-
A P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE
FINITE ELEMENT ON TRIANGULAR GRIDS
SHANGYOU ZHANG
DEDICATED TO PROFESSOR PETER MONK ON THE OCCASION OF HIS
60THBIRTHDAY
Abstract. On triangular grids, the continuous Pk plus
discontinuousPk−1 mixed finite element is stable for polynomial
degree k ≥ 4. Whenk = 3, the inf-sup condition fails and the mixed
finite element convergesat an order that is two orders lower than
the optimal order. We enrichthe continuous P3 by adding some P4
divergence-free bubble functions,to be exact, one P4
divergence-free bubble function each componenteach edge. We show
that such an enriched P3-P2 mixed element is inf-sup stable, and
converges at the optimal order. Numerical tests arepresented,
comparing the new element with the P4-P3 element and theunstable
P3-P2 element.
AMS subject classifications. 65M60, 65N30, 76M10, 76D07.
Keywords. finite element, divergence-free element, Stokes
equations,triangular grid.
1. Introduction
It is a challenge to construct stable H1 conforming mixed finite
elementssatisfying the incompressible condition exactly, in
computing the Stokes orNavier-Stokes equations. That is, the
velocity is approximated by the con-tinuous piecewise polynomials
of degree k and the pressure is approximatedby the discontinuous
piecewise polynomials of one degree less. Here themethod is truly
conforming in the sense that the finite element velocity isthe H1
projection of the true solution in a polynomial subspace. A
break-through on the method was done by Scott and Vogelius in 1985
[18, 19] thatthe method is stable and consequently of the optimal
order of convergenceon 2D triangular grids, for the Pk-Pk−1
element, if k ≥ 4, a magic number.What is this magic number k in
3D? Or if there is such a magic number in3D? The problem remains
open, after it was posted explicitly for so manyyears [18].
Scott and Vogelius showed that the Pk-Pk−1 element is not stable
ongeneral triangular grids, if k < 4. However, on special
triangular grids,
1
-
low order elements may be stable. On Hsieh-Clough-Tocher
macro-elementgrids, where each base triangle is split into 3
triangles by connecting thebarycenter with three vertices, the
P2-P1 and the P3-P2 mixed elements arestable, cf. [1, 17, 22]. On
Powell-Sabin macro-element grids, where eachtriangle is split into
6 sub-triangles, even the P1-P0(subspace) element isstable [26].
When enriching the continuous Pk velocity space by some ra-tional
functions, Guzman and Neilan showed the enriched Pk-Pk−1 elementis
table for all k ≥ 1, [9, 10]. With additional continuity
constraints, Falkand Neilan showed that the Pk-Pk−1 element is
stable if the continuous Pkvelocity is also C1 at vertices and the
discontinuous Pk−1 pressure is C0 atvertices, cf. [8].
In 3D, the Pk-Pk−1 mixed element is stable for all k ≥ 3 on the
Hsieh-Clough-Tocher macro-element tetrahedral grids (where each
base tetrahe-dron is split into 4 sub-tetrahedra by connecting the
barycenter with fourvertices), cf. [25]. If splitting further a
base tetrahedron into 12 sub-tetrahedra(connecting the barycenter
with 4 vertices and 4 face-triangle barycenters),the Pk-Pk−1 is
stable for all k ≥ 2, cf. [29]. On the uniform tetrahedralgrids,
i.e., each cube is subdivided into 6 tetrahedra, the continuous Pk
withdiscontinuous Pk−1 mixed finite element is stable for all k ≥
6, cf. [28]. Withadditional constraints on the finite element
spaces, Neilan showed that thePk-Pk−1 element is stable for k ≥ 6
on general tetrahedral grids if the con-tinuous velocity finite
element is C2 continuous at all vertices and also C1continuous on
all edges, and the discontinuous pressure finite element func-tion
is C1 at all vertices and C0 on all edges, cf. [16]. But the
Scott-Vogeliusproblem is still open, on general tetrahedral grids.
On rectangular grids,this problem is simple that Qk,k−1
×Qk−1,k-Qk−1 element (and its nD ver-sion) is stable for all k ≥ 2,
where Qk,k−1 denotes the continuous piecewisepolynomials of
separated degrees k and k− 1 in its first variable and
secondvarialbe, respectively, cf. [14, 15, 27].
The mixed finite element of continuous P3 velocity and
discotinuous P2pressure is not stable on general triangular grids
in 2D, cf. [18, 19]. Inthis work, we enrich the continuous P3 space
by some divergence-free P4bubble functions. Such P4 bubble
functions do not provide additional ap-proximation power, but do
provide additional degrees of freedom to relaxthe locking problem
of the divergence-free constraint. Such a finite elementenrichment
technique is used before, many times. For example, mentionedabove,
Guzman and Neilan enrich the continuous P3 velocity by some
ratio-nal bubble functions to obtain an inf-sup stable mixed finite
element [9, 10].Here, instead of rational functions (whose
numerical integration formula areunknown) we use the P4 bubble
polynomials in this work. In the low ordermixed finite element
methods for the linear elasticity equation, the H(div)Pk finite
element space must be enriched by Pd+1 divergence-free
bubblefunctions in d-dimensional space, cf. [2, 3, 11, 12, 13].
2
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2. The enriched P3 divergence-free element
In this section, we define the P4-enriched P3 divergence-free
finite element.Its uni-solvence is shown.
We consider a model stationary Stokes problem: Find the velocity
u andthe pressure p on a 2D polygonal domain Ω, such that
(2.1)
−∆u+∇p = f in Ω,divu = 0 in Ω,
u = 0 on ∂Ω.
The standard variational form for (2.1) is: Find u ∈ H10 (Ω)2
and p ∈L20(Ω) := L
2(Ω)/C = {p ∈ L2 |∫Ω p = 0} such that
(2.2)a(u,v) + b(v, p) = (f ,v) ∀v ∈ H10 (Ω)2,
b(u, q) = 0 ∀q ∈ L20(Ω).
Here H10 (Ω)2 is the subspace of the Sobolev space H1(Ω)2 (cf.
