Stability and approximability of the P1-P0 element for Stokes
equations
Jinshui Qin∗ and Shangyou Zhang†
Abstract
In this paper we study the stability and approximability of the 1 - 0 element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for
Stokes equations. Although this element is unstable for all meshes, it does provide optimal
approximations for the velocity and the pressure in many cases. We establish a relation
between the stabilities of the 1 - 0 element (bilinear/constant on quadrilaterals) and the
1 - 0 element. We apply many stability results on the
1 - 0 element to the analysis of
the 1 - 0 element. We prove that the element has the optimal order of approximations
for the velocity and the pressure on a variety of mesh families.
As a byproduct, we also obtain a basis of divergence-free piecewise linear functions on
a mesh family on squares. Numerical tests are provided to support the theory and to show
the efficiency of the newly discovered, truly divergence-free, 1 finite element spaces in
computation.
Keywords. mixed finite elements, Stokes, divergence-free.
AMS subject classifications (2000). 65M60, 65N30, 76M10, 76D07.
1
Introduction
The 1 - 0 element, continuous piecewise linear approximation for the velocity and piecewise
constant approximation for the pressure, is probably the simplest finite element which could
preserve the incompressibility condition of incompressible fluids (see [1, 3, 6, 8, 20, 10, 19, 22
and 25] for more information on divergence-free elements for Stokes). Unfortunately, the element is unstable for any mesh since the dimension of the discrete velocity space is always less
than that of the pressure space (with Dirichlet boundary condition). However, this element
does provide optimal approximations for both the velocity and the pressure on many mesh
families. Some discussions on this element can be found in [8, 9, 18 and 23] and references
therein. In this paper, we concentrate on the triangular meshes made by crisscross-refinements
of quadrilateral meshes, by dividing each quadrilateral of quadrilateral meshes into four triangles by its two diagonals.
To make the notation in introduction clear, we define some notations first. Let V h ⊂ H1 (Ω)
and Ph ⊂ L20 (Ω) := L2 (Ω)/C denote the mixed finite element spaces of velocity and pressure
respectively; they are defined on a triangulation h of a polygonal domain Ω. We solve the
finite element approximation (u h , ph ) satisfying the mixed formulation of Stokes equations:
a(uh , v) − b(v, ph ) = (f, v), ∀v ∈ Vh ,
b(uh , q) = 0,
∗
†
∀q ∈ Ph .
10321 Yellow Pine Dr., Vienna, VA 22182. [email protected].
Department of Mathematical Sciences, University of Delaware, DE 19716. [email protected].
1
(1)
R
R
R
Here a(u, v) := Ω ∇u : ∇v, b(v, q) := Ω q div v, and
(f,
v)
:=
Ω f · v for all u, v ∈ Vh and
R
q ∈ Ph . For convenience, we define Nh = {q ∈ Ph | Ω q div v = 0, ∀v ∈ Vh } and the reduced
pressure space Mh as the L2 -orthogonal complement of Nh in Ph . The non constant functions
of Nh are called spurious pressure modes. The element V h × Ph is said to be stable if the
inf-sup constant
R
Ω p div v
γh (Vh , Ph ) := inf
sup
06=p∈Ph
06=v∈Vh kvk1,Ω kpk0,Ω
is bounded below byR a positive number independent of h, see [2] and [7]. Here, P h is assumed
a subset of L20 , i.e., Ω q = 0 for all q ∈ Ph . If Vh × Ph is not stable but Vh × Mh is, then the
element Vh × Ph is said to be reduced-stable. Accordingly, the inf-sup constant γ h (Vh , Mh ) is
called the reduced inf-sup constant of the element V h × Ph .
In this paper, we show a close relation between the stabilities of the 1 - 0 element (bilinear/constant quadrilateral element) and the 1 - 0 element. That is, the reduced stability
of the 1 - 0 element on a family of quadrilateral meshes is equivalent to that for 1 - 0 on
the crisscross-refinement meshes of the quadrilateral family. Therefore, many results on the
stability of the 1 - 0 element can be applied to 1 - 0 . For example, the reduced inf-sup
constant of the 1 - 0 element is Ch on the square meshes of the unit square, therefore the
reduced inf-sup constant of the 1 - 0 element on the crisscross meshes of the unit square is
Ch too. There are many results on the stability and approximability of the 1 - 0 element
(see [4, 5, 8, 20, 11, 12, 13, 14, 15, 16, 17 and 21].)
