The divergence-free finite elements for the stationary
Stokes equations
Shangyou Zhang∗
Abstract
We review nearly all types of divergence-free elements for the stationary Stokes equations, namely,
• the Pk+1 -Pk element on triangular grids for k ≥ 3,
• the Pk+1 -Pk element on tetrahedral grids for k ≥ 7,
• the Qk+1 -Qk element and the Qk+1,k × Qk,k+1 -Qk element on rectangular grids for k ≥ 2,
• the Qk+1 -Qk element on box grids for k ≥ 2,
• the Pk+1 -Pk element on Hsieh-Clough-Tocher triangular grids for
k ≥ 1,
• the Pk+1 -Pk element on Hsieh-Clough-Tocher tetrahedral grids for
k ≥ 2,
• the Pk+1 -Pk element on Powell-Sabin and Powell-Sabin-Heindl triangular grids for k ≥ 0,
• the Pk+1 -Pk element on criss-cross triangular grids for k ≥ 0.
In all these methods, we approximate the velocity by the continuous
piecewise-polynomials and the pressure by the discontinuous piecewisepolynomials, of one degree lower. Therefore the discrete solutions approximating the velocity are also divergence-free, i.e., incompressible pointwise. The classic iterated penalty method is presented which guarantees
the iterates converge to the optimal solution and produces the discrete
pressure functions as byproducts.
Keywords.
mixed finite element, Stokes equations, divergence-free
element, incompressible flow, quadrilateral element, tetrahedral element,
hexahedral element, Hsieh-Clough-Tocher, Powell-Sabin, Powell-SabinHeindl.
AMS subject classifications. 65M60, 65N30, 76M10, 76D07.
∗
Department of Mathematical Sciences, University of Delaware, DE 19716, U.S.A.
[email protected].
1
1
Introduction
The finite element method is one of the most effective numerical method
for solving partial differential equations. When applying the finite element
method, we rewrite the differential equation in a weak variational form, then
determine the Sobolev spaces for the problem. Accordingly, we choose piecewise polynomial subspaces on some grids over the domain, consisting of triangles, tetrahedra, quadrilaterals or tetrahedra, usually. We solve the weak
variational problem in such a finite dimensional subspace. When the grid size
tends to zero, the piecewise polynomial solutions would converge to the true
solution of the differential equation. A finite element is referred to such a
small triangle along with some functionals defining polynomials on it.
In computational fluid dynamics, the stability of finite element methods
for approximating the incompressible Stokes or Navier-Stokes flows has been
a challenge and of great interests. Many techniques and constructions were
done, mostly for low order finite elements, in the past thirty years, cf. [22, 6].
Rewriting the Navier-Stokes or the Stokes equations in the the weak variational forms, the primitive unknowns, the velocity and the pressure, belong to
Sobolev spaces H1 and L2 , respectively. Naturally, a finite element method
would be the Pk+1 -Pk element which approximates the velocity by continuous
Pk+1 piecewise-polynomials and approximates the pressure by discontinuous
Pk piecewise-polynomials. Then it is apparent the divergence of the finite element space for the velocity is a subspace or the finite element space for the
pressure, cf. (4.4). This is a truly conforming element as the incompressibility
condition is satisfied pointwise and the discrete solution for the velocity is a
projection within the space of divergence-free functions. A fundamental study
on the method was done by Scott and Vogelius in 1983 ([23, 24]) that the
method is stable and consequently of the optimal order on 2D triangular grids
for any k ≥ 3, provided the grids have no singular or nearly-singular vertex. A
2D vertex of a triangulation is singular if all edges meeting at the vertex form
two cross lines, see Figure 1. One can see ([23, 24]) that the inf-sup constant
γ degenerates to 0 if a vertex becomes singular (e.g., when Point C moves
to A in Figure 1) in the following inf-sup condition, i.e., the Babuška-Brezzi
condition:
(div v, q)
≥ γ,
(1.1)
inf sup
06=q∈Ph b∈Vh |v|H1 (Ω)3 kqkL2 (Ω)
where Vh and Ph are the mixed finite spaces for the velocity and pressure, cf.
(4.3) and (4.4).
For k ≤ 2, Scott and Vogelius showed that the P k+1 -Pk element would
not be stable, and may not produce approximate solutions on general 2D
triangular grids in [23, 24]. What is this magic number k in 3D? Scott and
2
Vogelius posted this question explicitly after discovering that k = 3 in 2D.
There was little progress in this direction, on divergence-free finite elements,
in the past 20 years. A main reason is that the classic theory on the mixed
finite elements is based on the inf-sup condition (1.1).
@
@
@
@
@
Q
Q
@ @
@
@
@
B
@
Q
@
A
Q
S
CS
S
S
Q
Q
S
S
S
S
Figure 1: Singular vertices (A and B) and a nearly-singular vertex (C).
The geometry of the 3D tetrahedral grids is much more complicated than
that of 2D. By adding a few edges and vertices locally, one can easily eliminate
singular vertices in 2D, see Figure 1. When a triangulation is singular-vertex
free, it is shown by Scott and Vogelius [23, 24] that the divergence of C 0 -Pk+1
vector space is exactly the space of C −1 -Pk modulus a constant. Following
the approach of Scott and Vogelius in 2D that we try to discover when the
divergence of the space of C0 -Pk+1 polynomials is the space of discontinuous
Pk polynomials, on tetrahedral grids, we found previously that this is true on
Hsieh-Clough-Tocher tetrahedral grids ([32]) for all k ≥ 2. For general tetrahedral grids, it is not so easy to identify and to eliminate all such singular vertices
and edges. For example, when doing multigrid refinements on tetrahedral grids
(cf. [31]), the known type of singular-edges (all face-triangles meeting at the
edge fall into two planes) and singular-vertices (all face-triangles meeting at
the vertex fall into three planes) cannot be avoided. Therefore, from computational purposes, one should extend the theory of Scott and Vogelius [23, 24]
to cover the case of singular-vertex.
This is done recently by the author in a series of work, cf. [33, 34, 36, 35,
37, 38]. Different from traditional analysis of mixed finite elements which is
based on the inf-sup condition ([22, 23, 24, 6]), a framework was set up in [33]
where the convergence of the Pk+1 -Pk element is based on the approximation
property of the C1 -Pk+2 finite element. We reestablish the theory for the ScottVogelius Pk+1 -Pk element without any restriction on singular vertex. We note
that in the extensions [34] and [36], unlike the cases in [23] and [32], the
divergence of the C0 -Pk+1 space is a proper subspace of the discontinuous P k
polynomials. The spurious modes in the discrete pressure space are filtered
3
out naturally in an iterative method, the classic iterated penalty method,
for the resulting finite element equations. This avoids the stabilization of
Pk+1 -Pk when singular or near-singular vertices are presented, proposed in
[15]. In addition, the discrete pressures are obtained as byproducts in the
iteration, greatly simplifying the computation. Therefore we call such mixed
finite elements, where the discrete space for the pressure is the divergence of
the velocity space, divergence-free finite elements, as there is only one kind of
finite element spaces in the computation, for the velocity only.
