Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol106.Issue2.1999:
Journal of Computational and Applied Mathematics 106 (1999) 325–344
www.elsevier.nl/locate/cam
Global superconvergence in combinations of
Ritz–Galerkin-FEM for singularity problems (
Zi-Cai Lia; ∗ , Qun Linb , Ning-Ning Yanb
a
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, ROC 80424, Taiwan
b
Institute of Systems Science, Academic Sinica, Beijing 100080, People’s Republic of China
Received 9 February 1998; received in revised form 21 December 1998
Abstract
This paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of
elliptic boundary value problems. The global superconvergence rates O(h2− ) can be achieved by means of the techniques
of Lin and Yan (The Construction and Analysis of High Ecient FEM, Hobei University Publishing, Hobei, 1996) for
di erent coupling strategies, such as the nonconforming constraints, the penalty integrals, and the penalty plus hybrid
integrals, where (¿0) is an arbitrarily small number, and h is the maximal boundary length of quasiuniform rectangles
ij used. A little e ort in computation is paid to conduct a posteriori interpolation of the numerical solutions, uh , only
on the subregion used in nite element methods. This paper also explores an equivalence of superconvergence between
this paper and Z.C. Li, Internat. J. Numer. Methods Eng. 39 (1996) 1839 –1857 and J. Comput. Appl. Math. 81 (1997)
c 1999 Elsevier Science B.V. All rights reserved.
1–17.
MSC: 65N10; 65N30
Keywords: Elliptic equation; Singularity problem; Superconvergence; Combined method; Coupling technique;
Finite-element method; The Ritz–Galerkin method; Penalty method; Hybrid method
1. Introduction
In this paper, the global superconvergence rates of gradient of the error on the entire solution
domain are established by combinations of the Ritz–Galerkin and nite-element methods (simply
written as RGM–FEMs). There exist many reports on superconvergence at speci c points, see [3,13–
15,17] and in particular in the monograph of Wahlbin [16]. The traditional superconvergence in
(
This work was supported by the research grants from the National Science Council of Taiwan under Grants No.
NSC86-2125-M110-009.
∗
Corresponding author.
E-mail address: [email protected] (Z.-C. Li)
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 7 9 - 5
326
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
[3,13–17] is devoted to speci c nodal solutions; this paper is devoted to the global superconvergence
in the entire subdomains for RGM–FEM, based on Lin’s techniques [10 –12] (also see [2]).
For solving singularity problems, optimal convergence O(h) of RGM–FEM is reported in [9];
superconvergence O(h2− ); 0 ¡ ≪ 1, of combinations of the Ritz–Galerkin and nite di erence
methods (RGM–FDM) is proven in [6] for average nodal derivatives, where h is the maximal
boundary length of nite elements or the maximal meshspacing of di erence grids. Note that this
kind of superconvergence is also equivalent to the global superconvergence in this paper.
Let the solution domain S be divided into the singular and regular subdomains S1 and S2 , respectively. Suppose that the regular domain S1 can be partitioned into quasiuniform rectangles, and that
the bilinear elements are chosen as in the nite-element method in S1 . Five di erent combinations are
discussed in this paper, to match the FEM with the Ritz–Galerkin method using singular solutions
that t the solution singularity best. Under the assumption, u ∈ H 3 (S1 ), the global superconvergence
rates can be achieved as
kuI − u˜ h k1; S1 + kuL − u˜ L k1; S2 = O(h2− );
u − 2 u˜ h
+ kuL − u˜ L k1; S2 = O(h2− );
2h
1; S1
0 ¡ ≪ 1;
2
where uI ; u˜ h and 2h
u˜h are the solution interpolant, the numerical solution and a posteriori interpolant
of u˜ h , respectively. k · k1; S1 is the Sobolev norm in space H 1 (S). Here uL and u˜ L are the true and
approximate expansions with L + 1 singular functions, respectively. Five combinations are discussed
in this paper together to provide a deep view of algorithm nature, and their comparisons.
Below, we rst describe ve combinations of the RGM–FEMs in the next section, and then derive
the global convergence rates in Sections 3 and 4 for di erent combinations. In the last section, we
give some comparisons and report some numerical experiments.
2. The combinations of RGM–FEM
Consider the Poisson equation with the Dirichlet boundary condition
@2 u @2 u
− u = −
+ 2
@x2
@y
u| = 0;
(x; y) on
;
!
= f(x; y);
(x; y) in S;
(2.1)
(2.2)
where S is a polygonal domain,
the exterior boundary @S of S, and f smooth enough. Let
the solution domain S be divided by a piecewise straight line 0 into two subdomains S1 and
S2 . The Ritz–Galerkin method is used in S2 , where there may exist a singular point, and the nitedi erence method is used in S1 . For simplicity, the subdomain S1 is again split into small quasiuniform rectangles ij only, where ij = {(x; y); xi 6x6xi+1 ; yj 6y6yj+1 }. This con nes the subdomain
S1 to be rectangles or the “L” shape, and those consisting of rectangles.
In S2 , we assume that the unique solution u can be spanned by
u=
∞
X
i=1
ai i ;
(2.3)
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
327
where ai are the expansion coecients, and i (i =1; 2; : : : ; ∞) are complete and linearly independent
base functions in the sobolev space H 1 (S2 ). { i } may be chosen as analytical and singular functions.
Then the admissible functions of combinations of the RGM–FEM are written as
−
v = v1
in S1 ;
v=
(2.4)
v+ = fL (a˜i ) in S2 ;
P
where v1 is the piecewise bilinear functions on S1 ; fL (ai )= Li=1 ai i and a˜i are unknown coecients
to be sought. If the particular solutions of (2.1) and (2.2) are chosen as i , the total number of
i used will greatly decrease for a given accuracy of solutions. Considering the discontinuity of the
admissible solutions on 0 , i.e.,
v+ 6= v−
on
(2.5)
0;
we de ne another space
H = {v | v ∈ L2 (S); v ∈ H 1 (S1 ) and v ∈ H 1 (S2 )};
where H 1 (S1 ) is the Sobolev space. Let Vh (⊆ H ) denote a nite-dimensional collection of the function
v in (2.4) satisfying (2.2). The combinations of the RGM–FEMs involving integral approximation
on 0 can be expressed by
∀v ∈ Vh ;
aˆh (uh ; v) = f(v);
where
aˆh (u; v) =
ZZ
3u3v ds +
f(v) =
ZZ
ˆ v);
3u3v ds + D(u;
(2.7)
S2
S1
ZZ
(2.6)
fv ds;
(2.8)
S
ˆ v) = Pc
D(u;
h
Zc
0
+
−
−
(u − u )(v − v ) dl −
Zc
@v+
−
+
−
Zc
@u+
0
u−
+
(v+ − v− ) dl
@n
n
v
(u+ − u− ) dl;
@n
n
0
is the outer normal of @S2 , and u− =x|(xi ;yj ) ∈
+
(2.9)
where @=@n on 0
= (u− (xi + hi ; yj ) −
0 ∩(xi +hi ;yj ) ∈ S1
−
u (xi ; yj ))=hi .
In the coupling integrals (2.9), Pc (¿ 0) is the penalty constant, is the penalty power, and two
parameters (¿0) and (¿0) satisfy + = 1 or 0. The rst term on the right-hand side of (2.9)
is called the penalty integral, and the second and third terms are called the hybrid integrals. Four
combinations of (2.6) are obtained from di erent parameters in (2.9). [5 –7].
(I)
(II)
(III)
(IV)
Combination I :
= 0 and = 1;
Combination II :
= 1 and = 0;
Symmetric Combination : = = 21 ;
Penalty Combination :
= = 0:
Besides, a direct continuity constraint at the di erence nodes Zk on
0
is also given in [5 –7, 9]
as
v+ (Zk ) = v− (Zk );
∀Zk ∈
0;
(2.10)
328
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
where the interface di erence nodes Zk are located just on
combination:
I (uhN ; v) = f(v);
where
I (u; v) =
ZZ
0.
We then obtain the nonconforming
∀v ∈ Vh ;
3u3v ds +
(2.11)
ZZ
3u3v ds
(2.12)
S2
S1
and Vh (⊂ H ) is a nite-dimensional collection of functions de ned in (2.4) satisfying both (2.10)
and (2.2).
ˆ v) on 0 are given by using the following integration rules:
The approximate integrals in D(u;
Z
0
dl ≈
=
Zc
dl =
0
N1
X
Zk−1 Zk
k=1
S
6
Z
ˆˆ dl
0
[2(Zk−1 )(Zk−1 ) + (Zk−1 )(Zk ) + (Zk )(Zk−1 ) + 2(Zk )(Zk )]; (2.13)
(k)
(k)
1
ˆ
ˆ are the piecewise
where 0 = Nk=1
0 ;
0 =Zk−1 Zk ; Zk−1 Zk denotes the length of Zk−1 Zk , and and
linear interpolatory functions on 0 . For the interior boundary 0 we have
Z
Z
Z
@u− +
@u− +
@u− +
(v − v− ) dl =
(v − v− ) dy
(v − v− ) d x
@n
@x
@y
0
0
0
≈
Zc
u−
0
x
Zc
u−
(v+ − v− ) dy +
Zc
u−
0
y
(v+ − v− ) d x
(v+ − v− ) dl:
n
Note that the above integration is also suited for the slant-up boundary
elements (see [6,7]).
De ne
=
(2.14)
0
kvk1 = kvk21; S1 + kvk21; S2
1=2
;
|v|1 = (|v|21; S1 + |v|21; S2 )1=2 ;
0,
while using triangular
(2.15)
where kvk1; S1 and |v|1; S1 are the Sobolev norm and semi-norm (see [1,16]). Optimal convergence
rates of numerical solutions kk1 = O(h) have been obtained in [8,9], where = u − uh is error of
the solution. In this paper, we pursue global superconvergence based on the new norms
2
Pc
(2.16)
kvkh = kvk21; S1 + kvk21; S2 + kv+ − v− k0; 0 ;
h
where the norms with discrete summation in S1 are assigned on 0 only
kv+
−
2
v− k0;
0
=
Zc
0
(v+ − v− )2 dl:
(2.17)
Note that the norms de ned in [5,6] also involve the discrete summation in S1 , denoted as k · k1; S1 .