If a mesh family is stable for the 1 - 0 element, then the crisscross-refinement of the
mesh family is reduced-stable for P 1 - 0 . Then we show that the 1 - 0 element has optimal
approximations for the velocity and pressure. There are a wide range of stable families for the
quadrilateral element 1 - 0 , see 24]. Therefore the numerical solutions of the element 1 - 0
are of the optimal order on the crisscross-refinements of all these families. The performance of
the 1 - 0 element on some general mesh families are also discussed in this paper.
As a byproduct of our analysis, we explicitly display a basis of all divergence-free continuous
piecewise linear polynomials defined on the crisscross meshes of the unit square. Each basis
function has a small support formed by a few triangles. Moreover, we show that the space
of these divergence-free functions has the optimal approximation property. The analysis is
supported by a numerical test.
There are 6 sections in this paper. In Section 2, we discuss the stability relation between
1 - 0 and
1 - 0 . In Section 3, we analyze the approximation properties of the 1 - 0 element
on the crisscross meshes of the unit square. We study the performance of the element on a
general mesh family in Section 4. In Section 5, we display a basis of divergence-free piecewise
linear functions. Finally in Section 6, we report some numerical results on the reduced stability
and approximability of the 1 - 0 element. We also illustrate the efficiency of using the newly
discovered divergence-free basis in computation by a simple example.
2
Stabilities of
1- 0
and
1- 0
elements
In this section, we shall study the relationship between 1 – 0 and 1 – 0 finite elements.
The Ω under consideration could be a general polygonal domain. We denote a quadrilateral
partition of Ω by h and its corresponding crisscross-refinement by h , which is obtained by
dividing each quadrilateral in h by its two diagonals.
For each triangle T ∈ h , let hT denote the diameter of T and ρT the diameter of the circle
inscribed in T . A family of triangulations of Ω is said to be regular if there is a positive θ
2
independent of h such that
ρT
≥ θ.
hT
(2)
We assume triangulations are regular in this paper.
Let Ŵ denote the unit square [0, 1]×[0, 1] with vertices ŵ 1 , ŵ2 , ŵ3 , and ŵ4 (counterclockwise
from the left-lower corner). For any quadrilateral W ∈ h with the vertices w1 , w2 , w3 , and
w4 (counterclockwise), there exists exactly one bilinear mapping F W ∈ Q̂1 (Ŵ )2 that maps Ŵ
onto W such that FW (ŵi ) = wi . Here


 X

Q̂k (Ŵ ) =
aij x̂i ŷ j | aij ∈ R


0≤i,j≤k
On the quadrilateral W we define finite element space
−1
Qk (W ) = {v ◦ FW
, ∀v ∈ Qk (Ŵ )},
where k ≥ 1. On
h,
we define
= v ∈ H 1 (Ω) | v|W ∈ Qk (W ), ∀W ∈
Mk−1 ( h ) = v ∈ L2 (Ω) | v|W ∈ Qk (W ), ∀W ∈
Mk0 (
h)
The finite element
1–
0
h
h
and its associated spaces are defined as
Ṽh = M01 (
h)
,
.
∩ H1 (Ω),
P̃h = M0−1 ( h ),
n
o
Ñh = q ∈ P̃h | b(v, q) = 0, ∀v ∈ Ṽh ,
(3)
M̃h = L2 -orthogonal complement of Ñh in P̃h .
Let Pk (T ) denote the polynomials with degree less than or equal to k defined on each
triangle T ∈ h . On h we define
Mk0 ( h ) = v ∈ H 1 (Ω) | v|T ∈ Pk (T ), ∀T ∈ h ,
Mk−1 ( h ) = v ∈ L2 (Ω) | v|T ∈ Pk (T ), ∀T ∈ h .
For the
1
–
0
element, we use the notations Vh , Ph , Nh , Mh , as defined by
Vh = M01 (
Ph = M0−1 (
h)
∩ H1 (Ω),
h ),
(4)
Nh = {q ∈ Ph | b(v, q) = 0, ∀v ∈ Vh } ,
Mh = L2 -orthogonal complement of Nh in Ph .
It is obvious that P̃h ⊂ Ph .