2
The C0-Pk and C0-Qk finite elements
To define a polynomial on a triangle or a rectangle, K, one needs to specify
certain function values and derivatives at some interpolation points on the
element. Such function or derivative evaluations are called linear functionals
on the element, denoted by a set ΣK . Together, (K, PK , ΣK ) is called a finite
element, where PK is a finite dimensional space on the domain K. A grid
Ωh consists of elements K such that diam(K) ≤ h, forming a non-overlapping
covering of Ω while the intersection of any two elements would be either empty
or a common low-dimensional boundary set. For example, the intersection of
two triangles can be either empty, or a common edge, or a common vertex.
Given a C0 (continuous) or C1 (continuously differentiable) function v on Ω, to
ensure the interpolation function of v by (K, P K , ΣK ), K ∈ Ωh , be C0 or C1 ,
the linear functionals ΣK have to be well designed that certain nodal values
must be placed on the boundary of K so that the common values are shared
by the two functions on the two sides of the common interelement boundary.
For example, on a triangle K = {(x, y)} shown in Figure 2, a linear polynomial Ih v ∈ P1 = PK , Ih v = a + bx + cy, is uniquely defined by the three
values of v at the three vertices. Further, as the restriction of I h v on an edge
is a 1D linear polynomial which is uniquely defined by the two end-edge values
of the function v. Therefore Ih v is a C0 function on Ω. This can be extended
to any Pk , k ≥ 1, cf. [11]. We note that the dimension of polynomials P k (the
space of all polynomials total degree k or less) in 2D matches the number of
nodal value points, shown in Figure 2:
(k + 1)(k + 2)
2
|ΣK | = 1 + 2 + · · · + (k + 1).
dim Pk =
In the same fashion, cf. [11], we define C 0 -Pk tetrahedral elements, shown
in Figure 3. We note that in this case, the polynomial dimension matches the
4
dim P1 = 3
s
s
JJ
dim P2 = 6
dim P3 = 10
s
JJ
dim P4 = 15
s
s
JJ
JJs
s
s
s
J
Js
J
s
s
s J s
J
J
J
s
s Js
J
J
J
s s s Js
Js s
s
Js s
s
s Js s
s s s Js
Figure 2: Degrees of freedom for C0 -Pk triangles, k ≥ 1.
number of nodal values too:
(k + 1)(k + 2)(k + 3)
6
(2)(3)
(k + 1)(k + 2)
|ΣK | = 1 +
+ ··· +
.
2
2
dim Pk =
dim P1 = 4
s
s
JJ
dim P2 = 10
s
JJ
dim P3 = 20
s
J
s
J s
J
Js
J
s
J
J
s
s Js
J
J
J
s
s
s
s Js
Js s
Js Figure 3: Degrees of freedom (on front triangle) for C 0 -Pk tetrahedra, k ≥ 1.
For rectangular grids in 2D and 3D, we need to use Q k , polynomials of
separate degree k or less in each variable. The interpolation nodal points are
shown in Figure 4 for 2D C0 -Qk elements (cf. [11]), where we put k + 1 nodes
on each edge to ensure the continuity across the edge and the uniqueness in
resolving the polynomial functions. The polynomial dimension matches to the
number of nodal values:
dim Qk = (k + 1)2 ,
|ΣK | = (k + 1) + · · · + (k + 1).
|
{z
}
(k+1)
Finally, we define the C0 -Qk finite element on rectangular cubes, cf. [11].
The interpolation nodal points are shown in Figure 5, where we put 2D dim Q k
points, (k + 1)2 , on each face rectangles. The polynomial dimension of Q k in
5
s
dim Q1 = 4
s
s s
dim Q2 = 9
s
s s
s
s
s
s s
s
dim Q4 = 16
s
s
s
s
s
s
s
s
s s
s
s
s s
s s
s
s s
s s
dim Q5 = 25
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
Figure 4: Degrees of freedom for C0 -Qk rectangles, k ≥ 1.
3D matches to the number of nodal values:
dim Qk = (k + 1)3 ,
|ΣK | = (k + 1)2 + · · · + (k + 1)2 .
|
{z
}
(k+1)
dim Q2 = 27
dim Q1 = 8
s
s
s
s
dim Q4 = 64
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
Figure 5: Degrees of freedom (on front rectangle) for C 0 -Qk boxes, k ≥ 1.
To be precise, we note that the construction of finite element spaces are
done by the reference element (K̂, P̂K , Σ̂K̂ ), and by affine mappings:
F (x̂) = x0 + Ax̂, F : K̂ → K,
where A is a 2 × 2 or 3 × 3 constant matrix, if K is a triangle or a tetrahedron,
or by bilinear or trilinear mappings if K is a quadrilateral or a hexahedron,
or by high-order polynomial mappings if K is a curved boundary element, cf.
[11].
3
The C1-Pk and C1-Qk finite elements
To make a piecewise polynomial function continuously differentiable, the maximal degree of the polynomials must be high to give enough degrees of freedom
for one polynomial to adapt its boundary values and derivatives, matching
those on the neighboring elements. On 2D triangular grids, 5 is the least degree for constructing the C1 piecewise functions. This is well-known as the
6
Argyris element, cf. [11] and Figure 6. In [34], we extend the construction of
the Argyris element to any degree polynomial k, for k ≥ 5. The construction
is depicted in Figure 6, that is, using the derivatives up to the second order at
each vertex, (k −4) normal derivatives at different internal edge-points, (k −5)
nodal values at different internal edge-points, and (k − 5)(k − 4)/2 at the standard internal triangle-points. It is easy to check the dimension matches the
number of linear functionals (ΣK ) in defining the finite element,
dim Pk =
(k + 1)(k + 2)
,
2
(k − 5)(k − 4)
2
(k + 1)(k + 2)
(k − 5)(k − 4)
=
.
= 6k − 9 +
2
2
|ΣK | = 3(1 + 2 + 3) + 3(k − 4) + 3(k − 5) +
rf P5 : dim = 21
j
rj
f P6 : dim = 28
rj
f P7 : dim = 36
@
@
@
@
@
@
@
@
@
r
@
@
@r
r @
r
@
@r
r
@
@
@r
r
r @
r
@
@
@
@
@
@
@
@
j
rf
j
r j
rf
r
j
r j
rf
r
r
j
r
@f
@f
@f
?
?
?
? ? ?