[7]) with zero
boundary trace, and the blinear forms are defined by
a(u,v) =
∫Ω∇u · ∇v dx,
b(v, p) = −∫Ωdivv p dx,
(f ,v) =
∫Ωf v dx.
Let Th be an initial triangulation of Ω. We refine each triangle
into fourcongruent triangles by connecting the three mid-edge
points. This way, onegrid is refined to the next level grid. We
denote each triangulation in thesequence of grids also by Th, where
h is the grid size. Here we introducethe multigrids [23, 24],
instead of general quasi-uniform grids, to avoid thetechnical
details of nearly-singular points. For an initial triangulation,
wemay have a few singular points [4, 18]. Here a singular point is
a point atwhich all edges of an triangulation fall into two
crossing lines at the point.There are exactly four types of
singular points, three boundary ones and oneinternal one, shown in
Figure 2.1. There is a minor constraint for the discretepressure
functions at the singular points. However, all singular points of
theinitial grid will stay singular and no new singular points
appear, after themultigrid refinement.
Let the P4-enriched P3 velocity space be, for K ∈ Th,
VK = {v ∈ P4(K)2 | divv ∈ P2}.(2.3)
Here Pk stands for the space of polynomials of degree k or less.
We notethat, for a P 24 vector, the divergence is a P3 polynomial,
not a P2 polynomial.That is, under the constraint, the x3, x2y, xy2
and y3 coefficients of thepolynomial of the divergence must be
zero. With these four constraints, we
3
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BBBB
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s
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Figure 2.1. There boundary singular points (left three)and an
internal singular point.
would expect the dimension of VK be
2 dimP4 − 4 = 26 = 2dimP3 + 6.This is to be proved in Lemma 2.1.
But we would like to give another(equivalent) definition of VK . As
the range of the divergence operator onP 23 space is P2 already,
the newly added P
24 functions would be divergence-
free. That is, the above 6 dimension space is spanned by the
following 6divergence-free P 24 polynomials:
curlx5, curlx4y, curlx3y2 =
(2x3y
−3x2y2), curlx2y3, curlxy4, curly5.
Also there, in Lemma 2.1, the 26 degrees of freedom in VK are
given byv(xi), three vertex values of two components,∫eiv · xj ds,
0-th, 1st, 2nd moments on three edges,∫
K v dx, 0-th moment on the element.
(2.4)
The mixed element spaces are defined by, also equivalently by
(2.3) and(2.12),
Vh ={vh ∈ C(Ω)2 | vh|K ∈ VK ∀K ∈ Th, and vh|∂Ω = 0
},(2.5)
Ph = {divvh | vh ∈ Vh} .(2.6)
Since∫Ω ph =
∫Ω divvh =
∫∂Ω uh = 0 for any ph ∈ Ph, we conclude that
Vh ⊂ H10 (Ω)2, Ph ⊂ L20(Ω),i.e., the mixed-finite element pair
is conforming.
The resulting system of finite element equations for (2.2) is:
Find uh ∈ Vhand ph ∈ Ph such that
(2.7)a(uh,v) + b(v, ph) = (f ,v) ∀v ∈ Vh,
b(uh, q) = 0 ∀q ∈ Ph.Traditional mixed-finite elements require
the inf-sup condition to guar-
antee the existence of discrete solutions. As (2.6) provides a
compatibilitybetween the discrete velocity and discrete pressure
spaces, the linear systemof equations (2.7) always has a unique
solution, independent of the inf-supcondition.
4
-
Proposition 2.1. There is a unique solution in the discrete
linear system(2.7).
Proof. As (2.7) is a square system, the uniqueness implies
existence. Let(uh, ph) be a solution for the homogeneous equations
(2.7). Let v = uh andq = ph in (2.7). We have
a(uh,uh) + b(uh, ph) = 0,
b(uh, ph) = 0.
Subtracting the second equation from the first one, we get
a(uh,uh) = 0 ⇒ uh = 0.
The first equation of (2.7) is now
b(v, ph) = 0, forall v ∈ Vh.(2.8)
By (2.6), we have some wh ∈ Vh such that ph = divwh. Let v = wh
in(2.8). It is then ∥divwh∥2L2 = 0. So ph = divwh = 0.
Further, by the second equation in (2.7) and the definition of
Ph in (2.6),we conclude that
(2.9) b(uh, q) = b(uh,−divuh) = ∥divuh∥2L2(Ω)2 = 0
and that
divuh = 0,
i.e. uh is divergence-free. In this case, we call the mixed
finite element adivergence-free element. It is apparent that the
discrete velocity solution isdivergence-free if and only if the
discrete pressure finite element space is thedivergence of the
discrete velocity finite element space, i.e., (2.6). In fact,it is
trivial to show ([5, 6, 26]), in the next theorem, that uh is the
uniquea(·, ·) orthogonal projection from the divergence-free space
Z to its subspaceZh, defined by
Z :={v ∈ H10 (Ω)2 | divv = 0
},(2.10)
Zh := {v ∈ Vh | divv = 0} .(2.11)
That is,
uh ∈ Zh, a(u− uh,vh) = 0 ∀vh ∈ Zh.
We note that the definition of Ph is abstract, which may not be
goodenough for computation (the basis of functions of Ph is
unknown.) But asthe pressure space is defined implicitly by the
velocity space, the unknownpressure solution can be implicitly
defined by a function in the velocity space,by an iterative method,
cf. [27]. Indeed, such a computation saves half ofcoding work on
the pressure finite element and one-third unknowns in theresulting
linear system of equations. But we do give another,
traditionaldefinition of the pressure space Ph below. By (2.6), Ph
may not be the full
5
-
space of discontinuous P2 polynomials, but a proper subspace if
singularvertices are present. The next definition describes Ph
precisely.
Ph = {ph ∈ L20(Ω) | ph|K ∈ P2 ∀K ∈ Th,(2.12)i0∑i=1
(−1)ivh|Ki(s) = 0 at a singular vertex s },
where s is one of the four types of vertexes depicted in Figure
2.1, and Kiare the i0 (= 1, 2, 3, 4) triangles around the singular
vertex s.
Lemma 2.1. The dimension of VK in (2.3) is 26. The 26 degrees of
freedomare listed in (2.4).