We call a polygonal region U of h (resp.
h ) a macroelement if it is formed by some
triangles of h (resp. quadrilaterals of h ). For a macroelement U of h (resp. h ), we define
U
U
U
U
U
U
U
localizations VU
h , Ph , Nh , and Mh (resp. Ṽh , P̃h , Ñh , and M̃h ) to U of the finite element
spaces Vh , Ph , Nh , and Mh (resp. Ṽh , P̃h , Ñh , and M̃h ) by just replacing Ω by U in (3) (resp.
(4)).
3
Lemma 2.1 Let h be a quadrilateral partition of a polygonal domain Ω. If
sponding crisscross-refinement for h , then
h
is the corre-
N˜h ⊂ Nh .
Proof We note a simple fact first. For any v ∈ V h , there exists a w ∈ Ṽh such that
b(v, q) = b(w, q), ∀q ∈ P˜h .
(5)
Similarly, for any w ∈ Ṽh , there is v ∈ Vh such that (5) holds.
Since any function w ∈ Ṽh is linear on all the edges of h , there exists a function v ∈ Vh
such that v − w vanishes on all the edges of h . Therefore, (5) follows by using the Green’s
formula.
From (5), the lemma follows.
In order to study the structure of Nh , we consider h as a macroelement partition of h .
For any quadrilateral U ∈ h , there is a one-dimensional space of spurious pressure modes,
with their support in U , associated with the singular vertex of U (the intersection of the two
diagonals of U , see Figure 1). For convenience, we denote the spurious pressure mode, shown
in the second picture of Figure 1, by δ U (values of δU are shown in Figure 1, where a, b, c, and
d are the lengths of the four interior edges). A detailed calculation of δ U can be found in [19].
b
c
-1
bc
a
1
cd
d
1
ab
-1
ad
Figure 1. Macroelement U and spurious pressure mode δ U .
Lemma 2.2 Let
h
be the crisscross-refinement of
h,
then
Nh = N˜h + span {δU | for all quadrilaterals U ∈
h
}.
Proof Clearly, δU ∈ Nh for any U ∈ h . Since NhU contains only linear combinations of χ U
(the characteristic function of U ) and δ U for each U ∈ h , any function q ∈ Nh must be in P˜h
provided q is orthogonal to all δU ’s. By (5), q ∈ Ñh . Hence, the lemma follows.
Corollary 2.1
Mh ∩ P˜h = M̃h .
Theorem 2.1 For a polygonal domain Ω, let h be a quadrilateral partition of Ω and
the corresponding crisscross-refinement. Assume h is regular. Then
Cγh (Ṽh , M̃h ) ≤ γh (Vh , Mh ) ≤ C −1 γh (Ṽh , M̃h ),
where C is independent of h.
4
h
be
Proof Let
be a macroelement partition of
h
h.
On each macroelement U ∈
dim VU
h = 2,
dim PhU = 4,
dim MhU = 2,
dim NhU = 2.
h,
we have
Following the arguments of Theorem 4.3.1 in [19], we can bound the local inf-sup constants
U
γh (VU
h by a positive number which is independent of h. Aph , Mh ) on all macroelements U in
plying the macroelement partition theorem (Theorem 3.2.1 in [19]), we know that
P the reduced
inf-sup constant γh (Vh , Mh ) is determined by the stability of V h × (Mh ∩ U ∈
NhU ) =
h
Vh × M̃h (by the corollary). This implies that V h × Mh has exactly the same stability as
Vh × M̃h . If it is shown that Vh × M̃h and Ṽh × M̃h have the same stability, then the theorem
is proven.
We first show that
γh (Vh , Mh ) ≤ C −1 γh (Ṽh , M̃h ).
(6)
For any q ∈ M̃h , there exists a function w ∈ Ṽh such that
b(w, q) = kqk20,Ω ,
C
kqk0,Ω .
kwk1,Ω ≤
γh (Ṽh , M̃h )
(7)
If we can construct a function v ∈ Vh such that
b(v, q) = kqk20,Ω ,
C
kvk1,Ω ≤
kqk0,Ω ,
γh (Ṽh , M̃h )
then (6) is proved.
Let Ih := (Ih , Ih ) : Ṽh → Vh be the interpolation operator such that Ih g and g agree at
every vertex in h for any g ∈ Ṽh . For any function g ∈ Ṽh , the interpolation error g − Ih g
vanishes at all the edges of the quadrilateral partition h . We will show that there is a constant
C independent of h such that
kg − Ih gk1,Ω ≤ Ckgk1,Ω ,
(8)
for any g ∈ Ṽh . If this is the case, by taking v = Ih w, then
kvk1,Ω ≤ kw − vk1,Ω + kwk1,Ω ≤ Ckwk1,Ω .