An equivalence between these two norms will be discovered in the last section.
The stability analysis is given in [9] for ve combinations. In this paper, the global superconver2
gence rates, ku − 2h
uh kh = O(h2− ); can be achived by all ve combinations for the quasiuniform
rectangles; detailed proofs are presented by Theorems 3.3 and 4.3 in the continued sections.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
329
Fig. 1. The 2 × 2 rectangles in fashion.
Fig. 2. Partition of the solution domain.
3. The nonconforming combination
Let S1 be partitioned into quasiuniform rectangles in 2 × 2 fashion, see Fig. 1. S is a boundary
layer consisting of ij , to separate S2 and S ∗ in Fig. 2, where
S1 = S ∗ ∪ S ;
S ∗ ∩ S = ∅
such that S ∗ ⊂⊂ S1 . To construct the following interpolant:
uˆ I; L =
−
uI = uI
uˆ I
uL+ = uL
in S ∗ ;
in S ;
in S2 ;
(3.1)
P
where uL = Li=1 a˜i i ; a˜i are the unknown coecients and uI is the piecewise bilinear interpolant of
the true solution on rectangulation of S1 . We choose S with width of just one rectangle ij , and
the function uˆ I in S is also the piecewise bilinear function with the corner values at Pi , de ned by
uˆ I (Pi ) =
u(Pi )
uL+ (Pi )
0;
if Pi ∈ 1 = S ∗ ∩ S ;
if Pi ∈ 0 ;
if Pi ∈ :
(3.2)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Hence,
uˆ I; L ⊂ Vh :
(3.3)
First, let us prove a basic theorem.
Theorem 3.1. There exist the error bounds between the numerical and interpolant solutions; uhN
given in (2:11) and uˆ I; L given in (3:1)
kuhN
Z Z
3(u − uI )3w ds
S1
Z Z
Z Z
3(u − uL )3w ds
3(uI − uˆ I )3w ds +
+
S2
S
Z
@u +
+
(w − w− ) dl ;
@n
1
− uˆ I; L k1 6 C sup
w ∈ Vh kwk1
(3.4)
0
where C is a bounded constant independent of h; L; u and w; and the norm k·k1 is de ned in (2:15).
Proof. For the true solution u, we have
I (u − uhN ; v) =
Z
0
@u +
(v − v− ) dl;
@n
∀v ∈ Vh :
(3.5)
Let w = uhN − uˆ I; L ∈ Vh . Then from de nition (3.1)
|w|21 6 I (uhN − uˆ I; L ; w) = I (u − uˆ I; L ; w) −
=
ZZ
S∗
−
Z
0
3(u − uI )3w ds +
@u +
(w − w− ) dl:
@n
ZZ
S
Z
0
@u +
(w − w− ) dl
@n
3(u − uˆ I )3w ds +
ZZ
S2
3(u − uL+ )3w ds
(3.6)
Since u − uˆ I = u − uI + uI − uˆ I in S , we have
ZZ
S∗
=
3(u − uI )3w ds +
ZZ
S1
ZZ
S
3(u − uI )3w ds +
3(u − uˆ I )3w ds
ZZ
S
3(uI − uˆ z )3w ds:
(3.7)
Desired results (3.4) are obtained from (3.5) – (3.7) and the Poincare–Friedricks inequality
kwk1 6C|w|1
with a bounded constant independent of w. This completes the proof of Theorem 3.1.
Below, we will derive the bounds of all terms on the right-hand side of (3.4). We then have the
following lemma.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Lemma 3.1. Let the rectangles
ij
331
be quasiuniform; then for w ∈ Vh
Z Z
3(uI − uˆ I )3w ds 6Ch−(1=2) kRL k0; 0 kwk1 :
(3.8)
S
Proof. From the Schwarz inequality,
Z Z
3(uI − uˆ I )3w ds 6|uI − uˆ I |1; S |w|1 :
(3.9)
S
Denoting = uI − uˆ I , we obtain from the inverse estimates
|uI − uˆ I |1; S = | |1; S 6Ch−1 k k0; S :
(3.10)
Note that is the piecewise bilinear function on S and the fact that (Zi ) = 0 for all element
nodes Zi 6∈ 0 in S , the maximal values of | | along any vertical and horizontal lines in S are just
located on 0 , i.e.,
k k20; S =
Z
S
Moreover, since
2 ds6Ch
Z
0
2 dl = ChkuI − uˆ I k20; 0 :
uI − uˆ I = uI − uˆ L = Rˆ L on 0 :
Since Rˆ L is the piecewise linear interpolant of the remainder RL (=u − uL ) on
triangle inequality
kuI − uˆ I k0; 0 = kRˆ L k0; 0
6 kRˆ L k0; 0 + h kRL − Rˆ L k;
(3.11)
0,
we obtain from the
0
6 kRL k0; 0 + Ch kRL k; 0 6CkRL k0; 0 ;
where ¿ 0 and → 0. Hence, we have from (3.10) and (3.11)
|uI − uˆ I |1; S = | |1; S 6Ch−1 h1=2 kRL k0; 0 = Ch−1=2 kRL k0; 0 :
(3.12)
(3.13)
Combining (3.9) and (3.13) yields (3.8). This completes the proof of Lemma 3.1.
Lemma 3.2. Let u ∈ H 3 (S1 ); then
Z Z
6Ch2 |u|3; S kwk1 ;
3(u
−
u
)3w
ds
I
1
S1
∀w ∈ Vh ;
(3.14)
where |u|3; S1 is the Sobolev seminorm over the space H 3 (S1 ).
Proof. We may follow Lin [10, p. 5], to prove (3.14), but rather provide a straightforward argument
below. Denote the second-order interpolant of u by
uI(2) =
2
X
ij xi y j = uI + 20 x2 + 02 y2 ;
(3.15)
i+j=0
where ij are coecients. By virtue of the following equality (shown later):
ZZ
ij
3(u − uI )3w ds =
ZZ
ij
3(u − uI(2) )3w ds;
(3.16)
332
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
we obtain
X Z Z
Z Z
3(u − uI )3w ds =
S1
X Z Z
=
ij
ij
ij
3(u −
3(u − uI )3w ds
ij
Z Z
(2)
3(u − uI )3w ds
=
uI(2) )3w ds
S1
6|u − uI(2) |1; S1 |w|1; S1 6Ch2 |u|3; S1 |w|1 :
This is desired result (3.14).
Now we show (3.16). Denote
ˆ = {(x; y); − 21 6x6 12 ; − 21 6y6 12 }:
Then by the linear transformations
xˆ = − 21 + (x − xi )=hi ;
the integrals on
ZZ
ij
(u −
yˆ = − 21 + (y − yj )=kj ;
(3.17)
ij
uI(2) )x wx
ZZ
h i kj
cxˆ d sˆ
ds = 2
(uˆ − ubI 2 )xˆ w
ˆ
hi
ZZ
kj
cxˆ d sˆ
(uˆ − ubI − ˆ20 xˆ2 + ˆ02 yˆ 2 )xˆ w
=
ˆ
hi
ZZ
ZZ
kj
cxˆ d xˆ
cxˆ d sˆ − 2 ˆ20
(uˆ − ubI )xˆ w
xˆ w
=
ˆ
ˆ
hi
ZZ
kj
cxˆ d s;
(uˆ − ubI )xˆ w
ˆ
=
ˆ
hi
(3.18)
where ˆ20 and ˆ02 are also constant. In the last equality of (3.18), we have used the following
equation for any bilinear functions wˆ on ˆ :
ZZ
ˆ
cxˆ d sˆ =
xˆw
Z
1=2
xˆ d xˆ
−1=2
!
Z
1=2
!
wˆ xˆ d yˆ = 0:
−1=2
(3.19)
By the inverse transformtion of (3.17),
ZZ
ˆ
cxˆ d sˆ =
(uˆ − uˆ I )xˆ w
hi
kj
ZZ
ij
(u − uI )x wx ds:
(3.20)
Combining (3.18) and (3.20) yields
ZZ
ij
(u − uI(2) )x wx ds =
Similarly, we have
ZZ
ij
(u −
uI(2) )y wy
ds =
ZZ
ZZ
ij
(u − uI )x wx ds:
(3.21)
(u − uI )y wy ds:
(3.22)
ij
Adding (3.21) and (3.22) leads to (3.16). This completes the proof of Lemma 3.2.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
333
Lemma 3.3. Assume u ∈ H 2 (S) and that there exists a bounded power (¿ 0) independent of
L; h; v such that
|v+ |l; 0 6CLl kv+ k0; 0 ;
where |v+ |l;
0
l = 1; 2; ∀v ∈ Vh ;
(3.23)
is the Sobolev seminorm over the space H l ( 0 ). Then for ∀w ∈ Vh
Z
@u +
−
6C(hL )2
@u
−
w
)
dl
(w
@n
@n
0;
0
0
kwk1 :
(3.24)
Proof. From the Schwarz inequality, we have
Z
@u
@u +
−
6
−
w
)
dl
(w
@n
@n
0;
0
0
kw+ − w− k0; 0 :
(3.25)
Since the continuity of w at the nodes on 0 ; w− is, in fact, regarded as the piecewise linear
interpolant of w+ . Hence we obtain from assumption (3.23),
kw+ − w− k0;
0
= kw+ − wˆ + k0; 0 6Ch2 kw+ k2;
0
6 Ch2 L2 kw+ k0; 0 6C(hL )2 kw+ k1; S2 6C(hL )2 kwk1 :
(3.26)
Desired results (3.24) follows from (3.25) and (3.26).
Now we provide an important theorem.
Theorem 3.2. Let u ∈ H 3 (S1 ) and (3:23) be given. Then there exist the error bounds between uhN
and uI; L ;
kuhN
(
2
− uˆ I; L k1 61 = C h |u|3; S1 + kRL k1; S2
Proof. We have
@u
+ (hL )
@n
2
0;
+h
0
Z Z
3(u − uL )3w ds 6ku − uL k1; S2 kwk1; S2 6kRL k1; S2 kwk1 :
−(1=2)
kRL k0;
0
)
(3.27)
(3.28)
S2
The bounds of other terms in the right-hand side of (3.4) are given in Lemmas 3.1–3.3, to lead to
(3.27) directly. This completes the proof of Theorem 3.2.