Proof. dimP 24 = 30. Adding the four constraints of P3
coefficients of thedivergence to the 26 degrees of freedom, we have
a square system of linearequations. The existence of the solution
is implied by the uniqueness, whichwill be proved next.
The proof is done in two steps, on the reference triangle K̂ and
on thegeneral triangle K. Let the reference triangle be
K̂ = {x̂ ≥ 0, ŷ ≥ 0, x̂+ ŷ ≤ 1}.(2.13)
With all 26 dof in (2.4) of v̂h be zero, v̂h = 0 on the boundary
of K̂, as eachcomponent of v̂|êi is a degree 4 polynomial with 5
zeros. Thus
v̂h = x̂ŷ(1− x̂− ŷ)(c1 + c2x̂+ c3ŷc4 + c5x̂+ c6ŷ
),(2.14)
for some constants c1, ... c6. We show these constants are all
zero. The x̂3
and ŷ3 coefficients of d̂ivv̂h are −c5 and −c3, respectively.
Thus c3 = c5 = 0.The x̂2y coefficient of d̂ivv̂h is−3c2−2c5−2c6 =
−3c2−2c6 = 0. Similarly, bychecking the x̂ŷ2 coefficient of
d̂ivv̂h, we have −2c2−2c3−3c6 = −2c2−3c6 =0. Together, we get c2 =
c6 = 0. Thus
v̂h = x̂ŷ(1− x̂− ŷ)(c1c4
).
Because the bubble function x̂ŷ(1 − x̂ − ŷ) is positive inside
K̂, by the 0thmoment of v̂h on K̂ in (2.4), c1 = c4 = 0. Thus v̂h =
0.
Now, we show the uniqueness on a general triangle K ∈ Th. Let FK
bean affine mapping from K̂ to K, cf. Figure 2.2, such that
FK(x̂, ŷ) = x0 +Bx̂ =
(x0y0
)+(
⃗x0x1 ⃗x0x2)(x̂
ŷ
).(2.15)
Now, if the 26 dof’s of function vh have value 0, then vh is
identically zeroon the three edges of K:
vh = λ1λ2λ3
(p(1)1
p(2)1
),
6
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K̂ :
@@@
@@@@
@x̂0x̂1
x̂2-FK = Bx̂+ x0
K :
ccc
cc�
������������
��
��
��
�x2
x0
x1
Figure 2.2. An affine mapping FK from the reference tri-angle K̂
to a general triangle K.
where λi are three area-coordinates on K, and p(i)1 are two P1
polynomials.
We define a Piola transform by
v̂h(x̂) = B−1vh(FK(x̂)),(2.16)
where B is defined in (2.15). Because vh is identically zero on
the boundaryof K, so is v̂h. On the other side, if vh ̸= 0 on the
boundary of K, we cannotuse the Piola transformation as it would
destroy the tangential continuityof H1 functions. The v̂h defined
in (2.16) can also be expressed as (2.14).Since the Piola transform
preserves the divergence,
divvh = traceB−T (∇̂Bv̂h(x̂))T = trace (∇̂v̂h(x̂))T =
d̂ivv̂h,
d̂ivv̂h is also a P2 function. Further, as B is invertible, two
linear combina-tions of 2 zero 0-moment component of vh have also
zero 0-moment. Thus,by the analysis on v̂h above, v̂h = 0. So is vh
= Bv̂h(F
−1K (x)) = 0.
3. Stability and convergence
In this section, we will prove the on-to mapping property of the
divergenceoperator, from the discrete velocity space to the finite
element pressurespace. Consequently we prove the inf-sup stability
condition and the optimalorder of convergence for the finite
element solution.
Remark 3.1. The inf-sup condition (3.11), i,e, Theorem 3.1, can
be provedas a corollary of Soctt-Vogelius’ Theorem 5.1 [18]. But we
give an inde-pendent proof. The difference between the proof here
and the Scott-Vogelius’proof is in the construction of uh in next
lemma, Lemma 3.1. We use bubble-enriched P3 polynomials
(equivalently P4 polynomials) to construct uh whileScott and
Vogelius used only P3 polynomials, cf. [21, Lemma 2.3] and
[18,Lemma 4.1]. Thus the Scott-Vogelius result requires the grid
size h suffi-ciently small, [18, Remark 5.1, Lemma 5.1, and (5.6)].
But the theory heredoes not have any restriction on h.
7
-
Lemma 3.1. There is a uh ∈ Vh, supported on two triangles K,K ′,
cf. Fig-ure 3.1, such that divuh has a nodal value 1 at x1 on the K
side, and nodalvalue 0 at the rest 11 P2 Lagrange nodes on the two
triangles, as long as thetwo edges x0x1 and x1x0,K′ do not fall
into a same line.
K̂
@@
@@
@@
@@
b bbb
bbbb
bbb K̂ ′
�
K
cc
ccc�
����������
������
��
AAAAAAAA
��
��
��
��
�x0
x1 = x2,K′
x2 = x1,K′
x0,K′
K ′bb
bb
b-
divuh = 1b
bb
bb
b
Figure 3.1. C−1-P2 Lagrange nodes on two neighboringtriangles
and the reference mapping.
Proof. On the reference triangle K̂ in Figure 3.1, we find all
such vectorsû01, which vanish on the boundary of (K̂ ∪ K̂ ′) and
whose divergence is theP2 polynomial having value 1 at vertex x̂1
only.
d̂ivû01(x̂, ŷ) = 2x̂2 − x̂, on K̂.
Note that there are precisely three divergence-zero vectors.
They are thecurl of functions
x̂3ŷ2, x̂2ŷ2 and x̂2ŷ3.
So, on K̂,
û01(x̂, ŷ) = x̂ŷ
((0
2x̂− 1
)+ c1
(2x̂2
−3x̂ŷ
)+c2
(2x̂−2ŷ
)+ c3
(3x̂ŷ−2ŷ2
)).
(3.1)
On the other reference triangle K̂ ′ in Figure 3.1, we need
d̂ivû01 = 0. Sothe vector must be a linear combination of the
curls of, on K̂ ′,
(1− x̂)2(1− ŷ)3, (1− x̂)2(1− ŷ)2 and (1− x̂)3(1− ŷ)2.