Figure 6: The degrees of freedom for some C 1 -Pk spaces, k ≥ 5.
We note that, for the piecewise polynomial spaces of the Argyris element
and of the higher-order elements constructed in Figure 6, they may not the
whole C1 -Pk space, but subspaces only. Morgan and Scott constructed the
whole C1 -Pk finite element spaces on general triangular grids in [18], where
careful selections of second order derivatives at vertices are made. However,
for the purpose of analyzing divergence-free finite elements, we only need to
have a subspace (of the C1 -Pk space) with the optimal-order approximation.
Unlike the situation in 2D on triangular grids (cf. [18]), the dimension
and a complete local-basis of the C1 -Pk space on a general tetrahedral grid
are not available. In fact, even a local-basis for a subspace of the C 1 -Pk space
is difficult to construct. The first construction was done by Ženišek in [30],
building a subspace of C1 -P9 polynomials. The construction was simplified
by the author, and extended to all polynomial degree k ≥ 9, in [35]. By the
7
Table 1: The nodal degrees of freedom for a C 1 -Pk element (Figure 7).
Type
#
vertex
(A):
4
edge
(B):
6
(C):
6
(D):
6
(E):
4
(F):
4
(G):
1
triangle
tetrahedron
Description
function and derivatives up to
order 4
function values at (k − 9)
edge-points
2 normal derivatives at
(k − 6) edge-points
3 2nd-order normal
derivatives at (k − 7)
edge-points
(k − 8)(k − 7)
function values
2
inside each face triangle
(k − 6)(k − 5)
normal
2
derivatives inside each face
triangle
(k − 7)(k − 6)(k − 5)
values
6
inside the tetrahedron
deg of freedom
35
k−9
2(k − 8)
3(k − 7)
(k − 8)(k − 7)
2
(k − 6)(k − 5)
2
dim Pk−8
local basis of the C1 -Pk subspace, we derive the optimal-order approximation
properties which are enough to ensure the approximation properties of C 0 -Pk
divergence-free elements.
In Table 1, we list the set ΣK of functionals for defining a C1 -Pk element
in 3D: (K, PK = Pk , ΣK ), cf. [11]. For details, we refer to [35]. In Figure 7,
we also plot the nodal degrees of freedoms. For a simple check, we can add up
all the degrees of freedom listed in Table 1:
dim Pk . =
(k + 1)(k + 2)(k + 3)
,
6
|ΣK | = 4(35) + 6(k − 9) + 6(2)(k − 8) + 6(3)(k − 7) + 4
(k − 6)(k − 5)
2
(k − 6)(k − 5) k 3 − 18k 2 + 107k − 210
+
2
6
1 3
11
(k
+
1)(k
+
2)(k + 3)
= k + k2 + k + 1 =
.
6
6
6
+4
The construction of C1 polynomials on rectangles and boxes is relatively
simpler. But after the first 2D C1 -Q3 element was constructed on rectangles,
known as the Bogner-Fox-Schmit rectangle, in 1965 [8], there is nearly no
further work in the finite element community, cf. [37]. In [37], we construct
8
m
j
g
d
q
J
J
J
(A):
m
j
g
d
q
(B):
J
J
J
j
g
d
q Jm
J
h Jh
J
h
Jh
Jh
h
J
h h h J
(D):
r
J
J
J
I
J I
@
@
@
@
J
J I
@
I
@ @J
@
J
@@ I
@@ J
I
(C):
J
Jr
J
J
r
(E):
J
J
J
J
J
6J
6 6J
6 6 6J
(F):
J
r
J
J
J
J
J
J
(G): J
b
J
b J
...
J
b J
b
J
J
Figure 7: Degrees of freedom (on front triangle and internal) for C 1 -Pk elements, k ≥ 9.
the whole C1 -Qk spaces for all k ≥ 3 on both rectangular or box grids, in 2D
and 3D. We describe the C1 -Qk elements of [37] in the following.
Let {φ̂i (x)} be the (k + 2) basis functions for the C 1 splines of polynomial
degree (k + 1) on [0, 1]:
φ̂00 (0) = 1,
φ̂l (
j−1
) = 1,
k−1
φ̂0k+1 (1) = 1,
j = 1, 2, ..., k,
(3.1)
and φ̂l is zero when evaluated by the other (k + 1) functionals in (3.1). For
example, the 4 cubic spline basis functions for k = 2 are
φ̂0 (x) = x3 − 2x2 + x, φ̂1 (x) = 2x3 − 3x2 + 1,
φ̂2 (x) = −2x3 + 3x2 ,
φ̂3 (x) = x3 − x2 .
9
(3.2)
The basis functions for k = 3 are
φ̂0 = −2x4 + 5x3 − 4x2 + x,
φ̂00 (0) = 1,
φ̂1 = −8x4 + 18x3 − 11x2 + 1,
φ̂3 = −8x4 + 14x3 − 5x2 ,
φ̂1 (0) = 1,
1
φ̂2 ( ) = 1,
2
φ̂3 (1) = 1,
φ̂4 = 2x4 − 3x3 + x2 ,
φ̂04 (1) = 1.
φ̂2 = 16x4 − 32x3 + 16x2 ,
Q4 : dim = 25
Q3 : dim = 16
sm
sm
sm
s
sm
sm
sm
Q5 : dim = 36
6
s
sm
s
s-
s
?
(3.3)
sm
s
s
s
m
s
m
6
s
6
s
sm
s
s
s-
s
s
s
?
s
?
ss
m
Figure 8: The family of C1 -Qk+1 , k ≥ 2, rectangles.
The basis functions in 2D and 3D are constructed by the tensor products
of (3.1), depicted in Figures 8 and 9. Then they are mapped to individual
elements, for example, [xm , xm+1 ] × [yn , yn+1 ], a rectangle element in the grid
Ωh , by
(
m
0 < i < k + 1,
φ̂i ( x−x
h ),
φi (x) =
x−xm
hφ̂i ( h ), i = 0, k + 1,
(
(3.4)
n
),
0
<
j
<
k
+
1,
φ̂j ( y−y
h
φj (y) =
n
hφ̂j ( y−y
h ), j = 0, k + 1.
It is shown in [37] that the C1 -Qk space is precisely the span of local basis
functions defined in (3.1–3.4), both in 2D and 3D. By the nodal interpolation
operator, it is standard to get the optimal order approximation properties for
the C1 -Qk spaces.