Theorem 3.2 provides the global errors between uhN and uˆ h; L only, but not the true global errors between uhN and u. The key idea in raising convergence rates is to constitute an a posteriori interpolant,
p uhN , based on uhN . The global errors ku − p uhN k1 can attain the same global superconvergence.
The solution of (2.11) is denoted as
uhN
=
(uhN )− in S1 ;
(uLN )+ in S2 :
(3.29)
Based on Lin’s techniques [10] for the FEM solutions (uhN )− in S1 , the bi-quadratic interpolant,
2
(uhN )− , can be formed in 2 × 2 neighbouring rectangles shown in Fig. 1. The a posteriori process
2h
334
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
is made only in S1 . Denote
p v =
2 −
v
2h
+
v
in S1 ; ∀v ∈ Vh ;
in S2 ; ∀v ∈ Vh :
(3.30)
We obtain the following theorem.
Theorem 3.3. Let all conditions in Theorem 3:2 hold. Then there exist the error bounds
ku − p uhN k1 6C1 ;
(3.31)
where 1 is given in Theorem 3:2.
Proof. By noting the interpolation operation p in (3.30) we have
ku − p uhN k21 6 ku − p uˆ h; L k21 + kp (uˆ h;L − uhN )k21
2
2
(uˆ I − uI )k21; S ∪S∗
uI k21; S1 + k2h
= ku − 2h
+ kp (uˆ h;L − uhN )k21; S1 + kRL k21; S2 ;
(3.32)
where S∗ ∈ S ∗ is a continued extension of S due to the 2 × 2 rectangles.
Based on the equivalence of nite-dimensional norms, we have
2
|2h
v|l;
ij
6C|v|l;
ij
;
to lead to
2
vkl; S1 6Ckvkl; S1 ;
k2h
∀v ∈ Vh :
(3.33)
Also from Theorem 3.2,
kp (uˆ h; L − uhN )k1; S1 + kRL k1; S2 6 C(kuˆ h; L − uhN k1; S1 + kRL k1; S2 )
6 Ckuˆ h; L − uhN k1 6C1 :
(3.34)
Let v = uˆ I − uI . We then have from (3.13)
2
k2h
(uˆ I − uI )k1; S ∪S∗ 6Ckuˆ I − uI k1; S ∪S∗ = Ckuˆ I − uI k1; S 6Ch−(1=2) kRL k0; 0 :
(3.35)
2
uI is the bi-quadratic interpolant of u, then we have
Since 2h
2
ku − 2h
uI k1; S1 6Ch2 |u|3; S1 :
(3.36)
Combining (3.32), (3.34) – (3.36) yields the desired results (3.31). This completes the proof of
Theorem 3.3.
Based on Theorems 3.2 and 3.3 we obtain the following corollary.
Corollary 3.1. Let the conditions in Theorem 3:1 hold. Assume the expansion term L is chosen
such that
|RL |1; S2 = O(h2 );
kRL k0; 0 = O(h5=2 ):
(3.37)
335
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Then
kuhN − uˆ I; L k1 = O(h2 ) + O(h2 L2 )
(3.38)
ku − p uhN k1 = O(h2 ) + O(h2 L2 ):
(3.39)
and
Also if such L is so small as to satisfy (see [4])
L = O(|ln h|);
(3.40)
kuhN − uˆ I; L k1 = O(h2− )
(3.41)
ku − p uhN k1 = O(h2− );
(3.42)
then
and
where → 0 as h → 0.
Corollary 3.1 displays the global superconvergence in the entire solution domain S. In particular
in S1 .
2 N
ku − 2h
uh k1; S1 = O(h2− ):
2
In fact, the a posteriori quadratic interpolation 2h
on the solution uhN costs a little more computation. The analytic results can be applied directly to Motz’s problem, see Section 6.
4. The penalty combination
In this section, we will derive the error bounds for the solution uhP by (2.6) as = = 0. We rst
give a basic theorem in the norm k·kh de ned in (2.16).
Theorem 4.1. There exist the error bounds between the solution uhP and uˆ I; L de ned in (3:1);
kuhP
Z Z
3(u − uI )3w ds
S1
Z Z
Z Z
3(u − uL )3w ds
3(uI − uˆ I )3w ds +
+
S2
S
Z
@u
+
(w+ − w− ) dl ;
@n
1
− uI; L kh 6 C sup
w ∈ Vh kwkh
(4.1)
0
where C is a bounded constant independent of h; L; u and w.
Proof. By the following arguments similarly in Theorem 3.1, we also have:
aˆh (u − uhP ; v) =
Z
0
@u +
(v − v− ) dl;
@n
∀v ∈ Vh :
(4.2)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Then for w = uhP − uˆ I; L ∈ Vh ;
C0 kwk2h 6aˆh (uhP
− uˆ I; L ; w) = aˆh (u − uˆ I; L ; w) −
Z
@u +
(w − w− ) dl;
@n
0
(4.3)
where C0 (¿ 0) is constant independent of h; L; u and w. We have
aˆh (u − uˆ I; L ; w) =
=
ZZ
S1
3(u − uˆ I; L )3w ds +
S1
3(u − uˆ I; L )3w ds +
ZZ
+
ZZ
S2
ZZ
S2
ZZ
ˆ − uˆ I; L; w)
3(u − uˆ I; L )3w ds + D(u
S△
3(uˆ I − uI )3w ds
ˆ − uˆ I; L ; w):
3(u − u L )3w ds + D(u
(4.4)
The penalty term
Z
c
ˆ − u˜ I; L; w) = Pc (uL+ − uˆ−I )(w+ − w− ) dl = 0;
D(u
h 0
(4.5)
because of (3.2) and integration rule (2.13). Combining (4.4) and (4.5) leads to the desired results
(4.1).
The bounds in Theorem 4.1 are analogous to those in Theorem 3.1, but the key di erence is that
the space Vh does not o er the explicit relation between w+ and w− along 0 . Hence the bounds of
the last term in (4.1) should be estimated in di erent manner from the above. We can prove the
following lemma.
Lemma 4.1. Let (3:23) be given; then for v ∈ Vh ;
kv+ − v− k0; 0 6C{kv+ − v− k0; 0 + (hL )2 kvk1; S2 }
(4.6)
kv+ − v− k0; 0 6C{kv+ − v− k0; 0 + (hL )2 kvk1; S2 }:
(4.7)
and
Lemma 4.2. Let (3:23) be given then for v ∈ Vh ;
Z
@u +
−
6C
@u
(w
−
w
)
dl
@n
@n
0
(h=2 + (hL )2 )kwkh :
0;
0
Proof. We have from Schwarz’s inequality and (4.6)
Z
@u
@u +
−
+
−
(w − w ) dl 6
@n
kw − w k0; 0
0; 0
0 @n
@u
+
−
6C
@n
{kw − w k0;
0;
0
0
+ (hL )2 kwk1; S2 }
(4.8)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
@u
(=2)
6C
kwkh + (hL )2 kwk1; S2 }
@n
{h
0; 0
@u
=2
2
6C
@n
{h + (hL ) }kwkh :
0;
(4.9)
0
This completes the proof of Lemma 4.2.
Similarly, we obtain the following theorem from Lemmas 4.2, 3.1, 3.2 and (3:28).
Theorem 4.2. Let all conditions in Theorem 3:1 hold. Then there exist the error bounds of uhP
kuhP − uˆ I; L kh = 2
(
2
6 C h |u|3; S1 + kRL k1; S2
@u
+
@n
0;
(h
=2
2
+ (hL ) ) + h
0
−1=2
kRL k0;
0
)
:
(4.10)
Moreover for the a posteriori interpolant p uhP ; we also have the following theorem.
Theorem 4.3. Let all conditions in Theorem 3:1 hold. Then
u − p uP
6C2 :
h h
(4.11)
Proof. We have
u − p uP
6ku − p uˆ I; L kh +
p (uˆ I; L − uP )
:
h h
h
h
(4.12)
Since interpolation rule (3.30) gives
2 − −
v ) = (v+ )
(2h
then
(p v)+ = (p v)−
This leads to
Also,
on Pi ∈
on Pi ∈
(4.13)
0;
0:
ku − p u I; L k0; 0 = 0:
(4.14)
kp (uˆ I; L − uhP )k0; 0 = kuˆ I; L − uhP k0; 0 :
(4.15)
We then have
)
(
1=2
P
c
2
ku − p uˆ I; L kh 6 C
u − 2h uˆ I; L
1; S1 + ku − uL k1; S2 +
ku − p uˆ I; L k0; 0
h
2
uˆ I; L
1; S1 + kRL k1; S2 }
6 C{
u − 2h
6 C{
u − 2 uI
+
2 (uˆ I; L − uI )
2h
1; S1
2h
∗
1; S ∪S
+ kRL k1; S2 }:
(4.16)
From (3.35) and (3.36)
ku − p uˆ I; L kh 6C{h2 |u|3; S1 + h−1=2 kRL k0; 0 + kRL k1; S2 }:
(4.17)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
On the other hand, we obtain from (4.15) and Theorem 4.2
kp (uˆ I; L −
uhp )kh
(
)
1=2
2
P
c
p
p
p
6 C
2h (uˆ I; L − uh )
1; S1 + kuI; L − uh k1; S2 +
kp (uˆ I; L − uh )k0; 0
h
(
6 C kuˆ I; L −
uhp k1; S1
+
kuL+
−
uhp k1; S2
+
Pc
h
1=2
kuˆ I; L −
6 Ckuˆ I; L − uhp kh 6C2 :
uhp k0;
0
)
(4.18)
Combining (4.17) and (4.18) leads to (4.11).
Corollary 4.1. Let all conditions in Theorem 3:1 hold. Also assume Eq. (3:37) be given. Then
ku − p uhp kh = O(h2 ) + O(h2 L2 ) + O(h=2 ):
(4.19)
Moreover if ¿4 and (3:40) are given;
ku − p uhp kh = O(h2− );
0 ¡ ≪ 1:
(4.20)
The penalty combination leads to the same global convergence rates as the nonconforming combination does because
ku − p uhp k1 6ku − p uhp kh = O(h2− ):
(4.21)
Moreover, we have for ¿4
ku − p uhp k0; 0 6Ch=2 ku − p uhp kh = O(h2+(=2)− ):
(4.22)
Finally let us derive the relation of the penalty coupling to the nonconforming constraints (2.10).