û01(x̂, ŷ) = (1− x̂)(1− ŷ)(c4
(−3(1− x̂)(1− ŷ)
2(1− ŷ)2)
+c5
(−2(1− x̂)2(1− ŷ)
)+ c6
(−2(1− x̂)2
3(1− x̂)(1− ŷ)
)).
(3.2)
8
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We map these two vector functions to triangles K and K ′,
preserving thedivergence. Because û01 = 0 on the four outside
edges of triangles K̂ andK̂ ′, after the Piola transformation, the
piecewise P4 vector function remainszero on the four outside edges
of triangles K and K ′. Then we only matchthe interface values of
two P4 vector functions at the common edge x1x2.Each P4 function
has 3 internal Lagrange nodal-values on an edge. We endup with 6
equations for the matching on x1x2. Though we have 6 degreesof
freedom in (3.1) and (3.2), the system does not have a unique
solution,as the curl of the C1-P5 Argyris normal derivative basis
at the mid-point ofx1x2 is a solution of the homogeneous system. We
have either no solution,if x0x1 and x1x0,K′ are colinear, or
infinitely many solutions, if the two linesare different. We will
show the detail next.
The reference mapping from K̂ to K is defined in (2.15). On K,
u01 isdefined by the Piola transformation,
u01(x) = Bû01(F−1K (x)).(3.3)
The reference mapping from K̂ ′ to K ′ is(xy
)= FK′(x̂, ŷ) = x0,K′ +B
′(1− x̂1− ŷ
),
where B′ =(
⃗x0,K′x1,K′ ⃗x0,K′x2,K′). In order to keep the divergence,
sim-
ilar to the Piola transformation (3.3), we let, on K ′,
u01(x, y) = −B′û0,1(F−1K′ (x)).(3.4)
When restricted on the common edge x1x2, i.e., ŷ = 1− x̂, we
have
B
(2x̂2c1 + 2x̂c2 + 3(x̂− x̂2)c3
2x̂− 1 + 3(−x̂+ x̂2)c1 + 2(−1 + x̂)c2 + (−2 + 4x̂− 2x̂2)c3
)= −B′
(1
1
)(2x̂2c4 + 2x̂c5 + 3(x̂− x̂2)c6
3(−x̂+ x̂2)c4 + 2(−1 + x̂)c5 + (−2 + 4x̂− 2x̂2)c6
).
Matching coefficients of 1, x̂ and x̂2 of the two components, we
have thefollowing 6× 6 system of equations, in block matrix
form,
(0 0 00 −2 −2
)A
(0 0 00 −2 −2
)(
0 2 3−3 2 4
)A
(0 2 3−3 2 4
)(2 0 −33 0 −2
)A
(2 0 −33 0 −2
)
c1c2c3c4c5c6
=
010−200
,9
-
where the common matrix A is defined in (3.5).
A = B−1B′(
11
)=(
⃗x0x1 ⃗x0x2)−1 (
⃗x0,K′x1,K′ ⃗x0,K′x2,K′)( 1
1
)=(
⃗x0x1 ⃗x0x2)−1 (
⃗x0,K′x2 ⃗x0,K′x1)( 1
1
)=(
⃗x0x1 ⃗x0x2)−1 ((
⃗x0x1 ⃗x0x2)+(
⃗x0,K′x0 ⃗x0,K′x0))
=
(1
1
)+(B−1 ⃗x0,K′x0 B
−1 ⃗x0,K′x0)
=
(1 + z1 z1z2 1 + z2
).
(3.5)
Here
(z1z2
)= B−1 ⃗x0,K′x0. By adding the first block and the third block
to
the second block in the linear system, we have a simplified
linear system
0 0 0 0 −2z1 −2z10 −2 −2 0 −2− 2z2 −2− 2z22 2 0 2 + 2z1 2 + 2z1
00 0 0 2z2 2z2 02 0 −3 2 + 5z1 0 −3− 5z13 0 −2 3 + 5z2 0 −2−
5z2
c1c2c3c4c5c6
=
010−100
.(3.6)
There are two cases, z2 = 0 or z2 ̸= 0. z2 = 0 if and only if
⃗x0,K′x0 = c ⃗x0x1(i.e. x0,K′ is on the straight line x0x1.) If z2
= 0, by the fourth equation in(3.6), there is no solution.
When z2 ̸= 0, we let c6 = 0 (or a new constant). By the first
equation,if z1 ̸= 0, c5 = 0. But if z1 = 0, i.e., x0,K′ is on the
straight line x0x2, wehave another degree of freedom and we also
let it be zero, i.e., c5 = 0. Bythe fourth equation in (3.6), c4 =
−1/(2z2). By this time, the system (3.6)
10
-
is reduced to 0 −2 −2 | 12 2 0 | (1 + z1)/z22 0 −3 | (2 +
5z1)/(2z2)3 0 −2 | (3 + 5z2)/(2z2)
R1+R2−→
0 −2 −2 | 12 0 −2 | (1 + z1 + z2)/z22 0 −3 | (2 + 5z1)/(2z2)3 0
−2 | (3 + 5z2)/(2z2)
−R2+R3,(−3/2)R2+R4−→
0 −2 −2 | 12 0 −2 | (1 + z1 + z2)/z20 0 −1 | (3z1 − 2z2)/(2z2)0
0 1 | (−3z1 + 2z2)/(2z2)
R4+R3,2R4+R2,2R4+R1−→
0 −2 0 | (−3z1 + 3z2)/z22 0 0 | (1− 2z1 + 3z2)/z20 0 0 | 00 0 1
| (−3z1 + 2z2)/(2z2)
.Thus, we find a solution
c3 =−3z12z2
+ 1
c2 =3z12z2
− 32
c1 =1− 2z12z2
+3
2.
Lemma 3.2. There is a uh ∈ Vh, supported on two triangles K,K ′,
cf. Fig-ure 3.2, such that divuh has a nodal value 1 at x1 on both
K and K
′, andnodal value 0 at the rest 10 P2 Lagrange nodes on the two
triangles whenthe two edges x0x1 and x1x0,K′ fall into a same
line.