4
The divergence-free finite element method
We consider a model stationary Stokes problem: Find the velocity function
u and the pressure p on a 2D or a 3D polygonal domain Ω, which can be
10
f (xi ) :
fx (xi ), fy (xi ), fz (xi ) :
fxy (xi ), fxz , fyz , fxyz :
r
r
i
i
]
J *
J
@
R
@
]
J *
J
@
R
@
r
r
i
i
]
J *
J
@
R
@
]
J *
J
@
R
@
8 × 1 vertex values
8 × 3 first derivatives
8 × 4 mixed derivatives
Figure 9: The C1 -Q3 box (only the nodal freedoms on the front face.)
subdivided into triangles, tetrahedra, rectangles or boxes, such that
−∆u + ∇p = f
in Ω,
div u = 0
in Ω,
u=0
(4.1)
on ∂Ω.
d
The standard variational form for (4.1) is:
R Find u ∈ H 1,0 (Ω) (d = 2 or 3)
and p ∈ L2,0 (Ω) := L2 (Ω)/C = {p ∈ L2 | Ω p = 0} such that
a(u, v) + b(v, p) = (f , v), ∀v ∈ H1,0 (Ω)d ,
b(u, q) = 0,
∀q ∈ L2,0 (Ω).
(4.2)
Here H1,0 (Ω)d is the subspace of the Sobolev space H 1 (Ω)d (cf. [11]) with zero
boundary trace, and
Z
a(u, v) =
∇u · ∇v dx,
ΩZ
b(v, p) = −
div u p dx,
Z Ω
f v dx.
(f , v) =
Ω
We nestedly refine each grid of Ω to obtain a family of grids:
n
Ωh = R | R is a triangle, tetrahedron,
o
rectangle, or box of size hR , hR ≤ h .
For simplicity, we restrict our notations to 2D and will make comments for
the 3D case.
11
Let Qk,l be the space of polynomials of degree k or less in x and of degree
l or less in y:


l
k X
X

Qk,l =
qij xi y j .


i=0 j=0
As above, Qk = Qk,k , the space of polynomials of degree k or less in each
variable. Again, Pk is the space of polynomials of degree k, in 2D or 3D. Then
we define the divergence-free mixed element spaces by
Vh = {uh ∈ C(Ω) | uh |R ∈ Pk+1 ∀R ∈ Ωh , and uh |∂Ω = 0} ,
Vh = {uh ∈ C(Ω) | uh |R ∈ Qk+1 ∀R ∈ Ωh , and uh |∂Ω = 0} ,
(4.3)
Vh = {uh ∈ C(Ω) | uh |R ∈ Qk+1,k × Qk,k+1 , and uh |∂Ω = 0} ,
approximating the velocity, and by
Ph = {div uh | uh ∈ Vh } ,
(4.4)
approximating
theR pressure. R
R
Since Ω ph = Ω div uh = ∂Ω uh = 0 for any ph ∈ Ph , we conclude that
Vh ⊂ H1,0 (Ω)d ,
Ph ⊂ L2,0 (Ω),
i.e., the mixed-finite element pairs are conforming. The resulting system of
finite element equations for (4.2) is: Find u h ∈ Vh and ph ∈ Ph such that
a(uh , v) + b(v, ph ) = (f , v), ∀v ∈ Vh ,
b(uh , q) = 0,
∀q ∈ Ph .
(4.5)
Traditional mixed-finite elements require the inf-sup condition to guarantee the existence of discrete solutions. As (4.4) provides a compatibility
between the discrete velocity and discrete pressure spaces, the linear system
of equations (4.5) always has a unique solution, independent of the inf-sup
condition.
Proposition 4.1 ([33]) There is a unique solution in the discrete linear system (4.5) for any polynomial degree, i.e., k ≥ 1 in (4.3)-(4.4).
By the second equation in (4.5) and the definition of P h in (4.4), we conclude that
b(uh , q) = b(uh , − div uh ) = k div uh k2L2 (Ω)d = 0
(4.6)
12
and that div uh is also 0 everywhere. In this case, we call the mixed finite
element a divergence-free element. It is apparent that the discrete velocity
solution is divergence-free if and only if the discrete pressure finite element
space is the divergence of the discrete velocity finite element space, i.e., (4.4).
In fact, it is trivial to show ([22, 6, 5, 33]), in the next theorem, that u h is
the unique a(·, ·) orthogonal projection from the divergence-free space Z to its
subspace Zh , defined by
n
o
Z := v ∈ H1,0 (Ω)d | div v = 0 ,
(4.7)
Zh := {v ∈ Vh | div v = 0} .
(4.8)
Theorem 4.1 The unique solution uh of (4.5) is divergence-free, and is the
a(·, ·) orthogonal projection of u of (4.2), i.e.,
uh ∈ Z h ,
a(u − uh , v) = 0
∀v ∈ Zh .
Proof. By (4.6), uh is divergence-free. Letting v ∈ Zh in (4.2) and (4.5), we
get the above orthogonal projection equation.
We note that by (4.4), Ph is a subspace of discontinuous, piecewise polynomials of degree k. As singular vertices are present (see [23, 24]), P h is a proper
subspace. It is difficult to find a local basis for P h . But on the other side,
it is the special interest of the method that the space P h can be omitted in
computation and the discrete solutions approximating the pressure function in
the Stokes equations can be obtained as byproducts, as we shall see next. We
refer to [13, 6, 5, 26] for more information on the following iterative method.
Definition 4.1 (The iterated penalty method.) Let the initial iterate u 0h = 0
for the finite element Stokes equation (4.5). The rest iterates u nh are defined
iteratively to be the unique solution of
a(unh , vh ) + α(div unh , div vh ) = (f , vh ) + (div
n−1
X
ujh , div vh )
∀vh ∈ Vh ,
j=0
n = 1, 2, · · · . Here α is positive constant (between 1 and 10 usually.) At the
end of iteration, we let
pnh
= div
n
X
j=0
13
ujh .
Remark 4.1 In the iterated penalty method of Definition 4.1, we need only
to do computer coding for the continuous P k+1 , or Qk+1 , or Qk+1,k × Qk,k+1
element for the vector Laplacian like equations.
Remark 4.2 By Definition 4.1, at the convergence of iteration, div u nh = 0
and we obtain the solution uh of (4.5). Consequently, the unique solution p h
of (4.5) is obtained as a byproduct.
5
A convergence analysis
Though the finite equations (4.5) have a unique solution for any k ≥ 0, the
approximation cannot be guaranteed, unless k is large enough. We show the
convergence the C0 -Pk+1 and C0 -Qk+1 element by the optimal-order approximation property of C1 -Pk+2 or C1 -Qk+1 piecewise polynomials on the same
grid when solving a biharmonic equation in 2D or 3D.
We relate the Stokes equations (4.2) to the following biharmonic equation:
Find t ∈ H2,0 (Ω) (the closure of C∞,0 under H2 norm), such that
(∆t, ∆s) = −(curl f , s)
∀s ∈ H2,0 (Ω).
(5.1)
curl t = u.