Corollary 4.2. Let all conditions in Corollary 4:1 hold. Then when ¿4 the average jumps of the
solution v = p uhp at the element nodes Zi ∈ 0 have the convergence rates
E(v+ − v− ) =
N
1 X
|v+ (Zi ) − v− (Zi )| = O(h2+=2− );
N + 1 i=0
where N is the number of element nodes on
(4.23)
0.
Proof. Denote rst the trapezoidal rule
Zf
0
2
v dl =
N
X
Zk−1 Zk
k=1
2
[v2 (Zk−1 ) + v2 (Zk )]:
Since 21 (x2 + y2 )6(x2 + xy + y2 ), we have from (2.13),
Zf
0
v2 dl63
N
X
Zk−1 Zk
k=1
3
[v2 (Zk−1 ) + v(Zk−1 )v(Zk ) + v2 (Zk )] = 3
Zc
0
v2 dl:
(4.24)
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
339
Since the rectangles are quasiuniform, we obtain from the Schwarz inequality
N
X
v(Zk )
N +1
k=0
!2
6
N
X
k=0
=C
N
X
Zk−1 Zk 2
1
v2 (Zk )6C
[v (Zk−1 ) + v2 (Zk )]
N +1
2
k=1
Zf
2
v dl63C
Let v =
and w = u −
E(v+ − v− ) =
2
v2 dl = 3Ckvk0; 0 :
0
0
p uhp
Zc
p uhp
N
1 X
(p up )+ − (p up )− 6Ck(p up )+ − (p up )− k
h Zi
h Zi
h
h
0;
N + 1 i=0
0
= Ckw+ − w− k0; 0 6Ch=2 kwkh :
When ¿4,
E(v+ − v− )6Ch2+(=2)−
from Corollary 4.1. This completes the proof of Corollary 4.2.
Corollary 4.2 displays an interesting fact that the average jumps between (p uhp )+ and (p uhp )− are
O(h2+(=2)− ). Hence when → ∞, the penalty combination leads to the nonconforming combination.
Note that the large values of (¿4) do not incur the reduced convergence rates.
Remark. We may derive similarly the same global superconvergence O(h2− ) for Combinations I,
II and symmetric combination under the condition ¿2. Although their algorithms are more complicated, using smaller values of will lead to better stability.
5. Comparisons and computations
Now, let us clarify the relation between [5,6] and this paper, provide some numerical experiments
and make some remarks.
In [5,6], the semi-norm in discrete summation is de ned as
|v|1; S1 =
(
X ZZ
d
ij
ij
(vx2
+
vy2 ) ds
)1=2
(5.1)
;
where
ZZ
d
vx2 ds =
hi kj 2
(v (A) + vx2 (B));
2 x
vy2 ds =
h i kj 2
(v (C) + vy2 (D))
2 y
ij
ZZ
d
ij
and A; B; C and D are the mid-points of edges of
lemma.
(5.2)
ij ,
shown in Fig. 3. Then we have the following
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Fig. 3. The rectangle
ij .
Lemma 5.1. Let
u ∈ C 3 (S1 )
(5.3)
and assume the numerical solutions uh ∈ Vh have the errors
|u − uh |1; S1 = O(h2− );
06 ¡ 1:
(5.4)
Then the global errors between uh and the solution interpolant uI have also the error bounds
|uI − uh |1; S1 = O(h2− ):
(5.5)
Proof. For the piecewise bilinear functions, v ∈ Vh , we have
|v|21; S1 =
where
ZZ
XZ Z
ij
(vx2 + vy2 ) ds;
ij
h i kj 2
[vx (A) + vx2 (B) + vx (A)vx (B)];
3
ij
ZZ
h i kj 2
vy2 ds =
[v (C) + vy2 (D) + vy (C)vy (D)]:
3 y
ij
vx2 ds =
Since x2 + y2 + xy6 23 (x2 + y2 ), we obtain for v ∈ Vh ,
ZZ
vx2 ds6
ij
Hence
Zd
Z
h i kj 2
[vx (A) + vx2 (B)] =
2
|v|1; S1 6|v|1; S1 ;
We then have
Also,
vx2 ds:
ij
∀v ∈ Vh :
(5.6)
|uI − uh |1; S 6|uI − uh |1; S1 = |u − uh |1; S1 + |u − uI |1; S1 :
(5.7)
|u − uI |1; S1 = Ch2 |u|3; ∞; S1 = O(h2 );
(5.8)
where |u|3; ∞; S1 is also the Sobolev p-norm with p = ∞. Desired results (5.5) follow (5.4), (5:7)
and (5:8). This completes the proof of Lemma 5.1.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
341
Fig. 4. Partition of Motz’s problem with MS = 2.
Lemma 5.2. Let (5:3) and
|uI − uh |1; S1 = O(h2− );
06 ¡ 1
be given. Then
|u − uh |1; S1 = O(h2− ):
Proof. Similarly, from the arguments in Lemma 5.1 we have
√
|uI − uh |1; S1 6 3|uI − uh |1; S1 :
Then
|u − uh |1; S1 6|u − uI |1; S1 + |uI − uh |1; S1 6O(h2 ) +
=O(h2 ) + O(h2− ) = O(h2− ):
√
3|uI − uh |1; S1
(5.9)
Lemmas 5.1 and 5.2 imply that under assumption (5.3), the superconvergence in this paper and
in [5,6] is equivalent to each other. We then conclude that the order O(h2− ) also holds for superconvergence of the solutions of RGM–FEM in this paper, to the average nodal derivatives at the
edge mid-points of ij , so does for the maximal nodal derivatives in majority.
On the other hand, based on Lemma 5.1, the solutions in [6] by the combinations of Ritz–Galerkin
and nite-di erence methods also have the global superconvergence as (3.41) and (3.42). However,
the proofs in this paper using Lin’s techniques are simpler so as to be extend to other kinds of
singularity problems, i.e., in biharmonic equations and elastic plates with cracks. Details appear
elsewhere. Also note that the assumption u ∈ H 3 (S1 ) in Theorem 3.2 is weaker than (5.3).
In this section, numerical experiments are also carried out to con rm the global superconvergence
O(h2− ) made in Section 3 by the nonconforming combination. Other kinds of combinations can
also be veri ed by numerical experiments. Let us consider the typical Motz problem (see Fig. 4):
@ 2 u @2 u
u = 2 + 2 = 0 in S;
(5.10)
@x
@y
u|x¡0∧y=0 = 0;
@u
@y
y=1
@u
=
@y
u|x=1 = 500;
x¿0∧y=0
@u
=
@x
(5.11)
= 0;
x=−1
(5.12)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Table 1
Errors norms and condition number of nonconforming combination of RGM–FEM
Divisions
MS = 1
L+1=3
MS = 2
L+1=4
MS = 3
L+1=5
MS = 4
L+1=5
MS = 6
L+1=6
MS = 8
L+1=6
Max
kk0; S
kkh
||1
kk1
||1
kk1
2
ku − 2h
uh k1
8.10
3.77
42.8
15.6
15.9
12.6
13.0
–
3.23
1.29
22.4
9.66
9.75
8.72
8.82
6.90
33
1.33
0.559
14.3
4.19
4.22
3.93
3.97
–
82
0.779
0.315
10.6
2.33
2.35
2.22
2.23
1.83
163
0.358
0.141
6.97
1.04
1.05
1.01
1.02
0.834
462
0.214
0.0794
5.20
0.583
0.587
0.567
0.572
0.470
1000
Con.
8
where S is a rectangle (−16x61; 06y61). The origin (0; 0) is a singular point with the solution
behaviour u = O(r (1=2) ) as r → 0 due to the intersection of the Neumann and Dirichlet conditions.
Divide S by 0 into S2 and S1 . The subdomain S2 is chosen as a smaller rectangle (− 21 6x6 21 ;
06y6 12 ). Also the subdomain S1 is again split into uniform rectangular elements shown in Fig. 4.
The admissible functions are chosen as
−
v = v1 ;
L
X
v=
+
v
=
D˜ l r l+(1=2) cos(l + 21 );
(5.13)
l=0
where D˜ l are unknown coecients, and (r; ) are the polar coordinates with origin (0,0).
Let MS denote the di erence division number along DB. Based on the good matching between
L + 1 (the total number of basis functions used) and MS given in [4], we will choose
MS = 2 and L + 1 = 4;
MS = 3; 4 and L + 1 = 5;
MS = 6; 8 and L + 1 = 6:
(5.14)
Numerical solutions are conducted by the nonconforming combinations of RGM–FEM; and their
error norms and the coecients are provided in Tables 1 and 2. “Con.” in Table 1 denotes the
condition number of the associated matrix resulting from the nonconforming combination, and other
error norms are de ned by
kk0; S =
Z Z
S
2
2
ds
1=2
;
||1 = (||1; S1 + ||21; S1 )1=2 ;
max = max ||;
S
2
kk1 = (kk1; S1 + kk21; S1 )1=2 ;
||1 = (|uh − uI |21; S1 + |uL − u˜ L |21; S2 )1=2 ;
kk1 = (kuh − uI k21; S1 + kuL − u˜ L k21; S2 )1=2 ;
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Table 2
Approximate coecients by nonconforming combination of RGM–FEM
Coe s.
D˜ 3
D˜ 4
D˜ 5
–
–
–
13.759
−23:291
–
–
86.998
15.677
−15:336
2.936
–
400.847
87.278
16.350
−12:169
2.199
–
D˜ 2
D˜ 0
D˜ 1
403.724
81.257
2.670
399.364
86.172
400.593
MS = 1
L+1=3
MS = 2
L+1=4
MS = 3
L+1=5
MS = 4
L+1=5
MS = 6
L+1=6
MS = 8
L+1=6
401.023
87.585
16.773
−9:898
1.743
1.080
401.085
87.616
16.962
−9:097
1.601
0.740
True
401.162
87.656
17.238
−8:071
1.440
0.331
where = u − uhN and u˜ h = u˜ L in S2 . It is easy to see from the data in Tables 1 and 2 that
kkh = O(h);
||1 = O(h2− );
kk0; S = O(h2 );
max = O(h2− );
kk1 = O(h2− );
(5.15)
(5.16)
2
uh k1 = O(h2− );
ku − 2h
(5.17)
||1 = O(h2− );
(5.18)
kk1 = O(h2− );
|D0 − D˜ 0 | = O(h2 );
(5.19)
where D0 and D˜ 0 are the true and approximate coecients, respectively. In Table 1, the data for
2
ku − 2h
uh k1 are not available when using odd MS = 1; 3. Note that Eqs. (5.15) and (5.19) coincide
with the optimal results in [4]; Eqs. (5.16), (5.17) and (5:18) verify perfectly the analysis in Section 3
and this section, respectively.
