c 2003 Society for Industrial and Applied Mathematics
SIAM J. NUMER. ANAL.
Vol. 41, No. 2, pp. 624–640
P1 -NONCONFORMING QUADRILATERAL FINITE ELEMENT
METHODS FOR SECOND-ORDER ELLIPTIC PROBLEMS∗
CHUNJAE PARK† AND DONGWOO SHEEN‡
Abstract. A P1 -nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which
contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only
piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of
using our element is convenience in using rectangular or quadrilateral meshes with the least degrees
of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is
obtained. Also a nonparametric reference scheme is introduced in order to systematically compute
stiffness and mass matrices on each quadrilateral. An extension of the P1 -nonconforming element to
three dimensions is also given. Finally, several numerical results are reported to confirm the effective
nature of the proposed new element.
Key words. nonconforming finite elements, quadrilateral, elliptic problems
AMS subject classifications. 65N30, 65N12, 65N15
PII. S0036142902404923
1. Introduction. We are concerned with nonconforming finite element methods
for second-order elliptic problems. Nonconforming elements have been used effectively
especially in fluid and solid mechanics due to their stability. Recently, these elements
have attracted increasing attention from scientists and engineers in more wide areas,
as this type of element is potentially useful in parallel computing.
The use of finite elements for Stokes problems, which is fundamental in fluid mechanics, usually requires the discrete Babus̆ka–Brezzi condition (inf-sup condition) to
be satisfied by the velocity and pressure variables, generally set in the mixed finite element formulation; for instance, the standard P1 -P0 pair for triangular decompositions
or the Q1 -P0 pair for quadrilateral decompositions of the computational domain lead
to checkerboard solutions for pressure. However, if the nonconforming elements introduced in [3, 8, 15, 5] are used to approximate the velocity part instead of the usual
P1 or Q1 elements, the Babus̆ka–Brezzi condition is easily satisfied, and thus stable
solutions are obtained. Nonconforming finite element methods have been proved to be
effective for several parameter dependent elasticity problems in a stable fashion such
that the methods converge independently of the Lamé parameters, while standard
conforming methods fail to converge as the parameters tend to a locking limit; see
[2, 12, 13].
Moreover, in view of domain decomposition methods, the use of nonconforming
elements facilitates the exchange of information across each subdomain and provides
spectral radius estimates for the iterative domain decomposition operator [9].
The nonconforming simplicial finite element space of lowest degree introduced
by Crouzeix and Raviart [8] is identical to the corresponding conforming one (that
∗ Received by the editors March 29, 2002; accepted for publication (in revised form) October
21, 2002; published electronically May 6, 2003. This research was supported in part by the Korea
Research Foundation (KRF-2000-DS0004).
http://www.siam.org/journals/sinum/41-2/40492.html
† Institute for Pure and Applied Mathematics, University of California, Los Angeles, CA 90095.
Current address: Impedance Imaging Research Center, Kyunghee University, Seoul, Korea (cpark@
nasc.snu.ac.kr).
‡ Department of Mathematics, Seoul National University, Seoul 151–747, Korea ([email protected]).
624
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
625
is, P1 in both cases), and thus it is rather simple to understand. Although the triangular meshes are popular to use, in many cases one wishes to use quadrilateral
meshes with appropriate elements instead, when the problem geometry is of quadrilateral nature, especially in three dimensions. Concerning rectangular nonconforming
elements, Han [11] introduced a rectangular element with local degrees of freedom
being five, and Rannacher and Turek [15] introduced the rotated Q1 nonconforming
elements of two types: the first set of local degrees of freedom consists of the four
values at the midpoints of the edges, while the second one is composed of the four
average values over the edges. Recently, new nonconforming elements, which use only
the four values at the midpoints of the edges as degrees of freedom, have been announced by Douglas et al. [9], who in a sense combined and improved the two types of
local degrees of freedom for rotated Q1 elements, using high-order polynomials with
the degrees of freedom still being four. These elements were successfully applied to
solve Navier–Stokes problems by Cai, Douglas, and Ye [5]. A recent observation by
Arnold, Boffi, and Falk [1] implies that where the rectangular elements are applied to
truly quadrilateral meshes, the optimality in convergence will be lost. Thus for the
truly quadrilateral case, an extra element should be added [4], with the local degrees
of freedom being five; the extra degrees of freedom can be eliminated easily at an
element level since they are essentially bubble functions.
The purpose of this paper is to introduce P1 -nonconforming finite element spaces
on quadrilateral meshes which have the lowest degrees of freedom. The motivation for
our new element comes from the observation that any P1 function on a quadrilateral
can be uniquely determined at any three of the four midpoints of edges.
The degrees of freedom for our P1 -nonconforming quadrilateral element are about
half of those for the other rectangular nonconforming elements, and about a third of
those for the P1 triangular nonconforming space on the mesh with each quadrilateral
being divided into two triangles. Indeed, our P1 -nonconforming quadrilateral element
space turns out to be a subspace of P1 -nonconforming triangular element spaces by
dividing each quadrilateral into two triangles.
