasi-uniform, i.e., there exists a positive constant γ1 , independent
of h and T ∈ Th , such that h T ≥ γ1 h, then the estimate (3.9) also implies that Πhb is
uniformly bounded in the standard non weighted L 2 -norm. An alternative local mesh
restriction, which implies L 2 -stability, is the following condition.
There exists a constant δ ∈ [0, 1), independent of h, such that
|e | ≤ (1 + δ)|e |,
(3.10)
for all edges e, e ∈ i1 (Th ) sharing a common vertex.
The following consequence of this condition will be useful.
Lemma 3.4 Assume that the local mesh condition (3.10) holds. There is a constant
∈ [0, 1/3), and for each h and T ∈ Th a positive constant αT , such that
1 − |e | ≤ , e ∈ i (T ).
1
αT Proof For each T ∈ Th let
αT = (1 + δ/2) min |e |.
e∈i1 (T )
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Due to condition (3.10) this implies that |e | ≤ (1 + δ)(1 + δ/2)−1 αT for all e ∈
i1 (T ). Therefore, we can conclude that
1
|e |
1+δ
≤
,
≤
1 + δ/2
αT
1 + δ/2
which is equivalent to the desired bound with = δ/(2 + δ) < 1/3.
Lemma 3.5 Assume that the local mesh condition (3.10) holds. There is a positive
constant c1 , independent of h, such that for each v ∈ Vhb
sup
z∈Z h0
v, z
≥ c1 v L 2 () .
z L 2 ()
Proof
We let and αT be as in the statement of Lemma 3.4. For a given v =
b
0
0
i
e∈ (Th ) ae ψe ∈ Vh we choose z = E h h (v), where E h : Z h → Z h is defined by
1
E h φe = |e |φe . As a consequence the L 2 -norm of z is equivalent to the L 2 -norm of v.
It follows from these norm equivalences, combined with Lemmas 3.1 and 3.2, that
v, z = v, E h h (v) =
E h h (v) · v d x
=
T ∈Th
≥
T ∈Th
≥
⎡
⎣αT
T ∈Th T
h (v) · v d x +
T
⎤
(E h − αT I )h (v) · v d x ⎦
T
1 3
|e |
−
max 1 −
αT |T |
ae2
5 5 e∈i1 (T )
αT
i
3 1
−
5
5
T ∈Th
αT |T |
e∈1 (T )
ae2 ≥ c1 v L 2 () z L 2 () .
e∈i1 (T )
This completes the proof.
It is a direct consequence of Lemma 3.5 that the operator norm of
L(L 2 (; R2 )) is bounded by c1−1 .
We now can conclude with the following main result of this section.
Πhb
in
Theorem 3.6 The operator Πh is uniformly bounded in H 1 and satisfies the
commuting relation (2.7). Furthermore, if the triangulation is either quasi-uniform
or it satisfies the local condition (3.10), then Πh is also uniformly bounded in L 2 , i.e.,
(2.6) holds.
Proof To show that the operator Πh fulfills the condition (2.7), it is enough to show that
the operator Πhb satisfies the corresponding condition (3.2). However, since grad Q h ⊂
Z h0 we have
div Πhb v, q = −Πhb v, gradq = −v, gradq = div v, q.
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Furthermore, since Rh is bounded in H 1 , the desired H 1 boundedness of Πh will
follow from the corresponding bound on Πhb (I − Rh ). However, by shape regularity,
combined with (3.9) we have
Πhb (I − Rh )v2H 1 () ≤ c
≤c
T ∈Th
T ∈Th
b
2
h −2
T Πh (I − Rh )v L 2 (T )
2
h −2
T (I − Rh )v L 2 (T ) ,
and hence the H 1 -bound follows from (3.1). Finally, if the triangulation is quasiuniform or satisfies (3.10) then the L 2 -bound follows from the corresponding bounds
for the operators Πhb and Rh .
Remark 3.1 The two conditions guaranteeing L 2 -stability of the Fortin operator Πh
are rather different. Condition (3.10) is local, while quasi-uniformity is a global mesh
condition. In some sense, this resembles two of the mesh conditions which can be
used to obtain H 1 -stability of the L 2 -projection onto continuous finite elements. It is
a classical, and well known, result that quasi-uniformity implies H 1 -stability of this
operator, while an alternative local condition was derived in [3].
Remark 3.2 The analysis of the Taylor–Hood method given above leans heavily on
the assumption that there are no triangles in Th with more than one edge on the
boundary of . This assumption simplifies the analysis, but it is not necessary. For more
general triangulations, where some triangles may have two edges on the boundary, we
will still define the interpolation operators Πh and Πhb as above. The only necessary
modification is that the definition of the bubble space Vhb must be altered, since the
dimension of the space Z h0 , given by (3.3), in general is larger than the number of
interior edges. A possible choice is
Vhb = span { be te | e ∈ i1 (Th ) } ∪ { be n e | e ∈ i,2
1 (Th ) } ,
where te and n e are tangent and normal vectors to the edge e, and where i,2
1 (Th )
is the subset of i1 (Th ) consisting of the third edges of the triangles with two edges
on the boundary. Hence, the space Vhb is enriched by including “the normal edge
b
bubbles” associated the edges in i,2
1 (Th ). With this definition the space Vh has the
0
same dimension as Z h . In [15] the analysis above is extended to this more general
case. However, the proof in this setting is more technical than the analysis above, and
therefore it is omitted here.