Finally let us make a few remarks.
1. The rectangular elements are discussed on this paper. In fact, when the uniform right triangle
elements with the interior angle =4 are chosen, only the global superconvergence rates O(h3=2 ) can
be gained, based on the analysis on this paper and Lin and Yan’s [12]. Once the solution domain
is not very complicated, the rectangles and such simple triangles can be employed. Both global and
nodal superconvergence can be achieved simultaneously. Note that for the global superconvergence,
the a posteriori interpolant on the numerical solution costs a little more computation e ort.
2. The nonconforming combination will deal with the unknown constraints (2.10). Instead, we
may solicit other combinations. Corollary 4.2 implies that when is large, the penalty combination
approaches, indeed, the nonconforming combination. In fact, such a large weight technique has
already been applied in engineering computations. Among ve combinations, the nonconforming
combination is still basic.
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
3. The di erent coupling strategies are studied in [7,9] for Ritz–Galerkin–FEM and in [5,6] for
Ritz–Galerkin–FDM. The optimal convergence O(h) and the nodal superconvergence O(h2− ) have
been proven. In this paper the global superconvergence O(h2− ) has been exploited together. Note
that the high superconvergence of combinations yields the higher accuracy of both uh in S1 and the
coecients ai of the solution in S2 . The leading coecient a1 is, in fact, the stress intensity factor
at the singularity, which has important application in fracture mechanics.
Acknowledgements
The authors are grateful to the referee for his=her valuable comments and careful reading.
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www.elsevier.nl/locate/cam
Global superconvergence in combinations of
Ritz–Galerkin-FEM for singularity problems (
Zi-Cai Lia; ∗ , Qun Linb , Ning-Ning Yanb
a
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, ROC 80424, Taiwan
b
Institute of Systems Science, Academic Sinica, Beijing 100080, People’s Republic of China
Received 9 February 1998; received in revised form 21 December 1998
Abstract
This paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of
elliptic boundary value problems. The global superconvergence rates O(h2− ) can be achieved by means of the techniques
of Lin and Yan (The Construction and Analysis of High Ecient FEM, Hobei University Publishing, Hobei, 1996) for
di erent coupling strategies, such as the nonconforming constraints, the penalty integrals, and the penalty plus hybrid
integrals, where (¿0) is an arbitrarily small number, and h is the maximal boundary length of quasiuniform rectangles
ij used. A little e ort in computation is paid to conduct a posteriori interpolation of the numerical solutions, uh , only
on the subregion used in nite element methods. This paper also explores an equivalence of superconvergence between
this paper and Z.C. Li, Internat. J. Numer. Methods Eng. 39 (1996) 1839 –1857 and J. Comput. Appl. Math. 81 (1997)
c 1999 Elsevier Science B.V. All rights reserved.
1–17.
MSC: 65N10; 65N30
Keywords: Elliptic equation; Singularity problem; Superconvergence; Combined method; Coupling technique;
Finite-element method; The Ritz–Galerkin method; Penalty method; Hybrid method
1. Introduction
In this paper, the global superconvergence rates of gradient of the error on the entire solution
domain are established by combinations of the Ritz–Galerkin and nite-element methods (simply
written as RGM–FEMs). There exist many reports on superconvergence at speci c points, see [3,13–
15,17] and in particular in the monograph of Wahlbin [16]. The traditional superconvergence in
(
This work was supported by the research grants from the National Science Council of Taiwan under Grants No.
NSC86-2125-M110-009.
∗
Corresponding author.
E-mail address: [email protected] (Z.-C. Li)
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 7 9 - 5
326
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
[3,13–17] is devoted to speci c nodal solutions; this paper is devoted to the global superconvergence
in the entire subdomains for RGM–FEM, based on Lin’s techniques [10 –12] (also see [2]).
For solving singularity problems, optimal convergence O(h) of RGM–FEM is reported in [9];
superconvergence O(h2− ); 0 ¡ ≪ 1, of combinations of the Ritz–Galerkin and nite di erence
methods (RGM–FDM) is proven in [6] for average nodal derivatives, where h is the maximal
boundary length of nite elements or the maximal meshspacing of di erence grids. Note that this
kind of superconvergence is also equivalent to the global superconvergence in this paper.
Let the solution domain S be divided into the singular and regular subdomains S1 and S2 , respectively. Suppose that the regular domain S1 can be partitioned into quasiuniform rectangles, and that
the bilinear elements are chosen as in the nite-element method in S1 . Five di erent combinations are
discussed in this paper, to match the FEM with the Ritz–Galerkin method using singular solutions
that t the solution singularity best. Under the assumption, u ∈ H 3 (S1 ), the global superconvergence
rates can be achieved as
kuI − u˜ h k1; S1 + kuL − u˜ L k1; S2 = O(h2− );
u − 2 u˜ h
+ kuL − u˜ L k1; S2 = O(h2− );
2h
1; S1
0 ¡ ≪ 1;
2
where uI ; u˜ h and 2h
u˜h are the solution interpolant, the numerical solution and a posteriori interpolant
of u˜ h , respectively. k · k1; S1 is the Sobolev norm in space H 1 (S). Here uL and u˜ L are the true and
approximate expansions with L + 1 singular functions, respectively. Five combinations are discussed
in this paper together to provide a deep view of algorithm nature, and their comparisons.
Below, we rst describe ve combinations of the RGM–FEMs in the next section, and then derive
the global convergence rates in Sections 3 and 4 for di erent combinations. In the last section, we
give some comparisons and report some numerical experiments.
2. The combinations of RGM–FEM
Consider the Poisson equation with the Dirichlet boundary condition
@2 u @2 u
− u = −
+ 2
@x2
@y
u| = 0;
(x; y) on
;
!
= f(x; y);
(x; y) in S;
(2.1)
(2.2)
where S is a polygonal domain,
the exterior boundary @S of S, and f smooth enough. Let
the solution domain S be divided by a piecewise straight line 0 into two subdomains S1 and
S2 . The Ritz–Galerkin method is used in S2 , where there may exist a singular point, and the nitedi erence method is used in S1 . For simplicity, the subdomain S1 is again split into small quasiuniform rectangles ij only, where ij = {(x; y); xi 6x6xi+1 ; yj 6y6yj+1 }. This con nes the subdomain
S1 to be rectangles or the “L” shape, and those consisting of rectangles.
In S2 , we assume that the unique solution u can be spanned by
u=
∞
X
i=1
ai i ;
(2.3)
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
327
where ai are the expansion coecients, and i (i =1; 2; : : : ; ∞) are complete and linearly independent
base functions in the sobolev space H 1 (S2 ). { i } may be chosen as analytical and singular functions.
Then the admissible functions of combinations of the RGM–FEM are written as
−
v = v1
in S1 ;
v=
(2.4)
v+ = fL (a˜i ) in S2 ;
P
where v1 is the piecewise bilinear functions on S1 ; fL (ai )= Li=1 ai i and a˜i are unknown coecients
to be sought. If the particular solutions of (2.1) and (2.2) are chosen as i , the total number of
i used will greatly decrease for a given accuracy of solutions. Considering the discontinuity of the
admissible solutions on 0 , i.e.,
v+ 6= v−
on
(2.5)
0;
we de ne another space
H = {v | v ∈ L2 (S); v ∈ H 1 (S1 ) and v ∈ H 1 (S2 )};
where H 1 (S1 ) is the Sobolev space. Let Vh (⊆ H ) denote a nite-dimensional collection of the function
v in (2.4) satisfying (2.2). The combinations of the RGM–FEMs involving integral approximation
on 0 can be expressed by
∀v ∈ Vh ;
aˆh (uh ; v) = f(v);
where
aˆh (u; v) =
ZZ
3u3v ds +
f(v) =
ZZ
ˆ v);
3u3v ds + D(u;
(2.7)
S2
S1
ZZ
(2.6)
fv ds;
(2.8)
S
ˆ v) = Pc
D(u;
h
Zc
0
+
−
−
(u − u )(v − v ) dl −
Zc
@v+
−
+
−
Zc
@u+
0
u−
+
(v+ − v− ) dl
@n
n
v
(u+ − u− ) dl;
@n
n
0
is the outer normal of @S2 , and u− =x|(xi ;yj ) ∈
+
(2.9)
where @=@n on 0
= (u− (xi + hi ; yj ) −
0 ∩(xi +hi ;yj ) ∈ S1
−
u (xi ; yj ))=hi .
In the coupling integrals (2.9), Pc (¿ 0) is the penalty constant, is the penalty power, and two
parameters (¿0) and (¿0) satisfy + = 1 or 0. The rst term on the right-hand side of (2.9)
is called the penalty integral, and the second and third terms are called the hybrid integrals. Four
combinations of (2.6) are obtained from di erent parameters in (2.9). [5 –7].
(I)
(II)
(III)
(IV)
Combination I :
= 0 and = 1;
Combination II :
= 1 and = 0;
Symmetric Combination : = = 21 ;
Penalty Combination :
= = 0:
Besides, a direct continuity constraint at the di erence nodes Zk on
0
is also given in [5 –7, 9]
as
v+ (Zk ) = v− (Zk );
∀Zk ∈
0;
(2.10)
328
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
where the interface di erence nodes Zk are located just on
combination:
I (uhN ; v) = f(v);
where
I (u; v) =
ZZ
0.
We then obtain the nonconforming
∀v ∈ Vh ;
3u3v ds +
(2.11)
ZZ
3u3v ds
(2.12)
S2
S1
and Vh (⊂ H ) is a nite-dimensional collection of functions de ned in (2.4) satisfying both (2.10)
and (2.2).