In the Q1 -conforming quadrilateral element case, it is convenient to use a fixed
reference rectangle and basis from which, corresponding to each quadrilateral, a bilinear transformation can be used to calculate stiffness and mass matrices by pulling
back to the reference rectangle without losing the order of convergence. However, as
mentioned above, such a reference system does not guarantee optimal convergence
any more with existing nonconforming quadrilateral elements with only four degrees
of freedom [1]. We present a nonparametric reference scheme in section 4, which provides an efficient way of calculating the stiffness and mass matrices from a reference
rectangle without losing the order of convergence.
As discussed earlier, one of the motivations for seeking the P1 -nonconforming
quadrilateral element space is to try to use it for the approximation of the velocities
and the P0 space for the pressure as in [8, 11, 15, 4]. However, we remark that with
this combination the discrete inf-sup condition is not fulfilled, as there are only three
degrees of freedom for the normal components at the midpoints of a quadrilateral.
But the current element works well as a locking-free element for elasticity problems
[14].
The organization of the paper is as follows. In the next section we present two P1 nonconforming element spaces on quadrilateral meshes. Then section 3 describes an
interpolation operator and also deals with a brief analysis of convergence in the cases of
Dirichlet and Robin problems. Then a nonparametric reference scheme is introduced
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CHUNJAE PARK AND DONGWOO SHEEN
in section 4. The analysis carried out in the current paper has a counterpart in three
dimensions: P1 -nonconforming hexahedral finite elements will be briefly discussed in
section 5, detailed analyses being treated in a forthcoming paper. Finally, numerical
examples are illustrated in section 6.
2. The P1 -nonconforming element on quadrilateral meshes.
2.1. The P1 -nonconforming quadrilateral element. Let Ω be a simply
connected polygonal domain in R2 with boundary Γ. Let (Th )h>0 be a regular
family of decompositions (or triangulations) of Ω into convex quadrilaterals, where
h = maxQ∈Th hQ with hQ = diam(Q). For the standard definition of regular decomposition, we refer to [10]. Henceforth, in this paper, a quadrilateral will be implicitly
assumed to be convex.
For a general quadrilateral Q, denote by vj , 1 ≤ j ≤ 4, its four vertices and by
v
+v
mj , 1 ≤ j ≤ 4, the midpoints of edges of Q such that mj = j−12 j , 1 ≤ j ≤ 4, with
the identification v0 = v4 . Let P1 (Q) = Span{1, x, y}. The following lemmas are
trivial but useful in what follows.
Lemma 2.1. If u ∈ P1 (Q), then u(m1 ) + u(m3 ) = u(m2 ) + u(m4 ). Conversely,
if uj is a given value at mj , for 1 ≤ j ≤ 4, satisfying u1 + u3 = u2 + u4 , then there is
a unique u ∈ P1 (Q) such that u(mj ) = uj , 1 ≤ j ≤ 4.
Proof. The first half is trivial:
u(v4 ) + u(v1 ) u(v2 ) + u(v3 )
+
2
2
u(v1 ) + u(v2 ) u(v3 ) + u(v4 )
+
= u(m2 ) + u(m4 ).
=
2
2
u(m1 ) + u(m3 ) =
For the latter half, suppose that u1 + u3 = u2 + u4 and then choose a u ∈ P1 (Q) such
that u(mj ) = uj , j = 1, 2, 3. Then by the first half of the lemma, u1 +u3 = u2 +u(m4 ),
which implies that u(m4 ) = u4 , so that u(mj ) = uj , 1 ≤ j ≤ 4. Uniqueness is obvious.
Lemma 2.2. For 1 ≤ j ≤ 4, let ϕ
j ∈ P1 (Q) be defined such that
ϕ
j (mk ) =
1, k = j, j + 1
0, otherwise.
mod 4,
Then Span{ϕ
1 , ϕ
2 , ϕ
3 , ϕ
4 } = P1 (Q). Indeed, any three of ϕ
1 , ϕ
2 , ϕ
3 , ϕ
4 span P1 (Q).
Proof. Clearly Span{ϕ
1 , ϕ
2 , ϕ
3 , ϕ
4 } ⊂ P1 (Q). To show the other direction of
inclusion, it suffices to show that P1 (Q) ⊂ Span{ϕ
1 , ϕ
2 , ϕ
3 }; then rotational symmetry
would imply that any three of ϕ
1 , ϕ
2 , ϕ
3 , ϕ
4 span P1 (Q). Let u ∈ P1 (Q) be arbitrary.
Set
ψ = u(m1 )ϕ
1 + [u(m2 ) − u(m1 )]ϕ
2 + [u(m3 ) − u(m2 ) + u(m1 )]ϕ
3 .
Then it is immediate to see that ψ(mj ) = u(mj ), j = 1, 2, 3. Lemma 2.1 implies
that ψ(m4 ) = u(m4 ) and therefore ψ is identical to u. This proves that P1 (Q) ⊂
Span{ϕ
1 , ϕ
2 , ϕ
3 }.