(5.2)
It is well known [22] that
curl H2,0 (Ω) = Z
and
Here u and f are defined in (4.2). We will assume the 2D polygonal domain
provides a minimum elliptic regularity, i.e., for any g ∈ H −1 (Ω), the unique
weak solution s ∈ H2,0 (Ω) of the biharmonic equation ∆2 s = g satisfies
kskHr+2 (Ω) ≤ CkgkHr−2 (Ω) ,
r ≥ 1.
(5.3)
For example, Blum and Rannacher showed the H 4 (r = 2) elliptic regularity
in [4] for convex polygons with inner angles smaller than 126.28... o . We can
find some results on the elliptic regularity in [16].
The corresponding conforming finite element for (5.1) is the space of C 1
piecewise Pk+2 or Qk+1 polynomials on the grid Ωh :
1
(5.4)
h = sh ∈ C (Ω) | sh K ∈ Pk+2 (K) or Qk+1 (K) ∀K ∈ Ωh .
In the theory to be developed here, we do not need to know the dimension and
the construction of the whole space h . The optimal order of approximation for
a subspace Sh ⊂ h is needed to guarantee the optimal order of approximation
14
of the divergence-free element. By the constructions of local bases in Section
4, it is standard to get, cf. [11], that
inf ks − th kH2 ≤ Chmin{k,r} kskHr+2 (Ω)
∀s ∈ H2,0 ∩ Hr+2 , r ≥ 1.
th ∈Sh
(5.5)
We need the following assumption on the regular inversion of the divergence
operator too: The solution U ∈ H1,0 (Ω)d of
div U = F
(5.6)
satisfies
kUkHr (Ω)d ≤ CkF kHr−1 (Ω)
for some r ≥ 1,
(5.7)
where C is independent of r and U ∈ Hr (Ω)d ∩ H1,0 (Ω)d . We refer to [17, 2,
23, 28] and [33] for more results on (5.7).
We are ready to present the main theorem.
Theorem 5.1 ([33]) Let the smooth solution t in (5.1) have the elliptic regularity (5.3). Let the smooth solution U in (5.6) have the bounded regular
inversion (5.7). The unique solutions (u h , ph ), in Vh × Ph or in Ṽh × P̃h ,
of the discrete Stokes equations (4.5) approximate that of (4.2) in the optimal
order:
ku − uh kH1 (Ω)d + kp − ph kL2 (Ω) ≤ Chmin{k+1,r} kf kHr−1 (Ω)d ,
r ≥ 1.
(5.8)
Proof. Let th ∈ Sh be the finite element solution for the biharmonic problem
(5.1), i.e.,
(∆th , ∆sh ) = −(curl f , sh ) ∀sh ∈ Sh .
(5.9)
Subtracting (5.9) from (5.1), by Cea’s lemma [5, 11], (5.5) and (5.3),
|t−th |H2 (Ω) ≤ Chmin{k+1,r} k curl f kHr−2 (Ω) ≤ Chmin{k+1,r} kf kHr−1 (Ω)d . (5.10)
By Theorem 4.1 and (5.2), since curl t h ∈ Zh , it follows that
|u − uh |H1 (Ω)d = inf |u − v|H1 (Ω)d ≤ inf | curl t − curl sh |H1 (Ω)d
sh ∈Sh
v∈Zh
≤ | curl t − curl th |H1 (Ω)d ≤ C|t − th |H2 (Ω)
≤ Chmin{k+1,r} kf kHr−1 (Ω)d .
(5.11)
The L2 error of the velocity is bounded by the H 1 error, ensured by the
Poincaré inequality. Note that in (5.8), we used the inclusion relations:
curl Sh ⊂ Vh and curl Sh ⊂ Ṽh .
15
By (4.2) and (4.5), we have
(p − ph , div v) = (∇(u − uh ), ∇v)
∀v ∈ Vh or Ṽh .
(5.12)
Therefore we introduce p̃ ∈ L2,0 (Ω) such that
∀v ∈ H1,0 (Ω)d .
(p̃, div v) = (∇(u − uh ), ∇v)
(5.13)
Because div u = div uh = 0, the existence and the uniqueness of p̃ in (5.13) are
guaranteed by the inf-sup condition for the smooth functions ([22, 6]). Letting
F = p̃ in (5.6), we have an U ∈ H1,0 (Ω)d ∩ Hr (Ω)d such that div U = p̃ and
kUkH1 (Ω)d ≤ Ckp̃kL2 (Ω) .
Let v = U in (5.13).
kp̃kdL2 (Ω) ≤ |u − uh |H1 |U|H1 ≤ C|u − uh |H1 kp̃kL2 (Ω) .
(5.14)
By (5.12) and (5.13), because qh = div vh for some vh ∈ Vh , we get that
(p − ph − p̃, qh ) = 0
∀qh ∈ Ph .
Hence,
kp − ph − p̃kdL2 (Ω) = (p − ph − p̃, p − qh − p̃)
≤ kp − ph − p̃kL2 (Ω) kp − qh kL2 (Ω) + kp̃kL2 (Ω) ,
(5.15)
where qh ∈ Ph is arbitrary. Finally, letting F = p in (5.6), there is a U ∈
H1,0 (Ω)d ∩ Hr (Ω)d such that div U = p and
kUkH1 (Ω)d ≤ CkpkL2 (Ω) .
(5.16)
For the smooth function U in (5.16), we let v h ∈ Vh approximate U in optimal
order, for example, the edge-averaging interpolation of U defined in [25]. Let
qh = div vh . We conclude that
kp − qh kL2 (Ω) = k div(U − vh )kL2 (Ω) ≤ kU − vh kH1 (Ω)d
≤Chmin{k+1,r} kUkHr+1 (Ω)d ≤ Chmin{k+1,r} kpkHr (Ω)d
(5.17)
≤Chmin{k+1,r} kf kHr−1 (Ω)d ,
(5.18)
where the standard elliptic regularity on p for the Stokes equations (4.1) is
applied in the last step. Combining (5.14), (5.15) and (5.18), (5.8) is shown:
kp − ph kL2 (Ω) ≤ kp − ph − p̃kL2 (Ω) + kp̃kL2 (Ω)
≤ kp − qh kL2 (Ω) + 2kp̃kL2 (Ω)
≤ Chmin{k+1,r} kf kHr−1 (Ω)d + 2C|u − uh |H1 (Ω)d
≤ Chmin{k+1,r} kf kHr−1 (Ω)d .