ˆ v) on 0 are given by using the following integration rules:
The approximate integrals in D(u;
Z
0
dl ≈
=
Zc
dl =
0
N1
X
Zk−1 Zk
k=1
S
6
Z
ˆˆ dl
0
[2(Zk−1 )(Zk−1 ) + (Zk−1 )(Zk ) + (Zk )(Zk−1 ) + 2(Zk )(Zk )]; (2.13)
(k)
(k)
1
ˆ
ˆ are the piecewise
where 0 = Nk=1
0 ;
0 =Zk−1 Zk ; Zk−1 Zk denotes the length of Zk−1 Zk , and and
linear interpolatory functions on 0 . For the interior boundary 0 we have
Z
Z
Z
@u− +
@u− +
@u− +
(v − v− ) dl =
(v − v− ) dy
(v − v− ) d x
@n
@x
@y
0
0
0
≈
Zc
u−
0
x
Zc
u−
(v+ − v− ) dy +
Zc
u−
0
y
(v+ − v− ) d x
(v+ − v− ) dl:
n
Note that the above integration is also suited for the slant-up boundary
elements (see [6,7]).
De ne
=
(2.14)
0
kvk1 = kvk21; S1 + kvk21; S2
1=2
;
|v|1 = (|v|21; S1 + |v|21; S2 )1=2 ;
0,
while using triangular
(2.15)
where kvk1; S1 and |v|1; S1 are the Sobolev norm and semi-norm (see [1,16]). Optimal convergence
rates of numerical solutions kk1 = O(h) have been obtained in [8,9], where = u − uh is error of
the solution. In this paper, we pursue global superconvergence based on the new norms
2
Pc
(2.16)
kvkh = kvk21; S1 + kvk21; S2 + kv+ − v− k0; 0 ;
h
where the norms with discrete summation in S1 are assigned on 0 only
kv+
−
2
v− k0;
0
=
Zc
0
(v+ − v− )2 dl:
(2.17)
Note that the norms de ned in [5,6] also involve the discrete summation in S1 , denoted as k · k1; S1 .
An equivalence between these two norms will be discovered in the last section.
The stability analysis is given in [9] for ve combinations. In this paper, the global superconver2
gence rates, ku − 2h
uh kh = O(h2− ); can be achived by all ve combinations for the quasiuniform
rectangles; detailed proofs are presented by Theorems 3.3 and 4.3 in the continued sections.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
329
Fig. 1. The 2 × 2 rectangles in fashion.
Fig. 2. Partition of the solution domain.
3. The nonconforming combination
Let S1 be partitioned into quasiuniform rectangles in 2 × 2 fashion, see Fig. 1. S is a boundary
layer consisting of ij , to separate S2 and S ∗ in Fig. 2, where
S1 = S ∗ ∪ S ;
S ∗ ∩ S = ∅
such that S ∗ ⊂⊂ S1 . To construct the following interpolant:
uˆ I; L =
−
uI = uI
uˆ I
uL+ = uL
in S ∗ ;
in S ;
in S2 ;
(3.1)
P
where uL = Li=1 a˜i i ; a˜i are the unknown coecients and uI is the piecewise bilinear interpolant of
the true solution on rectangulation of S1 . We choose S with width of just one rectangle ij , and
the function uˆ I in S is also the piecewise bilinear function with the corner values at Pi , de ned by
uˆ I (Pi ) =
u(Pi )
uL+ (Pi )
0;
if Pi ∈ 1 = S ∗ ∩ S ;
if Pi ∈ 0 ;
if Pi ∈ :
(3.2)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Hence,
uˆ I; L ⊂ Vh :
(3.3)
First, let us prove a basic theorem.
Theorem 3.1. There exist the error bounds between the numerical and interpolant solutions; uhN
given in (2:11) and uˆ I; L given in (3:1)
kuhN
Z Z
3(u − uI )3w ds
S1
Z Z
Z Z
3(u − uL )3w ds
3(uI − uˆ I )3w ds +
+
S2
S
Z
@u +
+
(w − w− ) dl ;
@n
1
− uˆ I; L k1 6 C sup
w ∈ Vh kwk1
(3.4)
0
where C is a bounded constant independent of h; L; u and w; and the norm k·k1 is de ned in (2:15).
Proof. For the true solution u, we have
I (u − uhN ; v) =
Z
0
@u +
(v − v− ) dl;
@n
∀v ∈ Vh :
(3.5)
Let w = uhN − uˆ I; L ∈ Vh . Then from de nition (3.1)
|w|21 6 I (uhN − uˆ I; L ; w) = I (u − uˆ I; L ; w) −
=
ZZ
S∗
−
Z
0
3(u − uI )3w ds +
@u +
(w − w− ) dl:
@n
ZZ
S
Z
0
@u +
(w − w− ) dl
@n
3(u − uˆ I )3w ds +
ZZ
S2
3(u − uL+ )3w ds
(3.6)
Since u − uˆ I = u − uI + uI − uˆ I in S , we have
ZZ
S∗
=
3(u − uI )3w ds +
ZZ
S1
ZZ
S
3(u − uI )3w ds +
3(u − uˆ I )3w ds
ZZ
S
3(uI − uˆ z )3w ds:
(3.7)
Desired results (3.4) are obtained from (3.5) – (3.7) and the Poincare–Friedricks inequality
kwk1 6C|w|1
with a bounded constant independent of w. This completes the proof of Theorem 3.1.
Below, we will derive the bounds of all terms on the right-hand side of (3.4). We then have the
following lemma.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Lemma 3.1. Let the rectangles
ij
331
be quasiuniform; then for w ∈ Vh
Z Z
3(uI − uˆ I )3w ds 6Ch−(1=2) kRL k0; 0 kwk1 :
(3.8)
S
Proof. From the Schwarz inequality,
Z Z
3(uI − uˆ I )3w ds 6|uI − uˆ I |1; S |w|1 :
(3.9)
S
Denoting = uI − uˆ I , we obtain from the inverse estimates
|uI − uˆ I |1; S = | |1; S 6Ch−1 k k0; S :
(3.10)
Note that is the piecewise bilinear function on S and the fact that (Zi ) = 0 for all element
nodes Zi 6∈ 0 in S , the maximal values of | | along any vertical and horizontal lines in S are just
located on 0 , i.e.,
k k20; S =
Z
S
Moreover, since
2 ds6Ch
Z
0
2 dl = ChkuI − uˆ I k20; 0 :
uI − uˆ I = uI − uˆ L = Rˆ L on 0 :
Since Rˆ L is the piecewise linear interpolant of the remainder RL (=u − uL ) on
triangle inequality
kuI − uˆ I k0; 0 = kRˆ L k0; 0
6 kRˆ L k0; 0 + h kRL − Rˆ L k;
(3.11)
0,
we obtain from the
0
6 kRL k0; 0 + Ch kRL k; 0 6CkRL k0; 0 ;
where ¿ 0 and → 0. Hence, we have from (3.10) and (3.11)
|uI − uˆ I |1; S = | |1; S 6Ch−1 h1=2 kRL k0; 0 = Ch−1=2 kRL k0; 0 :
(3.12)
(3.13)
Combining (3.9) and (3.13) yields (3.8). This completes the proof of Lemma 3.1.
Lemma 3.2. Let u ∈ H 3 (S1 ); then
Z Z
6Ch2 |u|3; S kwk1 ;
3(u
−
u
)3w
ds
I
1
S1
∀w ∈ Vh ;
(3.14)
where |u|3; S1 is the Sobolev seminorm over the space H 3 (S1 ).
Proof. We may follow Lin [10, p. 5], to prove (3.14), but rather provide a straightforward argument
below. Denote the second-order interpolant of u by
uI(2) =
2
X
ij xi y j = uI + 20 x2 + 02 y2 ;
(3.15)
i+j=0
where ij are coecients. By virtue of the following equality (shown later):
ZZ
ij
3(u − uI )3w ds =
ZZ
ij
3(u − uI(2) )3w ds;
(3.16)
332
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
we obtain
X Z Z
Z Z
3(u − uI )3w ds =
S1
X Z Z
=
ij
ij
ij
3(u −
3(u − uI )3w ds
ij
Z Z
(2)
3(u − uI )3w ds
=
uI(2) )3w ds
S1
6|u − uI(2) |1; S1 |w|1; S1 6Ch2 |u|3; S1 |w|1 :
This is desired result (3.14).
Now we show (3.16). Denote
ˆ = {(x; y); − 21 6x6 12 ; − 21 6y6 12 }:
Then by the linear transformations
xˆ = − 21 + (x − xi )=hi ;
the integrals on
ZZ
ij
(u −
yˆ = − 21 + (y − yj )=kj ;
(3.17)
ij
uI(2) )x wx
ZZ
h i kj
cxˆ d sˆ
ds = 2
(uˆ − ubI 2 )xˆ w
ˆ
hi
ZZ
kj
cxˆ d sˆ
(uˆ − ubI − ˆ20 xˆ2 + ˆ02 yˆ 2 )xˆ w
=
ˆ
hi
ZZ
ZZ
kj
cxˆ d xˆ
cxˆ d sˆ − 2 ˆ20
(uˆ − ubI )xˆ w
xˆ w
=
ˆ
ˆ
hi
ZZ
kj
cxˆ d s;
(uˆ − ubI )xˆ w
ˆ
=
ˆ
hi
(3.18)
where ˆ20 and ˆ02 are also constant. In the last equality of (3.18), we have used the following
equation for any bilinear functions wˆ on ˆ :
ZZ
ˆ
cxˆ d sˆ =
xˆw
Z
1=2
xˆ d xˆ
−1=2
!