Given a decomposition Th of Ω into quadrilaterals, let NQ , NV , and NE denote
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
0
627
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
v
1
0
0
1
j
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
Fig. 1. Values at the midpoints of the basis function ϕj associated with the vertex vj .
the numbers of quadrilaterals, vertices, and edges, respectively. Then set
NQ
Th = {Q1 , Q2 , . . . , QNQ };
Qj = Ω,
j=1
V = {v1 , v2 , . . . , vNV } : the set of all vertices of Q ∈ Th ,
E = {e1 , e2 , . . . , eNE } : the set of all edges of Q ∈ Th ,
M = {m1 , m2 , . . . , mNE } : the set of all midpoints of e ∈ E.
i
In particular, let NVi , NEi , and NM
denote the numbers of interior vertices, edges, and
midpoints of Q ∈ Th , respectively. Our objective is to introduce a P1 -nonconforming
finite element space associated with the quadrilateral decomposition Th .
Set
N C h = {vh : Ω → R| vh |Q ∈ P1 (Q) for all Q ∈ Th ,
vh is continuous at every m ∈ M \ Γ},
N C h0 = {vh ∈ N C h | vh (m) = 0 for all m ∈ Γ ∩ M}.
For each vertex vj ∈ V, denote by E(j) the set of all edges e ∈ E such that one
of the endpoints is vj , and by M(j) the set of all midpoints m of edges in E(j). Let
ϕj ∈ N C h be such that
1 if m ∈ M(j),
ϕj (m) =
(2.1)
0 if m ∈ M \ M(j).
An example of such a function ϕj is shown in Figure 1. Notice that ϕ
k , 1 ≤ k ≤ 4,
given in Lemma 2.2 belong to the restriction of ϕj , j = 1, . . . , NV , to Q.
Remark 2.3. Obviously ϕj |Q (vj ) < 2 for all Q ∈ Th , with vj being one of its
vertices; moreover, ϕj |Q (vj ) = 3/2 if Q is a parallelogram. Therefore, if Th is decomposed into parallelograms, ϕj is continuous at vj for all j. However, ϕj may not be
continuous in general. Examples of basis functions in conforming and nonconforming
cases are depicted in Figure 3(b),(c) for a simple mesh given in Figure 3(a).
628
CHUNJAE PARK AND DONGWOO SHEEN
v
2
m3
v3
c
m
m2
4
v1
v4
m1
Fig. 2. The midpoints mj , 1 ≤ j ≤ 4, form a parallelogram in the quadrilateral with vertices
vj , 1 ≤ j ≤ 4.
2.2. The dimension and basis for N C h
0 . We proceed to investigate in the
dimension of N C h0 ; that of N C h will be discussed in the next subsection. Implicitly
the following assumption will be imposed on the decomposition in the rest of this
article, especially for Dirichlet problems, in order to exclude pathological cases.
Assumption I. Each interior edge has at least one interior vertex as its endpoint.
There are cases in which Assumption I is violated. For instance, some decompositions Th of Ω may contain elements whose four vertices lie on the boundary of Ω;
in these cases, the reduced decomposition Th , obtained by eliminating such elements
from Th , fulfills Assumption I.
An upper bound of dim(N C h0 ) is given in the following lemma.
Lemma 2.4. dim(N C h0 ) ≤ NVi .
i
Proof. For the degrees of freedom for N C h0 define d : N C h0 → RNE by
d(ϕ) := (d1 (ϕ), . . . , dNEi (ϕ))t ,
ϕ ∈ N C h0 ,
with dj (ϕ) = ϕ(mj ) for each interior midpoint mj , j = 1, . . . , NEi . If ϕ ∈ N C h0
Ni
E
satisfies dj (ϕ) = 0 for all j = 1, . . . , NEi , clearly ϕ = 0. This implies that {dj }j=1
h h
i
spans (N C 0 ) , the dual of N C 0 . Note that, for any j = 1, . . . , NE , the component
dj of d is nontrivial, since for each mj ∈ M \ Γ there exists a function ϕ such that
ϕ(mj ) = 0, as defined in (2.1) by Assumption I.
Due to Lemma 2.1, for each Qj ∈ Th having mj1 , mj2 , mj3 , and mj4 as its midpoints of edges as in Figure 2, one of the following linear restrictions should be
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
629
5
4
3
2
A
1
0
−1
−1
0
1
2
3
4
5
(a) An example of a mesh.
(b) The P1 -nonconforming basis whose
values are equal to 1 at the midpoints of
edges which meet with the vertex A, and
0 at all the other midpoints.
(c) The Q1 -conforming basis whose values are 1 at the vertex A, and 0 at all the
other vertices.
Fig. 3. Shapes of P1 -nonconforming (b) and Q1 -conforming (c) basis functions in the quadrilateral mesh shown in (a).
imposed:
dj1 + dj3 − dj2 − dj4 = 0
dj1 + dj3 − dj2 = 0
dj1 − dj2 = 0
if mjk ∈
/ Γ, 1 ≤ k ≤ 4,
if mj1 , mj2 , mj3 ∈
/ Γ and mj4 ∈ Γ,
if mj1 , mj2 ∈
/ Γ and mj3 , mj4 ∈ Γ,
which can be written formally as
(2.2)
Aj d = 0.
i
Here, Aj = (Aj,1 , . . . , Aj,NEi ) is a row vector in RNE with at most four nontrivial
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CHUNJAE PARK AND DONGWOO SHEEN
entries such that
Aj,j1 = Aj,j3 = −Aj,j2 = −Aj,j4 = 1
Aj,j1 = Aj,j3 = −Aj,j2 = 1
Aj,j1 = −Aj,j2 = 1
if mjk ∈
/ Γ, 1 ≤ k ≤ 4,
if mj1 , mj2 , mj3 ∈
/ Γ and mj4 ∈ Γ,
if mj1 , mj2 ∈
/ Γ and mj3 , mj4 ∈ Γ.