Proof. The proof repeats that for Lemma 3.1. We still give the
details.On the reference triangle K̂ in Figure 3.2, we find all
such vectors ûh,
which vanish on the boundary of (K̂ ∪ K̂ ′) and whose divergence
is the P2polynomial having value 1 at vertex x̂1 only.
d̂ivûh(x̂, ŷ) = 2x̂2 − x̂, on K̂.
Note that there are precisely three divergence-zero vectors.
They are thecurl of functions
x̂3ŷ2, x̂2ŷ2 and x̂2ŷ3.
11
-
K̂
@@
@@
@@
@@
b bbb
bbb
bbb K̂ ′ -
FK , FK′
K
����
����
��������CCCCCCCCx0
x1 = x2,K′
x2 = x1,K′
x0,K′
K ′b bb b b b
b b
�������
divuh = 1
Figure 3.2. C−1-P2 Lagrange nodes on two neighboringtriangles
and the reference mapping.
So, on K̂,
ûh(x̂, ŷ) = x̂ŷ
((0
2x̂− 1
)+ c1
(2x̂2
−3x̂ŷ
)+c2
(2x̂−2ŷ
)+ c3
(3x̂ŷ−2ŷ2
)).
(3.7)
On the other reference triangle K̂ ′ in Figure 3.2, similarly,
we have
ûh(x̂, ŷ) = (1− x̂)(1− ŷ)((
2ŷ − 10
)+ c4
(−3(1− x̂)(1− ŷ)
2(1− ŷ)2)
+c5
(−2(1− x̂)2(1− ŷ)
)+ c6
(−2(1− x̂)2
3(1− x̂)(1− ŷ)
)).
(3.8)
We map these two vector functions to triangles K and K ′,
preserving thedivergence, noting that they vanish on the outside of
two edges of the trian-gles where they are defined. Then we match
the interface values of two P4vector functions at the common edge
x1x2. The reference mapping from K̂to K is defined in (2.15). In
order to keep the divergence, we define the twoPiola
transformations in (3.3) and (3.4). When restricted on the
commonedge x1x2, i.e., ŷ = 1− x̂, we have
B
(2x̂2c1 + 2x̂c2 + 3(x̂− x̂2)c3
2x̂− 1 + 3(−x̂+ x̂2)c1 + 2(−1 + x̂)c2 + (−2 + 4x̂− 2x̂2)c3
)= −B′
(1
1
)(2x̂2c4 + 2x̂c5 + 3(x̂− x̂2)c6
1− 2x̂+ 3(−x̂+ x̂2)c4 + 2(−1 + x̂)c5 + (−2 + 4x̂− 2x̂2)c6
).
12
-
Matching coefficients of 1, x̂ and x̂2 of the two components, we
have thefollowing 6× 6 system of equations, in block matrix
form,
(0 0 00 −2 −2
)A
(0 0 00 −2 −2
)(
0 2 3−3 2 4
)A
(0 2 3−3 2 4
)(2 0 −33 0 −2
)A
(2 0 −33 0 −2
)
c1c2c3c4c5c6
=
(01
)−A
(01
)(
0−2
)−A
(0−2
)(00
)
,
where the common matrix A is defined in (3.5). As x0,K′ is on
the straightline x0x1, z2 = 0 in (3.5), and z1 ̸= 0 (unless the two
triangles degenerateto a line segment).
A =
(1 + z1 z1
0 1
).
By the row operations of (3.6), it follows that0 0 0 0 −2z1
−2z10 −2 −2 0 −2 −22 2 0 2 + 2z1 2 + 2z1 00 0 0 0 0 02 0 −3 2 + 5z1
0 −3− 5z13 0 −2 3 0 −2
c1c2c3c4c5c6
=−z10z1000
.
From the first equation, we choose c6 = 0 and c5 = 1/2. Then
adding thesecond equation to the third equation, we can let c1 = c3
= c4 = 0 in thenew third equation to satisfy it. By the second
equation, c2 = −c5 = −1/2.Thus, we find a solution uh which also
satisfies, cf. Figure 3.2,∫
Kdivuhdx =
∫K̂(2x̂2 − x̂)dx̂ = 0,(3.9) ∫
K′divuhdx =
∫K̂′(1− ŷ)(1− 2ŷ)dx̂ = 0.(3.10)
Theorem 3.1. For any qh ∈ Ph (2.6), there is a vh ∈ Vh (2.5),
such that
divvh = qh and ∥vh∥H1 ≤ C∥qh∥L2 ,(3.11)
where C is independent of h, but dependent on the first level
grid Th1.
Proof. As qh ∈ Ph ⊂ L20(Ω), there is an H1 vector v ∈ H10 (Ω)2
such that,by [4, 6],
divv = qh and ∥v∥H1 ≤ C∥qh∥L2 .
Let v1 = Ihv be the Scott-Zhang [20] interpolation of v, by the
26 degreesof freedom in (2.4) (so that the edge flux is preserved.)
Then ∥v1∥H1 ≤
13
-
C∥v∥H1 , and ∫Kdivv1dx =
∫∂K
v1 · nds =∫∂K
v · nds
=
∫Kdivvdx =
∫Kqhdx,
on every triangle K ∈ Th.Let q1 = qh−divv1 ∈ Ph. There are two
types of vertices in triangulation
Th, singular points and non-singular points. If s is an internal
singular point,shown in Figure 3.3, the other end point of an edge
having s as an end point(points s1, s2, s3, s4 in Figure 3.3)
cannot be a singular point, as the sum oftwo angles at s is already
π. At s, the Ph function q1 has three degrees offreedom, not four,
i.e.,
(q1|K1 − q1|K2 + q1|K3 − q1|K4)(s) = 0,(3.12)
cf. Figure 3.3.
s
BBBBBBB
""
""""
""""
BBBBBBB"
"""
""""
""
s1
K1
s2
s3
K2
s1
K4
s4
K3
Figure 3.3. An internal singular point s and its four
neigh-boring triangles.