16
Corollary 5.1 Assume that the elliptic regularity (5.3) the bounded regular
inversion (5.7) hold for Ω in 3D. The unique solution (u h , ph ) of the 3D Pk+1 Pk or Qk+1 -Qk element method of the discrete Stokes equations (4.5) approximate that of (4.2) in the optimal order:
ku − uh kH1 (Ω)d + kp − ph kL2 (Ω) ≤ Chmin{k+1,r} kf kHr−1 (Ω)d ,
r ≥ 1. (5.19)
Proof. The analysis remains the same except the biharmonic equation (5.1)
is now a vector equation. In 3D, the potential function t is not unique for
each u, unlike the 2D situation. However, by requiring div t = 0, we can still
have the uniqueness. For details, please read [36].
We remark that the theories, parallel to our series of work, for the continuous pressure Pk+1 -Pk elements on triangles, on tetrahedra, and the Q k+1 -Qk
elements on quadrilaterals are established in [9], [7] and [1], respectively, extending the Taylor-Hood element [22].
6
Some numerical tests
We choose, first for 2D tests, a simple exact solution so that the right hand
side function f for (4.1) is
−gyxx − gyyy − gxxx
,
(6.1)
f = −∆ curl g − ∇gxx =
gxxx + gxyy − gyxx
where
g = 28 (x − x2 )2 (y − y 2 )2 .
The continuous solution for the Stokes equations (4.1) is (see Figure 10)
u = curl g,
p = −gxx .
(6.2)
The grids Ωh are depicted in Figure 11, i.e., each squares are refined into
4 sub-squares each level. In Table 2 we list various norms of errors and orders
of convergence for the finite element solutions in the spaces V h × Ph defined
in (4.3)-(4.4) for the Qk+1,k × Qk,k+1 -Qk element. Here we do enough iterated
penalty iterations defined in Definition 4.1 until the iterative error is smaller
than the truncation error. We note that the Q k+1,k × Qk,k+1-Qk element is
much more efficient than the Qk+1 -Qk element. Stenberg and Suri showed in
[27] the stability, but the sub-optimal order of approximation, for the Q k+1 Qk−1 element for all k ≥ 1 in 2D. Bernardi and Maday showed the stability
and the optimal order of convergence for the Q k+1 -Pk element, cf. [3]. Given a
17
Figure 10: The exact solution, second component of u and p on the level 2
grid.
C0 -Qk+1 space for the velocity, it is preferred to use a maximal discrete space
for the pressure. It is apparently the Q k+1 -Qk element is better than the
Qk+1 -Qk−1 element and the Qk+1 -Pk element. And the Qk+1,k × Qk,k+1-Qk
element is even better as it perfectly matches the two discrete spaces, i.e., P h
in (4.4) is the whole C−1 -Qk space on the grid.
Figure 11: The first five levels of grids.
The H1 and L2 errors and convergence orders, reported in Table 2, are
consistent with the error bound proved in Theorem 5.1. In Table 2, the nodal
errors are of the optimal order too, but not proved theoretically yet. To be
precise, the order of convergence for the velocity when using the Q 3,2 × Q2,3 Q2 element, or using the Q4,3 × Q3,4 -Q3 element is one order, or two orders
higher than that predicted by the theory, respectively. This might be due to
the superconvergence of finite elements, cf. [29]. Or it might be caused by, in
addition, the special solution (6.2) used in the computation, noting that the
convergence orders for the pressure are not higher. We plot the error for the
second component of u on the level 3 grid in Figure 12, for the Q 3,2 × Q2,3 Q2 element method. The error of p of the method is shown in Figure 13.
In the tables for numerical errors, we use the notations e u = Ih u − uh and
ep = I h p − p h .
When we increase the polynomial degree k by one to 4, this time, both
the exact velocity solution and the exact pressure solution (6.1) are inside the
18
Table 2: The errors by spaces Ṽh × P̃h defined in (4.3)-(4.4), on Figure 11
grids.
level
|eu |H1
2
3
4
5
6
7
1.00989
0.10747
0.01274
0.00157
0.00019
0.00002
2
3
4
5
6
7
0.0147879
0.0007551
0.0000285
0.0000009
0.0000000
0.0000000
hn
|eu |l∞ hn
kep kL2
For the Q3,2 × Q2,3 -Q2 element
0.1231114
11.024
3.2
0.0091183 3.8
2.755
3.1
0.0005780 4.0
0.671
3.0
0.0000369 4.0
0.170
3.0
0.0000023 4.0
0.043
3.0
0.0000001 4.0
0.011
For the Q4,3 × Q3,4 -Q3 element
0.00403687
0.77588
4.3 0.00009680 5.4 0.11326
4.7 0.00000189 5.7 0.01526
4.9 0.00000003 5.9 0.00195
4.9 0.00000000 5.9 0.00024
5.0 0.00000000 6.0 0.00003
hm
kep kl∞
hm
2.0
2.0
2.0
2.0
2.0
2.82776
0.46705
0.08399
0.01257
0.00171
0.00022
2.6
2.5
2.7
2.9
2.9
2.0
2.8
2.9
3.0
3.0
3.0
0.19499
0.01649
0.00122
0.00008
0.00000
0.00000
2.8
3.6
3.7
3.9
3.9
4.0
finite element subspaces, Q5,4 × Q4,5 -Q4 . Then, on any grids, the numerical
solutions should be exact, the same as the exact solution in (6.2), if enough
iterated penalty iterations are done and if there is no round-off error. This
can be seen in Table 3.
Table 3: The errors by the Q5,4 × Q4,5 -Q4 element, on Figure 11 grids.
level
2
3
4
|eu |H1
0.0000106532
0.0000000029
0.0000000000
hn
5.3
11.8
13.8
kep kL2
0.0007548
0.0000005
0.0000000
hm
2.8
10.3
9.5
In Table 4, we list the errors and convergence orders when using the Q 2,1 ×
Q1,2 -Q1 element. A little surprising, the results are of optimal orders too, but
they are not covered by our theory. Further studies are needed to understand
and explain the results in Table 4.
We next test the influence of singular vertex in the Scott-Vogelius P k+1 -Pk
triangular elements. Three families of grids Ω h are used in the computation,
19
( 0.0 0.0)
( 0.0 1.0)
0.94E-02
0.90E-02
0.87E-02
0.83E-02
0.79E-02
0.75E-02
0.71E-02
0.67E-02
0.64E-02
0.60E-02
0.56E-02
0.52E-02
0.48E-02
0.44E-02
0.40E-02
0.37E-02
0.33E-02
0.29E-02
0.25E-02
0.21E-02
0.17E-02
0.13E-02
0.96E-03
0.58E-03
0.19E-03
-0.19E-03
-0.58E-03
-0.96E-03
-0.13E-02
-0.17E-02
-0.21E-02
-0.25E-02
-0.29E-02
-0.33E-02
-0.37E-02
-0.40E-02
-0.44E-02
-0.48E-02
-0.52E-02
-0.56E-02
-0.60E-02
-0.64E-02
-0.67E-02
-0.71E-02
-0.75E-02
-0.79E-02
-0.83E-02
-0.87E-02
-0.90E-02
-0.94E-02
( 1.0 0.0)
( 1.0 1.0)
Figure 12: The error of the second component of u on the level 3 grid (Q 3,2 ×
Q2,3 -Q2 ).
shown Figure 14. The first two families of grids, Figure 14(1) and Figure 14(2),
have a same level 1 grid. But (1) uses the longest-edge bisection refinement,
while (2) uses the standard multigrid refinement. We note that the level n
grid of Figure 14(1) has 4n−1 singular vertices. Every level of grid of Figure
14(2) has one singular vertex, (0.5, 0.5). But none of grids in Figure 14(3) has
any singular vertex.