We therefore see that
dim(N C h0 ) = dim((N C h0 ) )
(2.3)
≤ dim{d = (d1 , . . . , dNEi )t ; Aj d = 0, j = 1, . . . , NQ }.
We proceed to see whether Aj , j = 1, . . . , NQ , are linearly independent vectors or
not. For this, assume that for some proper subset J {1, 2, . . . , NQ },
(2.4)
cj Aj = 0,
j∈J
with cj = 0 for all j ∈ J. Set ΩJ = j∈J Qj . Then there exist an interior midpoint
ml ∈ ∂ΩJ ∩ M \ Γ and Qk ⊂ ΩJ for which ml is a midpoint of an edge of Qk , since
ΩJ Ω. From the linear restriction concerning Qk , Ak d = 0, we see that Ak has a
nonzero entry in the lth column; moreover, Ak is the unique vector that has a nonzero
value in the lth entry among all Aj , j ∈ J, since ml is at the boundary of ΩJ . Thus
(2.4) implies that ck = 0, which is a contradiction. Therefore, we have
i
any NQ − 1 elements from {A1 , A2 , . . . , ANQ } are linearly independent in RNE .
Let A = (At1 , . . . , AtNQ −1 )t be the (NQ − 1) × NEi matrix whose jth row is Aj .
Then the collection of (2.2) for j = 1, . . . , NQ − 1 can be written formally in the
matrix form
Ad = 0,
with the rank of A being NQ − 1. Notice from (2.3) that
(2.5)
dim(N C h0 ) ≤ dim{d = (d1 , . . . , dNEi )t ; Ad = 0}.
A
Let B be an (NEi − (NQ − 1)) × NEi matrix such that Ā = ( B
) is invertible: such
a matrix exists as rank(A) = NQ − 1. Setting (ψ1 , . . . , ψNEi −(NQ −1) )t = Bd, we have
i
Ād = (0, . . . , 0, ψ1 , . . . , ψNEi −(NQ −1) )t ∈ RNE .
N i −(N −1)
Q
E
This implies that {ψj }j=1
spans {d = (d1 , . . . , dNEi )t ; Ad = 0}, since Ā is
invertible. Therefore, from (2.5), we see that
dim(N C h0 ) ≤ NEi − (NQ − 1).
Recall Euler’s formula for a simply connected domain, NV − NE + NQ = 1, which is
equivalent to NVi −NEi +NQ = 1. The following lemma is thus obtained: dim(N C h0 ) ≤
NEi − (NQ − 1) = NVi .
The dimension and basis functions for N C h0 are given in the following theorem.
Theorem 2.5. Let ϕj be the function defined in (2.1) with interior vertex vj ∈
V \ Γ, j = 1, . . . , NVi . Then {ϕ1 , ϕ2 , . . . , ϕNVi } forms a basis for N C h0 . Therefore,
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
631
dim(N C h0 ) = NVi . That is, the degrees of freedom for N C h0 is equal to the number of
interior vertices in Th .
NVi
cj ϕj = 0. Choose a vertex vl located at the boundary Γ.
Proof. Suppose j=1
Since Ω is connected, there exists an interior vertex vk adjacent to vl ∈ Γ. Let m be
the midpoint of vl vk . Then we have
i
0=
NV
cj ϕj (m) = ck .
j=1
The coefficients cj of the ϕj ’s corresponding to all the vertices adjacent to Γ will vanish
in this manner. Then, stripping out all the boundary elements, we continue the above
argument to the next layer to show again that all the coefficients cj of the ϕj ’s corresponding to all the vertices adjacent to that boundary layer vanish. We can continue
the argument to show that all the coefficients vanish until the domain is exhausted.
Thus {ϕ1 , ϕ2 , . . . , ϕNVi } is linearly independent. Moreover, {ϕ1 , ϕ2 , . . . , ϕNVi } forms a
basis for N C h0 since dim(N C h0 ) ≤ NVi by Lemma 2.4, and therefore dim(N C h0 ) = NVi .
This completes the proof.
Remark 2.6. Let Th be the triangulation of Ω into triangles by dividing each
quadrilateral into two triangles. Consider the P1 -nonconforming simplicial element
space N
C h on T . We then observe that N C h ⊂ N
C h . Moreover,
0
h
0
0
dim(N
C h0 ) = NEi + NQ = NVi + 2NQ − 1 = dim(N C h0 ) + 2NQ − 1.
2.3. The dimension and basis for N C h . The dimension and basis for N C h
is then obtained by the arguments in the previous subsection with slight modifications.
Indeed, we have the following result.
Lemma 2.7. dim(N C h ) ≤ NE − NQ = NV − 1.