Let v1,2 ∈ (P3(K1)× P3(K2)) ∩H10 (K1 ∪K2) be such that
∇v12(s1) = ∇12v(s2) = ∇12v(s3) = 0, ∂ss2v12 = 1,
where ∂ss2 denotes the directional derivative along the
direction from s tos2. That is, on K1,
v1,2 = cλss1λ2s1s2 , c = (∂ss2λss1)
−1λs1s2(s)−2,
where λss1 is the linear function assuming 0 on edge ss1 and 1
at the oppositevertex s2. On K2, v12 is defined symmetrically.
Let
v12 = [q1|K1(s)v12]tss2 ,14
-
where tss2 is the unit (tangent) vector along the direction from
s to s2,cf. Figure 3.3. It follows that
divv12|K1(s) = q1|K1(s) [(tss2)1∂x1v12(s) + (tss2)2∂x2v12(s)]=
q1|K1(s)∂ss2v12(s) = q1|K1(s).
But, on K2, divv12(s) = q1|K1(s), not matching q1|K2(s) yet. At
the restnodes and the node s of (other) triangles, divv12|Kj (si) =
0. Also, due tothe tangent vector in the definition, via
integration by parts, we have∫
K1
divv12dx =
∫∂K1
v12 · nds =∫∂K1
0ds = 0.
The integral is also zero on K2. Similarly, we define
v23 = (q1|K2(s)− q1|K1(s))v23tss3 ,v34 = (q1|K3(s)− q1|K2(s) +
q1|K1(s))v34tss4 .
Together, we let
vs = v12 + v23 + v34.
By the construction, divvs(s) = q1(s), on K1, K2 and K3. By the
constraint(3.12), on K4,
divvs(s) = q1|K3(s)− q1|K2(s) + q1|K1(s) = q1|K4(s).
So the divergence of the constructed vs matches the four values
of q1 at thesingular point s. By the scaling argument, we have
∥vs∥H1(Ω) ≤ Ch max1≤i≤4
|q1|Ki(s)| ≤ C∥q1∥L2(∪4i=1Ki).
In the same way, we can define a discrete velocity at each
singular point,including the boundary singular points depicted in
Figure 2.1. Summingthese velocity functions, we name it v2 such
that
divv2(s) =
{q1(s) at all singular vertices on all triangles,
0 at rest vertices on all triangles,∫Kdivv2dx = 0 on all
triangles,
∥v2∥H1(Ω) ≤ C∥q1∥L2(Ω).
Let q2,1 = q1 − divv2 ∈ Ph. The rest vertices are non-singular.
But thereis a special type of nonsingular vertex, shown in Figure
3.4, that the triangleK and both its neighboring triangles at
vertex x, K ′ and K ′′, form a straightline passing through x. The
matching at x on K (must be done before thematching on the rest
triangles at x) can be done by the above method fortreating the
singular vertex. But we give another construction in Lemma3.2. Let
x be a non-singular vertex on the middle triangle K, cf. Figure
3.4.
15
-
x
BBBBBBB
""
""""
""""
BBBBBBB"
"""
""""
""
x1
K
x2
x3
K ′
x1
K ′′
x4
BBBB
Figure 3.4. A non-singular point x in K with two“straight-line”
neighboring triangles, K ′ and K ′′.
By lemma 3.2, (3.9) and (3.10), there is a vx,K,K′ ∈ Vh∩H10 (K∪K
′)2, suchthat
divvx,K,K′ |K(x) = divvx,K,K′ |K′(x) = q2,1|K(x),divvx,K,K′
|Ki(xj) = 0 at rest vertices xj on triangles Ki = K,K ′,∫
Ki
divvx,K,K′dx = 0 on all triangles Ki = K,K′,
∥vx,K,K′∥H1(Ω) ≤ Ch∣∣q2,1|K(x)∣∣ ≤ C∥q2,1∥L2(K∪K′).
There is likely no such “half-singular” vertex. But if there
are, we sum suchvelocity functions vx,K,K′ and name it v2,1.
Let q2 = q2,1 − divv2,1 ∈ Ph. q2 keeps all properties of q2,1
except hav-ing different nodal values at “half-singular” vertex
that q2|K(x) = 0 andq2|K′(x) = q2,1|K′(x)− q2,1|K(x).
For each x of the rest vertices on one of its associated
triangle K (itmust have at least one neighboring triangle K ′ not
forming a straight linewith K), by Lemma 3.1, there is a vector
vx,K ∈ Vh, supported on K andanother neighboring triangle K ′, such
that
divvx,K |K(x) = q2|K(x),divvx,K |K1(x1) = 0 at rest vertices x1
and triangles K1,∫
K1
divvx,Kdx = 0 on all triangles K1,(3.13)
∥vx,K∥H1(Ω) ≤ Ch∣∣q2|K(x)∣∣ ≤ C∥q2∥L2(K∪K′).
16
-
Here (3.13) holds because, in the construction of vx,K in Lemma
3.1, wehave vx,K ∈ C0(K ∪K ′)2 and divvx,K |K′ = 0 so that∫
Kdivvx,Kdx =
∫K∪K′
divvx,Kdx−∫K′
divvx,Kdx
=
∫∂(K∪K′)
vx,K · nds−∫K′
0dx
=
∫∂(K∪K′)
0ds− 0 = 0.
Once more, summing all these vx,K , we get a vector function v3
∈ Ph, suchthat divv3 matches all non-zero vertex values of q2, and
does not destroythe properties by the constructions v1 and v2.
Let q3 = q2 − divv3 ∈ Ph. On each triangle, q3 is P2
polynomialvanishing at three vertices and having mean value
∫K q3dx = 0. On a
triangle K, the dimension of q3 functions is 2, not 3. We
construct aP3 vector (bubble) on each triangle so that its
divergence matches twoof the mid-edge values, q3(m1) and q3(m2), of
q3 on the triangle. LetvK = BK(q3(m1)t1 + q3(m2)t2) where
BK = λ1λ2λ3,
A =(∇BK(m1) ∇BK(m2)
),(
t1 t2)= A−T ,
and λi is the linear function vanishing on edge ei and assuming
value 1on the opposite vertex of K. Note that, the matrix A is
invertible as thetriangle K is non-singular. By this construction,
we have
divvK(m1) = ∂x1BK(m1)(q3(m1)t1 + q3(m2)t2)1
+ ∂x2BK(m1)(q3(m1)t1 + q3(m2)t2)2
= q3(m1)∇BK(m1) · t1 + q3(m2)∇BK(m1) · t2= q3(m1) · 1 + q3(m2) ·
0 = q3(m1).