According to the arguments in the proof of Lemma 2.1 and (7), we have
b(v, q) = b(w, q) = kqk20,Ω .
C
kvk1,Ω ≤
kqk0,Ω .
γh (Ṽh , M̃h )
This proves (6) with the assumption (8).
Let U ∈ h be a macroelement. If we need to show that
kg − Ih gk0,U ≤ Ckgk0,U ,
(9)
|g − Ih g|1,U ≤ C|g|1,U ,
(10)
5
for any g ∈ Q1 (U ), any U ∈ h , and any h > 0 with constant C independent of h. Let E θ (Û )
denote the set of all the equivalence macroelements, see [1, 19 and 24], of the unit square Û
satisfying the regularity condition (2). Obviously, translations and dilations of U do not affect
(9) and (10). Therefore, for simplicity, we assume that the length of the longest diagonals of
each macroelement in Eθ (Û ) is one unit, and that the intersection of the two diagonals of any
macroelement in Eθ (Û ) has coordinates (0, 0). For any macroelement U in E θ (Û ), we denote
the intersection of its two diagonals by v 5 and the other four vertices by v1 , v2 , v3 , and v4 ,
clockwise. Hence, S = {(v1 , v2 , v3 , v4 , v5 ) | U ∈ Eθ (Û )} is a closed set in R10 . Let g ∈ Q1 (U )
be arbitrary and it values at vi be gi for i = 1, 2, 3, 4. For convenience, we denote the vector
(g1 , g2 , g3 , g4 )t by ḡ. Since (Ih g)(vi ) = gi for i = 1, 2, 3, 4 and (Ih g)(v5 ) is determined by g1 , g2 ,
g3 , and g4 , we have
kg − Ih gk20,U = ḡ t AU ḡ,
kgk20,U = ḡ t BU ḡ.
|g − Ih g|21,U = ḡ t CU ḡ,
|g|21,U = ḡ t DU ḡ.
Here BU is a symmetric positive definite and A U , CU , and DU are symmetric positive semidefinite. Clearly, the entries of AU , BU , CU , and DU are continuous functions of (v1 , v2 , v3 , v4 , v5 ).
Since S is a bounded closed set in R10 , there exists a constant C1 independent of h such
that
kg − Ih gk0,U ≤ C1 kgk0,U ,
(11)
for any U ∈ Eθ (Û ).
It is easy to see that only the constant functions make both sides of (10) zero. Hence, the
matrix DU always has exactly three positive eigenvalues which depend on (v 1 , v2 , v3 , v4 , v5 )
continuously. Since S is a bounded closed set in R 10 , the smallest nonzero eigenvalue of D U is
bounded away from zero by a positive number independent of h. Due to the same reason, the
largest eigenvalue of CU is bounded above by a constant independent of h. Therefore, there
exists a constant C2 independent of h such that
|g − Ih g|1,U ≤ C2 |g|1,U ,
(12)
for any U ∈ Eθ (Û ).
Combining (11) and (12), we obtain (9) and (10). Therefore, we have proven (6).
Using similar arguments, we can show that γ̃ h ≥ C γ̄h .
Theorem 2.2 Let h be a quadrilateral partition of a polygonal domain Ω and h be the
corresponding crisscross-refinement. If Ṽh × P̃h is stable, then Vh × Mh is stable. Moreover,
if (uh , p̄h ) ∈ Vh × Mh and (uh , ph ) ∈ Vh × Ph solve (1), then
ku − uh k1,Ω ≤ Chkuk2,Ω ,
(13)
kp − p̄h k0,Ω ≤ Ch(kuk2,Ω + kpk1,Ω ).
2
Here the constant C is independent of h, p̄ h = ph /Nh , and we assume that (u, p) ∈ H̊ (Ω) ×
H 1 (Ω) solves the Stokes equations on the continuous level.
Proof Stability of Vh × Mh is a direct consequence of Theorem 2.1. It is known that
M0−1 ( h )/R ⊂ Mh implies that the approximation properties of M h are as good as those
of Ph . Therefore, by the stability theorem established in [7], (13) follows (the error estimate
of velocity is decoupled from kpk1,Ω because of div uh = 0).