Proof. The arguments of the proof are essentially identical to those for Lemma 2.4
with minor modifications, but for the sake of the reader’s convenience we repeat most
of the arguments with proper modifications.
First, define d : N C h → RNE by
t
d(ϕ) := (d1 (ϕ), . . . , dNE (ϕ)) ,
ϕ ∈ N C h,
E
with dj (ϕ) = ϕ(mj ) for each midpoint mj , j = 1, . . . , NE . Then one sees that {dj }N
j=1
h h
spans (N C ) , the dual of N C .
For each Qj ∈ Th having mj1 , mj2 , mj3 , and mj4 as its midpoints of edges, the
following linear restriction should be imposed:
dj1 + dj3 − dj2 − dj4 = 0,
which can be written formally as
Aj d = 0,
where Aj = (Aj,1 , . . . , Aj,NE ) is a row vector in RNE with at most four nontrivial
entries such that
Aj,j1 = Aj,j3 = −Aj,j2 = −Aj,j4 = 1.
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CHUNJAE PARK AND DONGWOO SHEEN
Consequently,
dim(N C h ) = dim((N C h ) ) ≤ dim{d = (d1 , . . . , dNE )t ; Aj d = 0, j = 1, . . . , NQ }.
Next, assume that for a subset J {1, 2, . . . , NQ },
(2.6)
cj Aj = 0,
j∈J
with cj = 0 for all j ∈ J. Set ΩJ = j∈J Qj . Then there exist a midpoint ml ∈
∂ΩJ ∩ M and Qk ⊂ ΩJ for which ml is a midpoint of an edge of Qk . From the linear
restriction concerning Qk , Ak d = 0, we see that Ak has a nonzero entry in the lth
column; moreover, Ak is the unique vector that has a nonzero value in the lth entry
among all Aj , j ∈ J, since ml is at the boundary of ΩJ . Thus (2.6) implies that ck = 0,
which is a contradiction. Therefore, we have
{A1 , A2 , . . . , ANQ } are linearly independent in RNE .
Then by an argument quite identical to the proof of Lemma 2.4, we see that
dim(N C h ) ≤ NE − NQ .
Recall Euler’s formula for a simply connected domain, NV −NE +NQ = 1. The lemma
thus is obtained: dim(N C h ) ≤ NE − NQ = NV − 1.
The dimension and a basis functions for N C h are given in the following theorem.
Theorem 2.8. Let ϕj be the function defined in (2.1) with each vertex vj ∈
V, j = 1, . . . , NV . Choose any vertex vj0 ∈ V. Then {ϕ1 , ϕ2 , . . . , ϕNV } \ {ϕj0 } forms
a basis for N C h . Moreover, dim(N C h ) = NV − 1. That is, the degrees of freedom
for N C h is equal to the number of vertices in Th minus 1.
NV −1
Proof. Without loss of generality, assume vj0 = vNV . Suppose j=1
cj ϕj = 0.
Let vk be any vertex adjacent to vNV , and let m be the midpoint of vk vNV . Then
0=
N
V −1
cj ϕj (m) = ck ,
j=1
since m ∈ E(j) only if j = NV or k. Therefore we see that ck1 = 0 for all k1 such
that vk1 is a vertex of an edge e ∈ E(NV ). From all such vk1 ’s, we then proceed
to show that that ck2 = 0 for all k2 such that vk2 is a vertex of an edge e ∈ E(k1 ).
Since Ω is connected, by a finite repetition of the argument, we can conclude that all
cj , j = 1, . . . , NV − 1, are zeroes. Thus {ϕ1 , ϕ2 , . . . , ϕNV −1 } is linear independent and
forms a basis for N C h since dim(N C h ) ≤ NV − 1 by Lemma 2.7.
3. The interpolation operator and convergence analysis. In this section
we define an interpolation operator and analyze convergence. The case of Dirichlet
problems is considered and convergence results are obtained by using standard arguments. The case of Neumann problems, which is analogous to that of Dirichlet
problems, is then discussed in brief.
We first consider the following Dirichlet problem:
(3.1a)
(3.1b)
−∇ · α∇u + βu = f,
u = 0,
Ω,
Γ,
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
633
with α = (αjk ), αjk , β ∈ L∞ (Ω), j, k = 1, 2, 0 < α∗ |ξ|2 ≤ ξ t α(x)ξ ≤ α∗ |ξ|2 < ∞,
ξ ∈ R2 , β(x) ≥ 0, x ∈ Ω, and f ∈ H −1 (Ω). The weak problem is given as usual: find
u ∈ H01 (Ω) such that
a(u, v) = f, v ,
(3.2)
v ∈ H01 (Ω),
where a(u, v) = (α∇u, ∇v) + (βu, v), with (·, ·) being the L2 (Ω) inner product and
·, · the duality pairing between H −1 (Ω) and H01 (Ω).
Our P1 -nonconforming method for problem (3.1a) is stated as follows: find uh ∈
N C h0 such that
vh ∈ N C h0 ,
ah (uh , vh ) = f, vh ,
(3.3)
where
ah (u, v) =
aQ (u, v),
Q∈Th
with aQ : H 1 (Q) × H 1 (Q) → R being the restriction of a to Q.