Also divvK(m2) = q3(m2). As BK has two directional derivatives 0
atthe vertices, divvK(xi) = 0 at all three vertices. Also as BK |∂K
= 0,∫K divvK =
∫∂K ∇vK · n = 0. That is, the P2 polynomial q3 − divvK ,
on K, has 5 zero values at 5 Lagrange nodes and zero mean value.
Thusq3 = divvK on K. Summing all such vK of all triangles K ∈ Th,
we let itbe v4 that
divv4 = q3, ∥v4∥H1(Ω) ≤ Ch|q3|L∞(Ω) ≤ C∥q3∥L2(Ω).
Combining the four constructed vectors, we let
vh = v1 + v2 + v2,1 + v3 + v4.
17
-
Then divvh = ph, and, due to the finite over-lapping,
∥vh∥H1 ≤ ∥v1∥H1 + ∥v2∥H1 + ∥v2,1∥H1 + ∥v3∥H1 + ∥v4∥H1≤ ∥v1∥H1 +
∥v2∥H1 + ∥v2,1∥H1 + ∥v3∥H1 + C∥q3∥L2≤ ∥v1∥H1 + ∥v2∥H1 + ∥v2,1∥H1 +
C∥v3∥H1 + C∥q2∥L2≤ C∥v1∥H1 + C∥q1∥L2≤ C∥qh∥L2 .
Because of the homogeneous boundary condition, the divergence
operatormay not have a bounded right inverse in an high order
Sobolev norm, evenif the solution p is smooth, [4, Theorem 3.1]. So
we assume, by [4, Theorem3.1], for the solution p of (2.1), there
is a w ∈ H10 (Ω)2 ∩H4(Ω)2 such that
divw = p and ∥w∥H4 ≤ C∥p∥H3 .(3.14)
Theorem 3.2. Let the grids be defined by the multigrid
refinement of aninitial grid Th1. The finite element solution (uh,
ph) of (2.7) is quasi-optimalin approximating the exact solution
(u, p) of the Stokes equation (2.1), as-suming (3.14) holds,
∥u− uh∥H1 + ∥p− ph∥L2 ≤ C infvh∈Vh,qh∈Ph
(∥u− vh∥H1 + ∥p− qh∥L2)
≤ Ch3(∥u∥H4 + ∥p∥H3).
Proof. By (3.11), the following inf-sup condition holds,
infqh∈Ph
supvh∈Vh
(divvh, qh)
∥vh∥H1∥qh∥L2≥ C
with C independent of the grid size h. By the standard theory on
saddle-point approximation [5, 6], the quasi-optimality inequality
holds. By theScott-Zhang interpolation operator [20], we have
infvh∈Vh
∥u− vh∥H1 ≤ C∥u− Ihu∥H1 ≤ Ch3∥u∥H4 ,
infqh∈Ph
∥p− qh∥L2 ≤ C∥divw − div Ihw∥L2 ≤ Ch3∥w∥H4 ≤ Ch3∥p∥H3 .
4. Numerical tests
We solve the Stokes (2.1) on the unit square Ω = (0, 1)2, where
the exactsolution is
u = curl g, p = ∆g, where g = 28x2(1− x)2y2(1− y)2.(4.1)
The first grid is the northwest-southeast cut of the domain,
shown in Figure4.1. Then the standard multigrid refinement is
applied to generate higherlevels of grids, shown in Figure 4.1. For
such grids, there are only two
18
-
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@@
@@
@@
@
@@@
@@
@@
@
@@@
@@
@@
@
@@@
@@
@@
@
Figure 4.1. The level 1, 2 and 3 uniform grids.
Table 4.1. The errors, eu = uI −uh and ep = pI −ph, andthe order
of convergence, by the P3 + B4/P2 finite element(2.5) and (2.6),
for the problem (4.1).
∥eu∥L2 hn |eu|H1 hn ∥ep∥L2 hn dimVh1 1.7347540 0.0 14.82305 0.0
168.5918 0.0 432 0.4304411 2.0 6.21139 1.3 5.8976 4.8 1733
0.0328937 3.7 0.98969 2.6 1.1996 2.3 6234 0.0021759 3.9 0.13048 2.9
0.1674 2.8 22895 0.0001381 4.0 0.01654 3.0 0.0200 3.1 8691
singular-points, the northeast corner and the southwest corner,
i.e., (1, 1)and (0, 0).
We first apply the newly proposed P+3 -P2 mixed finite element
method(2.5). The output data are listed in Table 4.1. As proved in
Theorem 3.2,the finite element solution converges at the optimal
order.
Table 4.2. The errors, eu = uI −uh and ep = pI −ph, andthe order
of convergence, by the P4/P3 Scott-Vogelius finiteelement, for the
problem (4.1).
∥eu∥L2 hn |eu|H1 hn ∥ep∥L2 hn dimVh1 2.161128 0.0 16.855690 0.0
93.254046 0.0 502 0.156839 3.8 2.598032 2.7 9.471127 3.3 1623
0.006195 4.7 0.224538 3.5 1.065502 3.2 5784 0.000166 5.2 0.013377
4.1 0.055986 4.3 21785 0.000004 5.3 0.000760 4.1 0.003264 4.1
8450
In the second numerical test, we compute the solution (4.1)
again by theScott-Vogelius P4-P3 element, the lowest order stable
element of [18]. Theerror and the order of convergence are listed
in Table 4.2. The optimal orderof convergence is obtained
there.
In the third numerical test, we use an unstable mixed finite,
the contin-uous P3 velocity with discontinuous P2 pressure (subset,
Ph = div(C
0-P3)),to solve the Stokes equation (4.1). The resulting linear
system of equationsis solved by the iterated-penalty method, cf.
[26, 27]. The error and the
19
-
Table 4.3. The errors, eu = uI −uh and ep = pI −ph, andthe order
of convergence, by the P3-P2 finite element, for theproblem
(4.1).