6
For the 1 - 0 element, one stable mesh family was identified in [24]. On the crisscrossrefinements of all these stable families, the 1 - 0 element is reduced-stable and the numerical
solution is optimal.
The pressure p̄h ∈ Mh can be recovered from ph ∈ Ph by applying a simple postprocess
quadrilateral by quadrilateral. That is, on each quadrilateral U ∈ h ,
R
p δ
R h 2 U δU .
p̄h |U = ph |U − U
U δU
As a direct application of Lemma 2.2, Theorem 2.1, and the stability results of the element
1 - 0 in [4, 20 and 17], we get the following theorem.
Theorem 2.3 Let h , h = 1/n, be a square mesh of the unit square and
sponding crisscross-refinement. Then
h
be the corre-
dim Nh = n2 + 2.
γh (Vh , Mh ) = Ch,
However, the approximation properties of the numerical solution of the 1 - 0 element on
the crisscross-refinement (the mesh is also called crisscross mesh) of the square mesh of the
unit square can not be answered directly by applying the stability theorem established in [7]
since the reduced inf-sup constant is Ch. This will be answered in the next section.
3
Approximation properties on crisscross meshes
In this section, we show that the numerical solutions of (1) converge to the true solution of the
Stokes equations (with Dirichlet boundary condition) at a rate of h on the crisscross meshes
of the unit square. Moreover, we will display a way to recover the numerical solution for the
pressure. Since Vh × Mh is unstable, this means that Mh is still too large. Therefore, we need
to remove more modes from Mh while trying to preserve the approximation properties for the
remaining pressure space Sh . It is then necessary to define a new velocity space W h ⊂ Vh
such that not only is Wh × Sh stable but also the velocity from Wh × Sh is exactly the same
as uh from Vh × Mh . Of course, the space Wh should possess good approximation properties.
The crucial matter here is to determine the bad modes in M h . Let h = 1/n and n = 4k for
some positive integer k, and let h be the crisscross mesh of the unit square. For convenience, we
denote (α, β) the vertex in h with coordinates (αh, βh), and δα+1/2,β+1/2 the spurious pressure
mode associated with the singular vertex (α, β) (see Figure 2). Let 4h be a macroelement
partition of h such that every U ∈ 4h has 16 h × h squares. Therefore, U consists of 64
triangles. From Lemma 2.2, we know that N hU is the space spanned by the functions χ U (the
characteristic function with support U ), δ U (the checkerboard mode with support U , shown in
Figure 2), and δi+1/2,j+1/2 , where (i + 1/2, j + 1/2) is a singular vertex in U . Obviously, the
δi+1/2,j+1/2 ’s are not in Mh . Since the global checkerboard mode δ Ω ∈ span{δU , ∀U ∈ 4h },
we need to remove span{δU , ∀U ∈ 4h } from Mh .
7
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
Figure 2. δα+1/2,β+1/2 and δU .
Define
N¯h = span{1, δi+ 1 ,j+ 1 , δU , 1 ≤ i, j ≤ n − 1, ∀U ∈
2
4h },
2
Sh = L orthogonal complement of N¯h in Ph ,
Wh = v ∈ Vh | b(v, q) = 0, ∀q ∈ N¯h ,
2
ShU = χU Sh ,
WU
h = {v ∈ Wh | supportv ⊂ U } .
2
Theorem 3.1 Let h be a crisscross mesh of the unit square. Assume (u, p) ∈ H̊ (Ω)×H 1 (Ω)
is the true solution of the Stokes equations and ( ũh , p̃h ) ∈ Wh × Sh solves (1), then
ku − ũh k1,Ω ≤ Chkuk2,Ω ,
kp − p̃h k0,Ω ≤ Ch(kuk2,Ω + kpk1,Ω ).
Proof It is easy to verify that
Nh ⊂ N¯h ,
M01 (
2h )
1
∩ H̊ (Ω) ⊂ Wh ,
and
M0−1 (
4h )
⊂ Sh .
Therefore, Wh ×Sh has good approximation properties. It only remains to prove that W h ×Sh
is stable.
Define
U
N˜h = q ∈ ShU | b(v, q) = 0, ∀v ∈ WU
h .
U
Then Lemma 2.2 implies that N˜h contains only constant functions. Therefore, W h × Sh ∩
1
U
(∪U ∈
N˜h ) is stable by the fact that [M 01 ( 2h ) ∩ H̊ (Ω)] × M0−1 ( 4h ) is stable. Finally,
4h
by using the macroelement partition theorem, see [1, 19 and 24], we have that W h × Sh is
stable.