For our convergence analysis, define the projection Πh : H 2 (Ω) ∩ H01 (Ω) → N C h0
such that, for ϕ ∈ H 2 (Ω) ∩ H01 (Ω),
Πh ϕ(m) =
1
(ϕ(v1 ) + ϕ(v2 ))
2
for all m ∈ M,
where v1 and v2 are the two vertices of the edge in Th whose midpoint is m. Notice
that Πh is well defined. Indeed, with Q ∈ Th , vj , mj , 1 ≤ j ≤ 4, given as in Figure 2,
one has
Πh ϕ(m1 ) + Πh ϕ(m3 ) =
1
(ϕ(v1 ) + ϕ(v2 ) + ϕ(v3 ) + ϕ(v4 )) = Πh ϕ(m2 ) + Πh ϕ(m4 ).
2
Thus by Lemma 2.1, Πh ϕ ∈ P1 (Q). Clearly Πh ϕ is continuous at all midpoints of
edges of Th . Therefore Πh ϕ ∈ N C h0 .
Since Πh preserves P1 (Q) for all Q ∈ Th , standard interpolation approximation
results, not by using a reference element but by applying the Bramble–Hilbert lemma
to each actual element, lead to the finding that
(3.4)
||ϕ − Πh ϕ||L2 (Q) + h
||ϕ − Πh ϕ||H 1 (Q) ≤ Ch2 ||ϕ||H 2 (Ω) ,
Q∈Th
Q∈Th
ϕ ∈ H 2 (Ω) ∩ H01 (Ω).
(For instance, a slight modification to Exercise 3.1.2 in [6] using the result of [7] would
give the estimate.)
Also, letting γj = ∂Ω ∩ ∂Qj , γjk = ∂Qj ∩ ∂Qk , and denoting the midpoint of γj
and γjk by mj and mjk , respectively, define
Λh = {λ ∈ Πj,k P0 (γjk ) × Πj P0 (γj ) | λjk + λkj = 0, where λjk = λ|γjk , λj = λ|γj },
where P0 (S) denotes the set of constant functions on a set S. Then define the projection P0 : H 2 (Ω) → Λh so that if v ∈ H 2 (Ω),
∂vj
α
(3.5)
− P0 vj , z
= 0 for all z ∈ P0 (γ), γ = γjk or γj ,
∂νj
γ
634
CHUNJAE PARK AND DONGWOO SHEEN
where vj = v|Qj and νj is the unit outward normal to Qj . One then has
(3.6)

 ∂vj
α
− P0 v 
∂νj
2

L2 (∂Qj )
j

1
≤ Ch 2 ||v||2 .
With the broken energy norm
1
||ϕ||h = ah (ϕ, ϕ) 2 ,
we are now in a position to state the usual second Strang lemma [16, 17, 6].
Lemma 3.1. Let u ∈ H 1 (Ω) and uh ∈ N C h0 be the solutions of (3.2) and (3.3),
respectively. Then,
|ah (u, w) − f, w |
u − uh h ≤ C
.
inf u − vh + sup
wh
v∈N C h
w∈N C h
0
0
Notice that (3.4) implies that
(3.7)
inf
v∈N C h
0
u − vh ≤ Cu2 h.
Next, for the consistency error term, by a simple calculation one has
∂uj ah (u, w) − f, w =
α
,w
.
∂νj
∂Qj \γj
j
Since a function w in N C h0 is linear on each γjk and continuous at the midpoints, the
following useful orthogonality holds:
(3.8)
P0 uj , wj γjk + P0 uk , wk γkj = P0 uj , wj − wk γjk = 0
for all w ∈ N C h0 .
From the two orthogonalities (3.5) and (3.8),
∂uj
α
ah (u, w) − f, w =
(3.9)
− P0 u j , w − mj
,
∂νj
∂Qj \γj
j
where mj is chosen to be the average of w on Qj . Due to (3.4), (3.6), and a trace
theorem,
∂uj
α
−
P
u
,
w
−
m
0 j
j
∂νj
j
∂Qj 
 12
1
≤ C||u||2 h 2 
||w − mj ||L2 (Qj ) ||∇(w − mj )||L2 (Qj ) 
j

(3.10)
≤ C||u||2 h 
j
 12
||∇w||2L2 (Qj )  ≤ C||u||2 ||w||h h.
Consequently, applying the estimates (3.7) and (3.10), combined with (3.9), in Lemma
3.1 gives the usual energy-norm error estimate. The use of a duality argument is
635
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
y
η
v2
v1
v1
v 2 −d
m2
m
3
m1
Q
1
ξ
m
A: affine
m
Q
3
d
m2
1
1
−1
v3
m4
v4
d
v3
m4
−1
v
B=A ° S: bilinear
−1
v°3
1
°
Q
°
m
2
°
m
−1
4
2
v°1
1
°
m
3
−d
4
S: simple bilinear
S(x,y)=(x+d xy,y+d xy)
°
y
v°2
x
°
m
1
1
°
x
v°
4
Fig. 4. The nonparametric reference scheme: a general bilinear mapping B can be regarded as
the composition of a simple bilinear map S and an affine map A.
analogous to that in [9], and therefore we omit the details. To sum up, we have the
following theorem.