∥eu∥L2 hn |eu|H1 hn ∥ep∥L2 hn dimVh1 1.424589 0.0 10.660649 0.0
12.860813 0.0 282 0.325989 2.1 5.254361 1.0 6.636293 1.0 703
0.062660 2.4 2.033926 1.4 3.308077 1.0 2144 0.006759 3.2 0.408822
2.3 1.374595 1.3 7425 0.000787 3.1 0.090568 2.2 0.625108 1.1 27586
0.000096 3.0 0.021744 2.1 0.297753 1.1 10630
order of convergence are listed in Table 4.3. In this case, the
pressure con-verges two orders below the optimal order, to the true
solution. The finiteelement velocity is one order sub-optimal. But
as the inf-sup condition fails,we do not have a theory to cover the
above observation, i.e., the sub-optimalconvergence is not
proved.
Acknowledgment
We thank an anonymous referee for pointing out Remark 3.1, and
Figure3.4 (which was not discussed in an earlier version of the
paper.)
This research is partially supported by the NSFC project
11571023.
References
[1] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure
Stokes elements, inAdvances in Computer Methods for Partial
Differential Equations VII, ed. R. Vich-nevetsky and R.S.
Steplemen, 1992.
[2] D. N. Arnold and R. Winther, Mixed finite elements for
elasticity, Numer. Math.92 (2002) no. 3, 401–419.
[3] D. Arnold, G. Awanou and R. Winther, Finite elements for
symmetric tensors inthree dimensions, Math. Comp. 77 (2008), no.
263, 1229–1251.
[4] D. Arnold, L.R. Scott and M. Vogelius, Regular inversion of
the divergence operatorwith Dirichlet conditions on a polygon, Ann.
Sc. Norm. Super Pisa, C1. Sci., IV Ser.,15(1988), 169–192
[5] S.C. Brenner and L.R. Scott, The Mathematical Theory of
Finite Element Methods,Springer-Verlag, New York, 1994.
[6] F. Brezzi and M. Fortin, Mixed and hybrid finite element
methods, Springer, 1991.[7] P.G. Ciarlet, The Finite Element Method
for Elliptic Problems, North-Holland,
Amsterdam, 1978.[8] R. Falk and M. Neilan, Stokes complexes and
the construction of stable finite
elements with pointwise mass conservation, SIAM J. Numer. Anal.
51 (2013), no.2, 1308–1326.
[9] J. Guzman and M. Neilan, Conforming and divergence free
Stokes elements ongeneral triangular meshes, Math. Comp. 83 (2014),
no. 285, 15–36.
[10] J. Guzman and M. Neilan, Conforming and divergence-free
Stokes elements in threedimensions, IMA J. Numer. Anal. 34 (2014),
no. 4, 1489–1508.
[11] J. Hu and S. Zhang, A family of conforming mixed finite
elements for linear elasticityon triangle grids, arXiv:1406.7457
[math.NA].
20
-
[12] J. Hu and S. Zhang, A family of conforming mixed finite
elements for linear elasticityon tetrahedral grids, Sci. China
Math. 58 (2015), no. 2, 297–307.
[13] J. Hu and S. Zhang, Finite element approximations of
symmetric tensors on simpli-cial grids in Rn: the lower order case,
Math. Models Methods Appl. Sci. 26 (2016),no. 9, 1649–1669.
[14] Y. Huang and S. Zhang, A lowest order divergence-free
finite element on rectangulargrids, Frontiers of Mathematics in
China, 6 (2011), No 2, 253-270.
[15] Y. Huang and S. Zhang, Supercloseness of the
divergence-free finite element so-lutions on rectangular grids,
Communications in Mathematics and Statistics, 1(2013), 143-162.
[16] M. Neilan, Discrete and conforming smooth de Rham complexes
in three dimen-sions, Math. Comp. 84 (2015), no. 295,
2059–2081.
[17] J. Qin On the convergence of some low order mixed finite
elements for incompressiblefluids, Thesis, Pennsylvania State
University, 1994.
[18] L. R. Scott and M. Vogelius, Norm estimates for a maximal
right inverse of thedivergence operator in spaces of piecewise
polynomials, RAIRO, Modelisation Math.Anal. Numer. 19 (1985),
111–143.
[19] L. R. Scott and M. Vogelius, Conforming finite element
methods for incompressibleand nearly incompressible continua, in
Lectures in Applied Mathematics 22, 1985,221–244.
[20] L. R. Scott and S. Zhang, Finite element interpolation of
nonsmooth functionssatisfying boundary conditions , Math. Comp. 54
(1990), 483–493.
[21] M. Vogelius, A right-inverse for the divergence operator in
spaces of piecewise poly-nomials, Numer. Math. 41 (1983),
19–37.
[22] X. Xu and S. Zhang, A new divergence-free interpolation
operator with applicationsto the Darcy-Stokes-Brinkman equations,
SIAM J. Scientific Computing,32 (2010),no. 2, 855-874.
[23] S. Zhang, Optimal order non-nested multigrid methods for
solving finite elementequations I: On quasiuniform meshes, Math.
Comp. 55 (1990), 23–36.
[24] S. Zhang, Successive subdivisions of tetrahedra and
multigrid methods on tetrahe-dral meshes, Houston J. of Math., 21
(1995), 541–556.
[25] S. Zhang, A new family of stable mixed finite elements for
3D Stokes equations,Math. Comp. 74 (2005), 543–554.
[26] S. Zhang, On the P1 Powell-Sabin divergence-free finite
element for the Stokesequations, J. Comp. Math., 26 (2008),
456-470.
[27] S. Zhang, A family of Qk+1,k×Qk,k+1 divergence-free finite
elements on rectangulargrids, SIAM J. Num. Anal., 47 (2009),
2090-2107.
[28] S. Zhang, Divergence-free finite elements on tetrahedral
grids for k ≥ 6, Math.Comp. 80 (2011), 669–695.
[29] S. Zhang, Quadratic divergence-free finite elements on
Powell-Sabin tetrahedralgrids, Calcolo, 48 (2011), No 3,
211–244.
Department of Mathematical Sciences, University of Delaware,
Newark,DE 19716, USA. [email protected].
21