4 Uniform preconditioners
The interest in the construction of preconditioners for the discretizations of the singular
perturbation system (1.1) is partly motivated by the role of such systems in time
stepping schemes for the time dependent Stokes and Navier–Stokes equations. The
system (1.1) has a coefficient operator which formally is of the form
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A =
I − 2 −grad
,
div
0
with obvious weak interpretations of the differential operators derived from the weak
formulation (2.1). In the notation of [16], the corresponding block diagonal operator
B = diag((I − 2 )−1 , (−)−1 + 2 I ), can be referred to as a canonical preconditioner for A , since the uniform inf–sup condition (2.2) implies that the composition
B A is an isomorphism mapping the space (L 2 ∩ H01 ) × ((H 1 ∩ L 20 ) + −1 L 20 ) to
itself, with operator norms of B A and its inverse bounded independently of . We
refer to [16, Section 4] and [17, Section 3] for more details. Furthermore, as explained
in Sect. 2 above, if the Fortin operator Πh is uniformly bounded both in H 1 and L 2
then the corresponding discrete inf–sup condition (2.5) holds for the Taylor–Hood
method. Therefore, it follows from Theorem 3.6 that, if the triangulation {Th } is either
quasi-uniform or satisfies the local condition (3.10), then (2.5) holds. In particular, the
condition numbers of the corresponding discrete operators B,h A,h will be uniformly
bounded with respect to and h, cf. [16, Section 5]. This result is obtained without any
convexity assumption on the domain. Since all previous theoretical studies of uniform
preconditioners for the model (1.1) have relied on convexity, cf. Remark 2.1 above, we
will present a numerical experiment which confirm the theory above. More precisely,
the experiment indicates that the condition number of B,h A,h is basically unaffected
by such properties of the domain. The results below are obtained by using a canonical
preconditioner, i.e., the operator B,h is composed of exact inverses of discretizations
of the differential operators appearing in the definition of B . Of course, to obtain more
practical preconditioners, these inverses should again be replaced by suitable elliptic
preconditioners.
Fig. 1 The domains 1 , 2 ,
and 3
Table 1 Condition numbers for
the operators B,h A,h
Domain
1
2
3
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\h
2−2
2−3
2−4
2−5
1
14.3
15.3
18.5
22.0
0.1
10.3
11.9
12.7
13.2
0.01
6.2
6.7
8.0
9.9
1
17.1
17.2
17.1
17.1
0.1
10.3
11.8
12.7
13.2
0.01
6.1
6.7
8.0
9.9
1
13.2
13.4
13.5
13.6
0.1
10.3
11.9
12.8
13.2
0.01
6.1
6.7
8.1
9.9
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We consider the problem (1.1) on three two dimensional domains, referred to as
1 , 2 and 3 . Here 1 is the unit square, 2 is the L-shaped domain obtained by
cutting out the an upper right subsquare from 1 , while 3 is the slit domain where
a slit of length a half is removed from 1 , cf. Fig. 1. Hence, only 1 is convex.
The corresponding problems (1.1) were discretized by the lowest order Taylor–Hood
element on a uniform triangular grid. The condition numbers of the corresponding
discrete operators B,h A,h , for the three domains and for different values of and h,
are computed. The results are given in Table 1 below.
These results clearly indicate that the condition numbers of the operators B,h A,h are
not dramatically effected by lack of convexity of the domains.
References
1. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element method for Stokes equations. Calcolo 21,
337–344 (1984)
2. Bramble, J.H., Pasciak, J.E.: Iterative techniques for time dependent Stokes problems. Comput. Math.
Appl. 33(1–2), 13–30 (1997) (approximation theory and applications)
3. Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the L 2 projection in H 1 (). Math.
Comp. 71, 147–156 (2001)
4. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from L0
multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)
5. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
6. Cahouet, J., Chabard, J.-P.: Some fast 3D finite element solvers for the generalized Stokes problem.
Internat. J. Numer. Methods Fluids 8(8), 869–895 (1988)
7. Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal.
Numér. 9, 77–84 (1975)
8. Costabel, M., McIntosh, A.: On Bogovskiı̆ and regularized Poincaré integral operators for de Rham
complexes on Lipschitz domains. http://arxiv.org/abs/0808.2614v1
9. Falk, R.S.: A Fortin operator for two-dimensional Taylor–Hood elements. Math. Model. Numer. Anal.
42, 411–424 (2008)
10. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, vol 1. In:
Linearized Steady problems. Springer Tracts in Natural Philosophy, vol. 38. Springer, New York
(1994)
11. Geissert, M., Heck, H., Hieber, M.: On the equation divu = g and Bogovskiı̆’s operator in Sobolev
spaces of negative order. Oper. Theory Adv. Appl. 168, 113–121 (2006)
12. Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations: theory and algorithms.
Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
13. Kolev, T., Vassilevski, P.: Regular Decomposition of H(div) Spaces. Comput. Methods Appl. Math.
(to appear)
14. Lay, M.-J., Schumaker, L.L.: Spline functions on triangulations. In: Encyclopedia of Mathematics and
its Applications. Cambridge University Press, London (2007).
15. Mardal, K.-A., Schöberl, J., Winther, R.: A uniform inf-sup condition with applications to preconditioning. arXiv:1201.1513v1,2012.
16. Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations.
Numer. Linear Algebra Appl. 18, 1–40 (2011)
17. Mardal, K.-A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer.
Math. 98(2), 305–327 (2004)
18. Mardal, K.-A., Winther, R.: Erratum: “Uniform preconditioners for the time dependent Stokes problem”
[Numer. Math. 98(2), 2004, pp. 305–327]. Numer. Math. 103(1), 171–172 (2006)
19. Olshanskii, M.A., Peters, J., Reusken, A.: Uniform preconditioners for a parameter dependent saddle
point problem with application to generalized Stokes interface equations. Numer. Math. 105, 159–191
(2006)
20. Turek, S.: Efficient Solvers for Incompressible Flow Problems. Springer, Berlin (1999)
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