Z
1=2
!
wˆ xˆ d yˆ = 0:
−1=2
(3.19)
By the inverse transformtion of (3.17),
ZZ
ˆ
cxˆ d sˆ =
(uˆ − uˆ I )xˆ w
hi
kj
ZZ
ij
(u − uI )x wx ds:
(3.20)
Combining (3.18) and (3.20) yields
ZZ
ij
(u − uI(2) )x wx ds =
Similarly, we have
ZZ
ij
(u −
uI(2) )y wy
ds =
ZZ
ZZ
ij
(u − uI )x wx ds:
(3.21)
(u − uI )y wy ds:
(3.22)
ij
Adding (3.21) and (3.22) leads to (3.16). This completes the proof of Lemma 3.2.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
333
Lemma 3.3. Assume u ∈ H 2 (S) and that there exists a bounded power (¿ 0) independent of
L; h; v such that
|v+ |l; 0 6CLl kv+ k0; 0 ;
where |v+ |l;
0
l = 1; 2; ∀v ∈ Vh ;
(3.23)
is the Sobolev seminorm over the space H l ( 0 ). Then for ∀w ∈ Vh
Z
@u +
−
6C(hL )2
@u
−
w
)
dl
(w
@n
@n
0;
0
0
kwk1 :
(3.24)
Proof. From the Schwarz inequality, we have
Z
@u
@u +
−
6
−
w
)
dl
(w
@n
@n
0;
0
0
kw+ − w− k0; 0 :
(3.25)
Since the continuity of w at the nodes on 0 ; w− is, in fact, regarded as the piecewise linear
interpolant of w+ . Hence we obtain from assumption (3.23),
kw+ − w− k0;
0
= kw+ − wˆ + k0; 0 6Ch2 kw+ k2;
0
6 Ch2 L2 kw+ k0; 0 6C(hL )2 kw+ k1; S2 6C(hL )2 kwk1 :
(3.26)
Desired results (3.24) follows from (3.25) and (3.26).
Now we provide an important theorem.
Theorem 3.2. Let u ∈ H 3 (S1 ) and (3:23) be given. Then there exist the error bounds between uhN
and uI; L ;
kuhN
(
2
− uˆ I; L k1 61 = C h |u|3; S1 + kRL k1; S2
Proof. We have
@u
+ (hL )
@n
2
0;
+h
0
Z Z
3(u − uL )3w ds 6ku − uL k1; S2 kwk1; S2 6kRL k1; S2 kwk1 :
−(1=2)
kRL k0;
0
)
(3.27)
(3.28)
S2
The bounds of other terms in the right-hand side of (3.4) are given in Lemmas 3.1–3.3, to lead to
(3.27) directly. This completes the proof of Theorem 3.2.
Theorem 3.2 provides the global errors between uhN and uˆ h; L only, but not the true global errors between uhN and u. The key idea in raising convergence rates is to constitute an a posteriori interpolant,
p uhN , based on uhN . The global errors ku − p uhN k1 can attain the same global superconvergence.
The solution of (2.11) is denoted as
uhN
=
(uhN )− in S1 ;
(uLN )+ in S2 :
(3.29)
Based on Lin’s techniques [10] for the FEM solutions (uhN )− in S1 , the bi-quadratic interpolant,
2
(uhN )− , can be formed in 2 × 2 neighbouring rectangles shown in Fig. 1. The a posteriori process
2h
334
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
is made only in S1 . Denote
p v =
2 −
v
2h
+
v
in S1 ; ∀v ∈ Vh ;
in S2 ; ∀v ∈ Vh :
(3.30)
We obtain the following theorem.
Theorem 3.3. Let all conditions in Theorem 3:2 hold. Then there exist the error bounds
ku − p uhN k1 6C1 ;
(3.31)
where 1 is given in Theorem 3:2.
Proof. By noting the interpolation operation p in (3.30) we have
ku − p uhN k21 6 ku − p uˆ h; L k21 + kp (uˆ h;L − uhN )k21
2
2
(uˆ I − uI )k21; S ∪S∗
uI k21; S1 + k2h
= ku − 2h
+ kp (uˆ h;L − uhN )k21; S1 + kRL k21; S2 ;
(3.32)
where S∗ ∈ S ∗ is a continued extension of S due to the 2 × 2 rectangles.
Based on the equivalence of nite-dimensional norms, we have
2
|2h
v|l;
ij
6C|v|l;
ij
;
to lead to
2
vkl; S1 6Ckvkl; S1 ;
k2h
∀v ∈ Vh :
(3.33)
Also from Theorem 3.2,
kp (uˆ h; L − uhN )k1; S1 + kRL k1; S2 6 C(kuˆ h; L − uhN k1; S1 + kRL k1; S2 )
6 Ckuˆ h; L − uhN k1 6C1 :
(3.34)
Let v = uˆ I − uI . We then have from (3.13)
2
k2h
(uˆ I − uI )k1; S ∪S∗ 6Ckuˆ I − uI k1; S ∪S∗ = Ckuˆ I − uI k1; S 6Ch−(1=2) kRL k0; 0 :
(3.35)
2
uI is the bi-quadratic interpolant of u, then we have
Since 2h
2
ku − 2h
uI k1; S1 6Ch2 |u|3; S1 :
(3.36)
Combining (3.32), (3.34) – (3.36) yields the desired results (3.31). This completes the proof of
Theorem 3.3.
Based on Theorems 3.2 and 3.3 we obtain the following corollary.
Corollary 3.1. Let the conditions in Theorem 3:1 hold. Assume the expansion term L is chosen
such that
|RL |1; S2 = O(h2 );
kRL k0; 0 = O(h5=2 ):
(3.37)
335
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Then
kuhN − uˆ I; L k1 = O(h2 ) + O(h2 L2 )
(3.38)
ku − p uhN k1 = O(h2 ) + O(h2 L2 ):
(3.39)
and
Also if such L is so small as to satisfy (see [4])
L = O(|ln h|);
(3.40)
kuhN − uˆ I; L k1 = O(h2− )
(3.41)
ku − p uhN k1 = O(h2− );
(3.42)
then
and
where → 0 as h → 0.
Corollary 3.1 displays the global superconvergence in the entire solution domain S. In particular
in S1 .
2 N
ku − 2h
uh k1; S1 = O(h2− ):
2
In fact, the a posteriori quadratic interpolation 2h
on the solution uhN costs a little more computation. The analytic results can be applied directly to Motz’s problem, see Section 6.
4. The penalty combination
In this section, we will derive the error bounds for the solution uhP by (2.6) as = = 0. We rst
give a basic theorem in the norm k·kh de ned in (2.16).
Theorem 4.1. There exist the error bounds between the solution uhP and uˆ I; L de ned in (3:1);
kuhP
Z Z
3(u − uI )3w ds
S1
Z Z
Z Z
3(u − uL )3w ds
3(uI − uˆ I )3w ds +
+
S2
S
Z
@u
+
(w+ − w− ) dl ;
@n
1
− uI; L kh 6 C sup
w ∈ Vh kwkh
(4.1)
0
where C is a bounded constant independent of h; L; u and w.
Proof. By the following arguments similarly in Theorem 3.1, we also have:
aˆh (u − uhP ; v) =
Z
0
@u +
(v − v− ) dl;
@n
∀v ∈ Vh :
(4.2)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Then for w = uhP − uˆ I; L ∈ Vh ;
C0 kwk2h 6aˆh (uhP
− uˆ I; L ; w) = aˆh (u − uˆ I; L ; w) −
Z
@u +
(w − w− ) dl;
@n
0
(4.3)
where C0 (¿ 0) is constant independent of h; L; u and w. We have
aˆh (u − uˆ I; L ; w) =
=
ZZ
S1
3(u − uˆ I; L )3w ds +
S1
3(u − uˆ I; L )3w ds +
ZZ
+
ZZ
S2
ZZ
S2
ZZ
ˆ − uˆ I; L; w)
3(u − uˆ I; L )3w ds + D(u
S△
3(uˆ I − uI )3w ds
ˆ − uˆ I; L ; w):
3(u − u L )3w ds + D(u
(4.4)
The penalty term
Z
c
ˆ − u˜ I; L; w) = Pc (uL+ − uˆ−I )(w+ − w− ) dl = 0;
D(u
h 0
(4.5)
because of (3.2) and integration rule (2.13). Combining (4.4) and (4.5) leads to the desired results
(4.1).
The bounds in Theorem 4.1 are analogous to those in Theorem 3.1, but the key di erence is that
the space Vh does not o er the explicit relation between w+ and w− along 0 . Hence the bounds of
the last term in (4.1) should be estimated in di erent manner from the above. We can prove the
following lemma.
Lemma 4.1. Let (3:23) be given; then for v ∈ Vh ;
kv+ − v− k0; 0 6C{kv+ − v− k0; 0 + (hL )2 kvk1; S2 }
(4.6)
kv+ − v− k0; 0 6C{kv+ − v− k0; 0 + (hL )2 kvk1; S2 }:
(4.7)
and
Lemma 4.2. Let (3:23) be given then for v ∈ Vh ;
Z
@u +
−
6C
@u
(w
−
w
)
dl
@n
@n
0
(h=2 + (hL )2 )kwkh :
0;
0
Proof. We have from Schwarz’s inequality and (4.6)
Z
@u
@u +
−
+
−
(w − w ) dl 6
@n
kw − w k0; 0
0; 0
0 @n
@u
+
−
6C
@n
{kw − w k0;
0;
0
0
+ (hL )2 kwk1; S2 }
(4.8)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
@u
(=2)
6C
kwkh + (hL )2 kwk1; S2 }
@n
{h
0; 0
@u
=2
2
6C
@n
{h + (hL ) }kwkh :
0;
(4.9)
0
This completes the proof of Lemma 4.2.
Similarly, we obtain the following theorem from Lemmas 4.2, 3.1, 3.2 and (3:28).
Theorem 4.2. Let all conditions in Theorem 3:1 hold. Then there exist the error bounds of uhP
kuhP − uˆ I; L kh = 2
(
2
6 C h |u|3; S1 + kRL k1; S2
@u
+
@n
0;
(h
=2
2
+ (hL ) ) + h
0
−1=2
kRL k0;
0
)
:
(4.10)
Moreover for the a posteriori interpolant p uhP ; we also have the following theorem.