Theorem 3.2 Let h be the crisscross mesh of the unit square with h = 1/(4k) for some
2
positive integer k. If (u, p) ∈ H̊ (Ω) × H 1 (Ω) is the true solution of the Stokes equations, then
the solution (uh , ph ) ∈ Vh × Ph of (1) satisfies
ku − uh k1,Ω ≤ Chkuk2,Ω ,
ph = p̃h + N¯h ,
and the pressure can be recovered from p h by a postprocess.
Proof Clearly, the solution (u h , ph ) ∈ Vh × Ph of
a(uh , v) + b(v, ph ) = (f, v),
b(uh , q) = 0,
8
∀v ∈ Vh ,
∀q ∈ Ph
satisfies
a(uh , v) + b(v, ph ) = (f, v),
b(uh , q) = 0,
∀v ∈ Wh ,
(14)
∀q ∈ Sh .
Since uh ∈ Wh and ph = (ph /N¯h ) + nh for some nh ∈ N¯h , (14) implies
a(uh , v) + b(v, ph /N¯h ) = (f, v),
b(uh , q) = 0,
∀v ∈ Wh ,
∀q ∈ Sh .
Since the solution of (1) in Wh × Sh is unique, we get
and ph /N¯h = p̃h .
uh = ũh
By Theorem 3.1, the velocity uh and recovered pressure p̃h have optimal rates of convergence.
It is simple to recover p̃h from ph since N¯h is known.
4
Approximation properties on a general mesh family
The 1 – 0 element is further analyzed on a general mesh family in this section. On this family
of meshes, it is shown that the finite element solution for the velocity converges at an order h
and the pressure can be recovered by a simple postprocess.
The triangulation h is formed in the following way. First, we partition the polygonal
domain Ω into quadrilaterals. This quadrilateral partition is denoted by 4h . Secondly, each
quadrilateral in 4h is divided into four sub-quadrilaterals by linking the intersection of its
two diagonals to the middle point of each edge, so 2h is formed. Repeating the above process
to all quadrilaterals in 2h , we have h . Finally, partitioning each quadrilateral in h into
4 triangles by its two diagonals, we obtain the triangulation h . The first figure in Figure 3
shows how to partition a quadrilateral in 4h into 4 quadrilaterals in 2h . The second figure
shows a macroelement in 4h with 64 triangles in h .
Figure 3. Form
h
from
4h .
We will show that on the triangulation h , the numerical solution uh has an optimal rate
of convergence and a pressure with an optimal rate of convergence can be recovered from
ph . The measure to achieve this objective is quite similar to what we used for the crisscross
mesh. We first remove all possible “bad modes” from P h , and then consider the pressure
in the remaining pressure space. The key issue is to determine bad pressure modes in P h .
We know that spurious pressure modes associated with each singular vertex must be removed
from the pressure space Ph . However, this is not enough to guarantee the stability according
to our experience with the crisscross mesh. Since the crisscross mesh is a special case of h , we
definitely need to remove those pressure modes which may degenerate to local spurious modes.
Based on this consideration, we will remove all multiples of the mode (shown in Figure 4) on
each macroelement in 2h from the pressure space Ph .
9
-1
-1
-1
-1
1
1
1
1
-1
1
1
1
-1
-1
1
-1
Figure 4. “Bad pressure mode” over U 2h .
For convenience, we denote a quadrilateral in h by Uh , a quadrilateral in 2h by U2h ,
and a quadrilateral in 4h by U4h . We also denote the mode shown in Figure 4 by δ U2h . Let
δUh denote the spurious pressure mode associated with the singular vertex in U h ∈ h . We
can easily conclude that
Z
T
for any T ∈
Define
2h ,
any U2h ∈
δU2h = 0,
2h .
N¯h = span{1, δUh , δU2h , ∀Uh ∈
h , ∀U2h
∈
2h },
Sh = L orthogonal complement of N¯h in Ph ,
Wh = v ∈ Vh | b(v, q) = 0, ∀q ∈ N¯h ,
2
ShU4h = χU4h Sh ,
4h
WU
= {v ∈ Wh | supportv ⊂ U4h } .
h
2
Theorem 4.1 Let h be a regular triangulation defined above. If (u, p) ∈ H̊ (Ω) × H 1 (Ω) is
the true solution of the Stokes equations and ( ũh , p̃h ) ∈ Wh × Sh solves (1), then
ku − ũh k1,Ω ≤ Chkuk2,Ω ,
kp − p̃h k0,Ω ≤ Ch(kuk2,Ω + kpk1,Ω ).