Theorem 3.2. Let u ∈ H 1 (Ω) and uh ∈ N C h0 be the solutions of (3.2) and
(3.3), respectively. Then we have
||u − uh ||h ≤ Ch||u||H 2 (Ω) .
Moreover, if Ω is convex and f ∈ L2 (Ω), then we have
||u − uh ||L2 (Ω) ≤ Ch2 ||u||H 2 (Ω) .
Remark 3.3. The case of Robin problems is similar to that of Dirichlet problems,
replacing the space H01 (Ω) and N C h0 H 1 (Ω) and N C h , as usual.
Remark 3.4. For the case of mixed boundary value problems, the dimension and
basis functions can be computed and constructed analogously. Indeed, the dimension
and basis functions are between those for the Dirichlet and Robin boundary problems.
4. A nonparametric reference scheme. In this section we introduce a nonparametric reference scheme with which finite elements in general quadrilaterals can
be easily built from a fixed reference basis function space defined on a reference domain.
For given Q ∈ Th with vertices vj , 1 ≤ j ≤ 4, and midpoints of edges mj , 1 ≤ j ≤
4, as in Figure 4, there is a unique affine transformation A : R2 → Q such that
A(1, 0) = m1 , A(0, 1) = m2 , A(−1, 0) = m3 , A(0, −1) = m4 ,
636
CHUNJAE PARK AND DONGWOO SHEEN
since the four midpoints of any quadrilateral form a parallelogram. In fact, A is given
by
A(x̂, ŷ) =
v1 − v2 − v3 + v4
v1 + v2 − v3 − v4
v1 + v2 + v3 + v4
+
x̂ +
ŷ.
4
4
4
= A−1 (Q) and let m
Denote Q
j , 1 ≤ j ≤ 4, indicate the points (1, 0), (0, 1), (−1, 0),
(0, −1), respectively. Define ϕ
j ∈ Span{1, x̂, ŷ}, 1 ≤ j ≤ 4, such that
1, k = j, j + 1 mod 4,
ϕ
j (m
k) =
0, otherwise.
Then, by Lemma 2.2, P1 (Q) = Span{ϕ
j ◦ A−1 ; 1 ≤ j ≤ 4, }. This enables us to
construct a basis function space by using this fixed reference basis function space
may vary.
{
ϕj }4j=1 , although Q
may come from difficulty in calcuA possible drawback due to the variance of Q
However,
lating the integrals of products of basis functions and their gradients on Q.
◦
this will be overcome easily as follows. Let Q= [−1, 1]2 and denote its vertices by
◦
◦
v j , 1 ≤ j ≤ 4, as in Figure 4. Then there is a unique bilinear transformation B :Q→ Q,
◦
◦
◦ ◦
B(x, y ) = v1 + (v2 − v1 )
◦
◦
1− x
1− y
1− x 1− y
+ (v4 − v1 )
+ (v3 + v1 − v2 − v4 )
,
2
2
2
2
◦
so that B(v j ) = vj , 1 ≤ j ≤ 4. Indeed, S = A−1 ◦ B is given by
◦ ◦
◦
◦◦ ◦
◦◦
S(x, y ) = (x +d1 xy , y +d2 xy ),
where
(d1 , d2 ) = (v1 + v3 − v2 − v4 )
v1 − v3 − v2 + v4
v1 − v3 + v2 − v4
−1
.
◦
to those on Q by a change of variables,
Now, we can pull back the integrals on Q
j ◦ A−1 , j = 1, 2, to be two
using the transformation S. For example, suppose ϕj = ϕ
basis functions on Q. Then the integral on Q can be calculated as follows:
β(x, y)ϕ1 (x, y)ϕ2 (x, y) dxdy
Q
◦
◦ ◦
◦ ◦
◦ ◦
◦
1 (S(x, y ))ϕ
2 (S(x, y )) | det DB| d x d y.
= ◦ β(B(x, y ))ϕ
Q
5. The extension to three dimensions. We give only a brief remark to extend
the results in sections 2, 3, and 4, to three dimensions. For the sake of simplicity, let
R be a three-dimensional hexahedron, with mj , j = 1, . . . , 6, being the barycenters
of the six faces such that mj and mk are barycenters of opposite faces if j + k = 7.
Analogously to Lemma 2.1, if u ∈ P1 (R), then
u(m1 ) + u(m6 ) = u(m2 ) + u(m5 ) = u(m3 ) + u(m4 ).
Conversely, if uj is a given value at mj , for 1 ≤ j ≤ 6, satisfying u1 + u6 = u2 +
u5 = u3 + u4 , then there is a unique u ∈ P1 (R) such that u(mj ) = uj , 1 ≤ j ≤
637
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
Table 1
Degrees of freedom for Q1 -conforming, P1 -nonconforming, and other nonconforming elements.