In Table 5 we list various norms of errors for the finite element solutions.
The H1 and L2 errors and convergence orders on all three types of grids,
reported in Table 5, match the error bound proved in Theorem 5.1.
Table 5 confirms the theory that the convergence of P k+1 -Pk element is
independent of singular and nearly-singular vertices. In fact, the element
with the most singular vertices converges fastest, while the one without any
singular vertex converges slowest. We would emphasize again that the analysis
in this work does not use the inf-sup condition (1.1), while nearly all other
work on mixed finite elements for the Stokes equations are based on the infsup condition. Here we compute the inf-sup constant γ in (1.1), on the grids
shown in Figure 14(3) and Figure 1. We let the interior point C move toward
the center point A in Figure 1, becoming a singular vertex. We list the inf-sup
20
( 0.0 0.0)
( 0.0 1.0)
0.43E+00
0.39E+00
0.37E+00
0.35E+00
0.33E+00
0.31E+00
0.30E+00
0.28E+00
0.26E+00
0.24E+00
0.22E+00
0.20E+00
0.18E+00
0.16E+00
0.14E+00
0.12E+00
0.10E+00
0.83E-01
0.63E-01
0.44E-01
0.25E-01
0.56E-02
-0.14E-01
-0.33E-01
-0.52E-01
-0.72E-01
-0.91E-01
-0.11E+00
-0.13E+00
-0.15E+00
-0.17E+00
-0.19E+00
-0.21E+00
-0.23E+00
-0.25E+00
-0.26E+00
-0.28E+00
-0.30E+00
-0.32E+00
-0.34E+00
-0.36E+00
-0.38E+00
-0.40E+00
-0.42E+00
-0.44E+00
-0.46E+00
-0.48E+00
-0.50E+00
-0.52E+00
( 1.0 0.0)
( 1.0 1.0)
Figure 13: The error of p on the level 3 grid (Q 3,2 × Q2,3 -Q2 ).
constant in the second column of Table 6. In the table, A is the matrix for
the a(·, ·) inner product, B for b(·, ·), and M for the (·, ·) on P h . We can also
find the next smallest eigenvalue in the last column of Table 6.
Finally we do a 3D test on the Pk+1 -Pk elements for the stationary Stokes
equations on the unit cube, Ω = (0, 1) 3 . The grids are obtained by the standard
multigrid refinement, cf. [31]. The first three grids are depicted in Figure 15.
Table 4: The errors by the Q2,1 × Q1,2 -Q1 element, on Figure 11 grids.
level
3
4
5
6
7
|eu |H1
1.2110
0.3119
0.0783
0.0196
0.0049
hn
2.0
2.0
2.0
2.0
|eu |l∞
0.19947
0.04576
0.01100
0.00270
0.00067
hn
2.1
2.1
2.0
2.0
21
kep kL2
25.9107
10.5018
4.9049
2.4052
1.1964
hm
1.3
1.1
1.0
1.0
kep kl∞
6.6670
1.4463
0.3727
0.0927
0.0234
hm
2.2
2.0
2.0
2.0
(1)
(2)
(3)
Figure 14: Three computational grids with (1) many, (2) only one, (3) none,
singular vertex.
Figure 15: The first three levels of grids in the 3D test, Ω h .
The right hand side function f in 3D for (4.1) is
 
0
1
1

f = −∆ curl g  + ∇gxy
3
9
g


−g
− gyyy − gyzz + gxxz + gyyz + gzzz + gxxy /3
1  xxy
,
−gxxx − gxyy − gxzz + gxyy /3
=
3
gxxx + gxyy + gxzz + gxyz /3
where
g = 212 (x − x2 )2 (y − y 2 )2 (z − z 2 )2 .
22
(6.3)
Table 5: The P4 element on the three families of grids (Figure 14).
level
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
|eu |H1
hn
|eu |l∞
hn
kep kL2
hk
kep kl∞
n−1
On the grids with 4
singular vertices, Figure 14(1)
0.536873
0.0464484
1.39824
11.2370
0.044922 3.58 0.0026137 4.15 0.10879 3.68
1.5131
0.002583 4.12 0.0001097 4.57 0.00591 4.20
0.1269
0.000156 4.05 0.0000039 4.80 0.00034 4.10
0.0090
0.000009 4.01 0.0000001 4.90 0.00002 4.04
0.0006
On the grids with one singular vertex, Figure 14(2)
0.535961
0.0463440
1.39626
11.2232
0.046138 3.54 0.0025915 4.16 0.13677 3.35
1.5252
0.002822 4.03 0.0001103 4.55 0.00978 3.81
0.1281
0.000180 3.97 0.0000048 4.52 0.00067 3.85
0.0115
0.000011 3.99 0.0000002 4.60 0.00004 3.97
0.0009
On the grids with no singular vertex, Figure 14(3)
0.656464
0.0828363
1.77368
15.6980
0.078603 3.06 0.0071213 3.54 0.26038 2.77
3.0422
0.004891 4.01 0.0003334 4.42 0.01690 3.95
0.2719
0.000311 3.97 0.0000125 4.74 0.00109 3.95
0.0293
0.000019 3.99 0.0000005 4.64 0.00006 4.01
0.0023
The continuous solution for the Stokes equations (4.1) is
 
0
1
1
u = curl g  , p = gxy .
3
9
g
hk
2.89
3.58
3.81
3.91
2.88
3.57
3.48
3.60
2.37
3.48
3.21
3.63
(6.4)
As we are unable to plot a 3D function in 4D, we show the restriction of the
functions u (the first component) and p, on the plane z = 0.33 in Figure 17.
We note that the grids obtained by the intersection of tetrahedra in Ω h and
the plane consist of both rectangles and triangles, shown in Figure 16 and at
the bottom in Figures 17 and 18.
In Table 7 we list errors for the Pk+1 -Pk element for k = 0, 1, on various
level of grids Ωh . It is interesting to see that no convergence for both velocity
and pressure for the P1 -P0 element, and that a convergence for the velocity
only for the P2 -P1 element.