Proof It is expected that Nh ⊂ N¯h , if
structure, we can show that
M0−1 (
4h )
h
is a crisscross mesh. Since
⊂ Sh and [M01 (
2h )
U4h
h
has the special
1
∩ H̊ (Ω)] ⊂ Wh .
Therefore, Wh × Sh has good approximation properties. If we can prove that W h × Sh is
stable, then the proof is done.
Define
n
o
U
4h
N˜h 4h = q ∈ ShU4h | b(v, q) = 0, ∀v ∈ WU
.
h
U4h
We need to show that N˜h
contains only constant functions. If it does for any U 4h ∈ 4h ,
1
U
˜
then Wh × (Sh ∩ ∪U ∈
Nh 4h ) is stable by the fact that [M 01 ( 2h ) ∩ H̊ (Ω)] × M0−1 ( 4h )
4h
4h
is stable. Therefore, by using the macroelement partition theorem we can prove that W h × Sh
is stable.
U
U
We show that dim N˜h 4h = 1. Since δUh ∈ N¯h , if q ∈ N˜h 4h , then q is a constant on each
Uh in U4h . Hence, q must be a constant on each U 2h in U4h since we already removed δU2h
from the pressure space. Therefore, q must be a constant on each of the four U 2h in U4h . A
simple computation shows that q must be a constant on U 4h .
10
Theorem 4.2 Let
specially constructed
h
be the triangulation defined at the beginning of this section, from a
h , then the numerical solution (u h , ph ) ∈ Vh × Ph satisfies
ku − uh k1,Ω ≤ Chkuk2,Ω ,
2
and ph /N¯h = p̃h . Here we assume (u, p) ∈ H̊ (Ω) × H 1 (Ω) solves the Stokes equations.
Proof Use similar arguments as in the proof of Theorem 3.2.
5
A basis of divergence-free piecewise linear functions
In this section, we display a basis for Z h , the space of all divergence-free continuous piecewise
linear polynomials on the crisscross mesh h . All the basis functions have a very small local
support and the space Zh has optimal approximation properties.
Let the domain Ω be the unit square and h , h = 1/n a partition of Ω which contains
n × n equal small squares. The triangulation h is the crisscross-refinement of h .
Define
1
Vh =M01 ( h ) ∩ H̊ (Ω),
Zh = {v ∈ Vh | div v = 0} .
A simple calculation shows that
dim Vh = 4n2 − 4n + 2.
In order to study Zh , we define a “pressure” space Ph as
Ph = M0−1 (
h ).
The analysis of the properties of Zh can be carried out using the frame work of the analysis of
the 1 – 0 finite element for the Stokes equations with Dirichlet boundary conditions. Since
div Vh ⊂ Ph , we have
Zh = {v ∈ Vh | div v = 0} = {v ∈ Vh |b(v, q) = 0, ∀q ∈ Ph } .
Lemma 5.1 On the crisscross triangulation
h
with h = 1/n,
dim Zh = (n − 2)2 .
Proof Since dim Ph = 4n2 and dim Nh = n2 + 2 (see Theorem 2.3), the lemma holds.
In order to find a basis for Zh , we first consider a small macroelement U with size 3h × 3h
(see Figure 5).
7
8
13
12
4
5
6
11
10
1
9
2
3
Figure 5. Macroelement U .
11
We denote the interior vertices of U by 1, 2, · · · , 13, as shown in Figure 5, and denote
the nodal basis functions by φ1 , φ2 , · · · , φ13 accordingly. Let VU
h denote the subspace of Vh
such that all the functions in VU
have
supports
contained
in
U
. We look for functions v =
h
U
13
Σi=1 (ui , vi )φi ∈ Vh such that div v = 0 in U . Namely, we need to solve a system of 36 linear
equations in 26 unknowns. Since dim Z U
h = 1, we know that this system has a one-dimensional
solution space. After some algebraic computations, we find that the solution space is
U
ZU
h = span{z },
where zU = (ξ, η) and
ξ =φ2 − φ8 + φ10 + φ11 − φ12 − φ13 ,
η = − φ4 + φ6 − φ10 + φ11 − φ12 + φ13 .