Elements
Q1 -conforming element
P1 -NC element
Other NC elements
42
9
9
24
82
49
49
112
162
225
225
480
322
961
961
1984
642
3969
3969
8064
1282
16129
16129
32512
2562
65025
65025
130560
6. This fact therefore leads to the conclusion that the local degrees of freedom for
the three-dimensional nonconforming hexahedral element is four. Indeed, the space
Span{1, x, y, z} serves as the basis for the local nonconforming hexahedral element
space for each hexahedron.
Concerning the global basis, consider a standard decomposition Th of a threedimensional domain Ω into the union of hexahedrons Rj with vertices pk and barycenters ml . At each vertex pk , the global basis function ϕk is then defined analogously
to the two-dimensional case: ϕk |Rj ∈ P1 (Rj ), ϕk (ml ) = 1 if ml is the barycenter of a
face whose vertex contains pk ; ϕk (ml ) = 0 otherwise.
Then extensions of the rest of sections 2, 3, and 4 to three dimensions will be
valid with suitable modifications.
6. Numerical results. In this section we present several numerical results to
compare lowest-order quadrilateral elements which are either conforming or nonconforming. More precisely, six different elements are examined here including the P1 nonconforming quadrilateral element and the standard Q1 -conforming element. We
also test the two rotated Q1 -nonconforming elements introduced by Rannacher and
Turek [15] with the degrees of freedom being the four midpoint values at the midpoints of edges and the four average values over edges. In addition, comparisons are
made with the elements given by Douglas et al. [9], the local basis of which is of the
form Span{1, x, y, θl (x) − θl (y)}, l = 1, 2, where the θl is given by
2 5 4
l = 1,
t − 3t ,
θl (t) =
7 6
4
t2 − 25
6 t + 2 t , l = 2.
The following Dirichlet boundary problem is employed:
−u = f,
Ω,
u = 0,
∂Ω,
with the domain Ω = [0, 1]2 and the exact solution u(x, y) = sin(2πx) sin(2πy)(x3 −
y 4 + x2 y 3 ), the function f being generated.
In every figure the logarithmic errors with base 2 are plotted against the logarithmic values of degrees of freedom again with base 2. With the uniform mesh as in
Figure 5(a), the numerical errors are given in Figure 6. Convergence behaves more
or less in optimal fashion for every element. Notice that the degrees of freedom for
P1 -nonconforming and Q1 -conforming are nearly half of those of other nonconforming
elements, as shown in Table 1.
We observed that the optimal convergence patterns break for nonconforming elements if the nonuniform mesh depicted in Figure 5(b) is used with the standard bilinear reference scheme, since the nonconforming spaces do not contain the linear space
as explained in [1]. In Figure 7 we show the error behaviors for the P1 -nonconforming
element method, using the nonparametric reference scheme introduced in section 4,
638
CHUNJAE PARK AND DONGWOO SHEEN
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
(a) Uniform mesh.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Nonuniform mesh.
Fig. 5. The uniform and nonuniform meshes on Ω.
-2
1
P1 nonconforming
Q1 conforming
Rotated Q1; value DOF case
Rotated Q1; integral DOF case
DSSY; θ1 case
DSSY; θ2 case
-4
0
2
log of L error of ∇ u to the base 2
-8
- 10
-1
-2
-3
2
log of L2 error of u to the base 2
-6
P1 nonconforming
Q conforming
1
Rotated Q1; value DOF case
Rotated Q1; integral DOF case
DSSY; θ case
1
DSSY; θ case
- 12
-4
- 14
-5
- 16
- 18
2
4
6
8
10
12
log of DOF to the base 2
(a) ||u − uh ||L2 (Ω) .
14
16
18
-6
2
4
6
8
10
12
log of DOF to the base 2
14
16
18
(b) ||∇u − ∇uh ||L2 (Ω) .
Fig. 6. L2 (Ω) errors of uh and ∇uh (in logarithmic scale) on the uniform mesh.
and compare them with those for the Q1 -conforming element method with the standard bilinear reference scheme applied. These two cases perform as well as we can
expect, and the convergence rates are drawn in Figure 7. Our nonparametric reference
scheme, which seems to be specific to the P1 -nonconforming quadrilateral element,
does not work for the other known nonconforming quadrilateral elements mentioned
in the paper; hence it does not seem fair to report such results here, some of which
can be found in [14].
Several experiments were performed with the Robin problem. The errors, omitted
here, behave quite similarly to those for the case of Dirichlet problems, as discussed
above. Some reports can be found in [14].
639
P1 -NONCONFORMING QUADRILATERAL ELEMENTS
-2
1
P1 nonconforming with a nonparametric scheme
Q1 conforming with the bilinear reference scheme
-4
0
-6
-1
log of L error of ∇ u to the base 2
-8
- 10
-2
-3
2
log of L2 error of u to the base 2
P nonconforming with a nonparametric scheme
1
Q1 conforming with the bilinear reference scheme
- 12
- 14
-4
-5
- 16
2
4
6
8
10
log of DOF to the base 2
12
14
16
-6
2
(a) ||u − uh ||L2 (Ω) .
4
6
8
10
log of DOF to the base 2
12
14
16
(b) ||∇u − ∇uh ||L2 (Ω) .
Fig. 7. L2 (Ω) errors of uh and ∇uh (in logarithmic scale) on the nonuniform mesh.
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