In Table 8 we list errors for the Pk+1 -Pk element for k = 2, 3, 4. We note
that our theory ensures the convergence only for k ≥ 8. But for the uniform
23
Table 6: The inf-sup constants for the P 4 -P3 element on grids in Figure 1.
C→A
(0.50, 0.50)
(0.50, 0.51)
(0.50, 0.52)
(0.50, 0.54)
(0.50, 0.58)
(0.50, 0.66)
(0.50, 0.74)
(0.50, 0.82)
(0.50, 0.90)
(0.50, 0.94)
(0.50, 0.98)
γ=
p
λ1 (M −1 B T A−1 B)
Singular! 0.00000
0.00613
0.01226
0.02453
0.04907
0.09786
0.14457
0.18405
0.20582
0.20346
0.16764
p
λ2 (M −1 B T A−1 B)
0.41990
0.41869
0.41603
0.40953
0.39586
0.36959
0.34539
0.32317
0.30140
0.25604
0.17348
grids on the cube used in this computation, the polynomial degree k could be
much lower.
7
Some low-order divergence-free finite elements
The Pk+1 -Pk finite element is stable and provides the optimal order solution
for k ≥ 3 on general triangular grids. But for special grids, the polynomial degree can be lower. In [20], Qin showed that the P 2 -P1 and P3 -P2 elements are
also stable on Hsieh-Clough-Tocher triangular grids, cf. [12]. Given a trian-
Figure 16: The cut on the third level grid Ω h by plane z = 0.33.
24
Figure 17: The exact solution, the first component of u and p in (6.4), restricted on z = 0.33.
Figure 18: The errors for the first component of u and p for the P 5 -P4 element
on the level 2 grid, restricted on z = 0.33.
gulation, the Hsieh-Clough-Tocher triangular grid is generated by connecting
three vertices to the bary-center of each triangle, shown in Figure 19.
→
→
Figure 19: Constructing Hsieh-Clough-Tocher triangular grids.
For constructing 3D Hsieh-Clough-Tocher grids, each tetrahedron is subdivided into four by connecting the bary-center with 4 vertices, shown in Figure
20.
It can be shown that on the Hsieh-Clough-Tocher grids in 2D and 3D, cf
[20, 32], the divergence of the C0 -Pk+1 function space is precisely the C−1 Pk space modulus constants, for any k ≥ 0. However, when the degree of
polynomial k is too low, we do not have approximation properties for the
finite element spaces. k ≥ 1 and k ≥ 2 in 2D and 3D, on Hsieh-Clough-Tocher
grids, are required, respectively, shown by Qin and Zhang [20, 32]. Of course,
25
Table 7: The errors for the Pk+1 -Pk (k = 0, 1) element on Figure 14 grids.
|eu |H1
hn
|eu |l∞
hn
kep kL2
hk
kep kl∞ hk
The P1 -P0 element
3 4.15901
1.12500
10.70967
29.91138
4 4.76608
– 1.26172
– 26.33037
–
91.85602
–
5 4.77452
– 1.25096
– 58.69458
– 225.61878
–
The P2 -P1 element
3 2.75202
0.44361
17.46410
97.76562
4 1.35240 1.02 0.13309 1.73 16.41843 0.08 133.52599
–
5 0.64181 1.07 0.03400 1.96 14.53410 0.17 138.50007
–
x3
c
d
e
b
x2
a
x1
Figure 20: Constructing the 3D Hsieh-Clough-Tocher grids.
when k is larger (k ≥ 3 in 2D, k ≥ 7 in 3D), the theory on general grids would
guarantee the optimal-order of convergence for the element.
For the lowest polynomial degree k = 0, Qin and Zhang found a local basis
for the C0 -P1 divergence-free space on the uniform criss-cross grids shown in
Figure 21. Consequently, it shows that the P 1 -P0 divergence-free element is
stable and of optimal order. Such an analysis can be extended to 3D, on the
uniform criss-cross grids shown in Figure 22.
On another type of grids, the polynomial degree k can be 0 as well. It is
shown in [33] that the P1 -P0 divergence-free element converges with the optimal order too, on the 2D Powell-Sabin triangulations ([19]). Similar to the
Hsieh-Clough-Tocher grids, a 2D Powell-Sabin grid is constructed by subdividing each triangle of a regular triangulation into 6 subtriangles, shown in
Figure 23. An appropriate internal point is chosen on each triangle, which
is connected to the three vertices of the triangle and the three neighboring
26
Table 8: The errors for the Pk+1 -Pk element on Figure 14 grids.
|eu |H1 hn
|eu |l∞ hn
kep kL2 hk
kep kl∞ hk
The P3 -P2 element
1 1.27852
0.20145
0.25618
0.49327
2 2.00055
– 0.24271
– 7.51700
– 65.94546
–
4 0.09805 2.4 0.00466 3.4 1.25564 1.4 15.67054 1.6
5 0.01424 2.8 0.00037 3.6 0.73651 0.7
4.44028 1.8
The P4 -P3 element
1 1.55317
0.21257
0.36283
1.00000
– 55.39757
–
2 0.85394 0.8 0.07851 1.4 3.36404
3 0.09648 3.1 0.00687 3.5 0.85591 1.9 10.30833 2.4
4 0.00661 3.8 0.00038 4.1 0.69055 0.3
2.30741 2.1
5 0.00044 3.9 0.00002 4.4 0.69039 0.0
2.10600 0.1
The P5 -P4 element
1 1.11184
0.13294
0.39278
0.96637
2 0.33257 1.7 0.02801 2.2 1.33425
– 33.37696
–
3 0.01510 4.5 0.00110 4.7 0.69419 0.9
3.36344 3.3
4 0.00052 4.9 0.00003 5.3 0.69075 0.0
2.09780 0.6
Figure 21: The uniform 2D criss-cross grids.
internal points.
A numerical study on the P1 -P0 divergence-free was done in [33]. The
numerical results for solving the problem (6.1) are listed in Table 9, showing
the optimal order of convergence of the method.
The analysis in [33] on the P1 -P0 divergence-free element works for the 2D
Powell-Sabin-Heindl triangulations ([10, 14]) as well. A Powell-Sabin-Heindl
triangulation refines every triangle into 12 subtriangles, shown in Figure 24.
Similar to the 2D case [33], we can show the convergence of the P 1 -P0
element on 3D Powell-Sabin grids (Figure 25).
Acknowledgments. This work was initially supported by the National Sci-
27
Figure 22: The uniform 3D criss-cross grids.
→
→
Figure 23: Constructing Powell-Sabin triangular grids.
ence Foundation Award 9625907.
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28
Table 9: The Powell-Sabin divergence-free C 0 -P1 element for (6.1).
Level
2
3
4
5
6
7
# triangles
48
192
768
3072
12288
49152
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32