See Figure 6 for graphs of ξ and η. Since there are exactly (n − 2) 2 different macroelement
with size 3h × 3h in h — the set of all these macroelements is named by h —we find a basis
for Zh . By the results of Section 4, we have
Theorem 5.1 Let
h
be the crisscross mesh of the unit square with h = 1/n. Then
U
z | ∀U ∈ h ,
2
form a basis of Zh . Furthermore, if u ∈ H̊ (Ω) and div u = 0, then
inf ku − vk1,Ω ≤ Chkuk2,Ω .
v∈Zh
Figure 6. The shape of the two components of z U .
6
Numerical tests
In this section, we report some results of numerical experiments. First we calculate the reduced inf-sup constant of the element 1 - 0 on a reduced-stable family, which is a crisscrossrefinement of a stable quadrilateral family for the 1 - 0 element, see [24]. We consider problem (1) when Ω is the unit square. The unit square is first partitioned into n × n small
squares, 2h = 1/n. The partition is denoted by 2h . The h is then defined by partitioning
each square of 2h into 5 quadrilaterals(see Figure 7). Finally, the triangulation h is the
crisscross-refinement of h as depicted in Figure 7.
12
Figure 7. One macro-quadrilateral element and its crisscross-refinement
By the work [24], we know the 1 - 0 element is stable on h . Therefore, 1 - 0 is
reduced-stable on h , and dim Nh = 5n2 + 1 (Lemma 2.2). Although there is a relatively
large space of spurious pressure modes involved, all these modes can be filter out from the
numerical approximation ph quite easily, quadrilateral by quadrilateral. Of course, the velocity
approximation uh is of the optimal order, and no recovery is needed.
The reduced inf-sup constant γh (Vh , Nh ) and dim Nh for 1/h = 2n = 2, 4, 6, · · · , 16 are
reported in Table 1. The reduced inf-sup constant is clearly bounded below, and the bound is
a relatively large number.
!
!
!
"
Table 1. Inf-sup constants for 1 - 0 .
h
γh (Vh , Mh )
dim Nh
1/2
0.31808
6
0.31314
21
1/4
0.31378
46
1/6
1/8
0.31481
81
1/10
0.31615
126
1/13
0.31653
181
1/14
0.31660
246
1/16
0.31693
321
!
!
Finally, we use the newly discovered div-free P 1 elements on the criss-cross grids (cf. Figure
5) to solve the following model Stokes equations:
a(u, v) − b(v, p) = (f, v), ∀v ∈ V,
b(uh , q) = 0,
∀q ∈ P,
defined on the unit square Ω = [0, 1]2 , where
f = −∆curlg + ∇gxx =
−gyxx − gyyy − gxxx
.
gxxx + gxyy − gyxx
with g = 64(x − x2 )2 (y − y 2 )2 . The exact solution is
u = curlg.
We depict the first component of u (uh in fact) in Figure 8.
13
(15)
Figure 8. The first component of uh on level 4 grid.
Thanks to the newly discovered div-free basis, the problem (15) is reduced to, find u h ∈ Zh
such that
a(uh , v) = (f, v),
∀v ∈ Zh ,
(16)
where Zh is the div-free P1 finite element space defined in Section 5. We note that, comparing to the traditional non-positive definite finite element systems, we have a positive definite
matrix for the discrete linear system (16). Further, the number of unknowns in the finite element equations (16) is much less than one-eighth of that in the standard mixed finite element
equations, see the fourth and the fifth columns of Table 2. From the fifth column of Table 2,
it is apparent that the div-free finite element solution converges at the optimal order, O(h 2 ).
At the end, in Figure 9, we plot one component of error u − u h on the grid level 4. We can
see that the nodal error is much larger at the criss-cross points (the center of each square),
compared with that at the square vertices.
Level
3
4
5
6
Table 2. Convergence of div-free P1 finite elements.
h
# elements dim Vh + dim Ph dim Zh ku − uh kl∞
1/4
64
146
4 0.21398925
1/8
256
546
36 0.07262499
1/16
1024
2114
196 0.02063174
1/32
4096
8322
900 0.00549276
Figure 9. The first component of the error u − u h .
Acknowledgement The author would like to thank Professor Douglas N. Arnold for his
encouragement, advice, and support during the years of the author’s study at the Pennsylvania
14
State University.
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