Theorem 4.3. Let all conditions in Theorem 3:1 hold. Then
u − p uP
6C2 :
h h
(4.11)
Proof. We have
u − p uP
6ku − p uˆ I; L kh +
p (uˆ I; L − uP )
:
h h
h
h
(4.12)
Since interpolation rule (3.30) gives
2 − −
v ) = (v+ )
(2h
then
(p v)+ = (p v)−
This leads to
Also,
on Pi ∈
on Pi ∈
(4.13)
0;
0:
ku − p u I; L k0; 0 = 0:
(4.14)
kp (uˆ I; L − uhP )k0; 0 = kuˆ I; L − uhP k0; 0 :
(4.15)
We then have
)
(
1=2
P
c
2
ku − p uˆ I; L kh 6 C
u − 2h uˆ I; L
1; S1 + ku − uL k1; S2 +
ku − p uˆ I; L k0; 0
h
2
uˆ I; L
1; S1 + kRL k1; S2 }
6 C{
u − 2h
6 C{
u − 2 uI
+
2 (uˆ I; L − uI )
2h
1; S1
2h
∗
1; S ∪S
+ kRL k1; S2 }:
(4.16)
From (3.35) and (3.36)
ku − p uˆ I; L kh 6C{h2 |u|3; S1 + h−1=2 kRL k0; 0 + kRL k1; S2 }:
(4.17)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
On the other hand, we obtain from (4.15) and Theorem 4.2
kp (uˆ I; L −
uhp )kh
(
)
1=2
2
P
c
p
p
p
6 C
2h (uˆ I; L − uh )
1; S1 + kuI; L − uh k1; S2 +
kp (uˆ I; L − uh )k0; 0
h
(
6 C kuˆ I; L −
uhp k1; S1
+
kuL+
−
uhp k1; S2
+
Pc
h
1=2
kuˆ I; L −
6 Ckuˆ I; L − uhp kh 6C2 :
uhp k0;
0
)
(4.18)
Combining (4.17) and (4.18) leads to (4.11).
Corollary 4.1. Let all conditions in Theorem 3:1 hold. Also assume Eq. (3:37) be given. Then
ku − p uhp kh = O(h2 ) + O(h2 L2 ) + O(h=2 ):
(4.19)
Moreover if ¿4 and (3:40) are given;
ku − p uhp kh = O(h2− );
0 ¡ ≪ 1:
(4.20)
The penalty combination leads to the same global convergence rates as the nonconforming combination does because
ku − p uhp k1 6ku − p uhp kh = O(h2− ):
(4.21)
Moreover, we have for ¿4
ku − p uhp k0; 0 6Ch=2 ku − p uhp kh = O(h2+(=2)− ):
(4.22)
Finally let us derive the relation of the penalty coupling to the nonconforming constraints (2.10).
Corollary 4.2. Let all conditions in Corollary 4:1 hold. Then when ¿4 the average jumps of the
solution v = p uhp at the element nodes Zi ∈ 0 have the convergence rates
E(v+ − v− ) =
N
1 X
|v+ (Zi ) − v− (Zi )| = O(h2+=2− );
N + 1 i=0
where N is the number of element nodes on
(4.23)
0.
Proof. Denote rst the trapezoidal rule
Zf
0
2
v dl =
N
X
Zk−1 Zk
k=1
2
[v2 (Zk−1 ) + v2 (Zk )]:
Since 21 (x2 + y2 )6(x2 + xy + y2 ), we have from (2.13),
Zf
0
v2 dl63
N
X
Zk−1 Zk
k=1
3
[v2 (Zk−1 ) + v(Zk−1 )v(Zk ) + v2 (Zk )] = 3
Zc
0
v2 dl:
(4.24)
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
339
Since the rectangles are quasiuniform, we obtain from the Schwarz inequality
N
X
v(Zk )
N +1
k=0
!2
6
N
X
k=0
=C
N
X
Zk−1 Zk 2
1
v2 (Zk )6C
[v (Zk−1 ) + v2 (Zk )]
N +1
2
k=1
Zf
2
v dl63C
Let v =
and w = u −
E(v+ − v− ) =
2
v2 dl = 3Ckvk0; 0 :
0
0
p uhp
Zc
p uhp
N
1 X
(p up )+ − (p up )− 6Ck(p up )+ − (p up )− k
h Zi
h Zi
h
h
0;
N + 1 i=0
0
= Ckw+ − w− k0; 0 6Ch=2 kwkh :
When ¿4,
E(v+ − v− )6Ch2+(=2)−
from Corollary 4.1. This completes the proof of Corollary 4.2.
Corollary 4.2 displays an interesting fact that the average jumps between (p uhp )+ and (p uhp )− are
O(h2+(=2)− ). Hence when → ∞, the penalty combination leads to the nonconforming combination.
Note that the large values of (¿4) do not incur the reduced convergence rates.
Remark. We may derive similarly the same global superconvergence O(h2− ) for Combinations I,
II and symmetric combination under the condition ¿2. Although their algorithms are more complicated, using smaller values of will lead to better stability.
5. Comparisons and computations
Now, let us clarify the relation between [5,6] and this paper, provide some numerical experiments
and make some remarks.
In [5,6], the semi-norm in discrete summation is de ned as
|v|1; S1 =
(
X ZZ
d
ij
ij
(vx2
+
vy2 ) ds
)1=2
(5.1)
;
where
ZZ
d
vx2 ds =
hi kj 2
(v (A) + vx2 (B));
2 x
vy2 ds =
h i kj 2
(v (C) + vy2 (D))
2 y
ij
ZZ
d
ij
and A; B; C and D are the mid-points of edges of
lemma.
(5.2)
ij ,
shown in Fig. 3. Then we have the following
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Fig. 3. The rectangle
ij .
Lemma 5.1. Let
u ∈ C 3 (S1 )
(5.3)
and assume the numerical solutions uh ∈ Vh have the errors
|u − uh |1; S1 = O(h2− );
06 ¡ 1:
(5.4)
Then the global errors between uh and the solution interpolant uI have also the error bounds
|uI − uh |1; S1 = O(h2− ):
(5.5)
Proof. For the piecewise bilinear functions, v ∈ Vh , we have
|v|21; S1 =
where
ZZ
XZ Z
ij
(vx2 + vy2 ) ds;
ij
h i kj 2
[vx (A) + vx2 (B) + vx (A)vx (B)];
3
ij
ZZ
h i kj 2
vy2 ds =
[v (C) + vy2 (D) + vy (C)vy (D)]:
3 y
ij
vx2 ds =
Since x2 + y2 + xy6 23 (x2 + y2 ), we obtain for v ∈ Vh ,
ZZ
vx2 ds6
ij
Hence
Zd
Z
h i kj 2
[vx (A) + vx2 (B)] =
2
|v|1; S1 6|v|1; S1 ;
We then have
Also,
vx2 ds:
ij
∀v ∈ Vh :
(5.6)
|uI − uh |1; S 6|uI − uh |1; S1 = |u − uh |1; S1 + |u − uI |1; S1 :
(5.7)
|u − uI |1; S1 = Ch2 |u|3; ∞; S1 = O(h2 );
(5.8)
where |u|3; ∞; S1 is also the Sobolev p-norm with p = ∞. Desired results (5.5) follow (5.4), (5:7)
and (5:8). This completes the proof of Lemma 5.1.
Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
341
Fig. 4. Partition of Motz’s problem with MS = 2.
Lemma 5.2. Let (5:3) and
|uI − uh |1; S1 = O(h2− );
06 ¡ 1
be given. Then
|u − uh |1; S1 = O(h2− ):
Proof. Similarly, from the arguments in Lemma 5.1 we have
√
|uI − uh |1; S1 6 3|uI − uh |1; S1 :
Then
|u − uh |1; S1 6|u − uI |1; S1 + |uI − uh |1; S1 6O(h2 ) +
=O(h2 ) + O(h2− ) = O(h2− ):
√
3|uI − uh |1; S1
(5.9)
Lemmas 5.1 and 5.2 imply that under assumption (5.3), the superconvergence in this paper and
in [5,6] is equivalent to each other. We then conclude that the order O(h2− ) also holds for superconvergence of the solutions of RGM–FEM in this paper, to the average nodal derivatives at the
edge mid-points of ij , so does for the maximal nodal derivatives in majority.
On the other hand, based on Lemma 5.1, the solutions in [6] by the combinations of Ritz–Galerkin
and nite-di erence methods also have the global superconvergence as (3.41) and (3.42). However,
the proofs in this paper using Lin’s techniques are simpler so as to be extend to other kinds of
singularity problems, i.e., in biharmonic equations and elastic plates with cracks. Details appear
elsewhere. Also note that the assumption u ∈ H 3 (S1 ) in Theorem 3.2 is weaker than (5.3).
In this section, numerical experiments are also carried out to con rm the global superconvergence
O(h2− ) made in Section 3 by the nonconforming combination. Other kinds of combinations can
also be veri ed by numerical experiments. Let us consider the typical Motz problem (see Fig. 4):
@ 2 u @2 u
u = 2 + 2 = 0 in S;
(5.10)
@x
@y
u|x¡0∧y=0 = 0;
@u
@y
y=1
@u
=
@y
u|x=1 = 500;
x¿0∧y=0
@u
=
@x
(5.11)
= 0;
x=−1
(5.12)
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Table 1
Errors norms and condition number of nonconforming combination of RGM–FEM
Divisions
MS = 1
L+1=3
MS = 2
L+1=4
MS = 3
L+1=5
MS = 4
L+1=5
MS = 6
L+1=6
MS = 8
L+1=6
Max
kk0; S
kkh
||1
kk1
||1
kk1
2
ku − 2h
uh k1
8.10
3.77
42.8
15.6
15.9
12.6
13.0
–
3.23
1.29
22.4
9.66
9.75
8.72
8.82
6.90
33
1.33
0.559
14.3
4.19
4.22
3.93
3.97
–
82
0.779
0.315
10.6
2.33
2.35
2.22
2.23
1.83
163
0.358
0.141
6.97
1.04
1.05
1.01
1.02
0.834
462
0.214
0.0794
5.20
0.583
0.587
0.567
0.572
0.470
1000
Con.
8
where S is a rectangle (−16x61; 06y61). The origin (0; 0) is a singular point with the solution
behaviour u = O(r (1=2) ) as r → 0 due to the intersection of the Neumann and Dirichlet conditions.
Divide S by 0 into S2 and S1 . The subdomain S2 is chosen as a smaller rectangle (− 21 6x6 21 ;
06y6 12 ). Also the subdomain S1 is again split into uniform rectangular elements shown in Fig. 4.
The admissible functions are chosen as
−
v = v1 ;
L
X
v=
+
v
=
D˜ l r l+(1=2) cos(l + 21 );
(5.13)
l=0
where D˜ l are unknown coecients, and (r; ) are the polar coordinates with origin (0,0).
Let MS denote the di erence division number along DB. Based on the good matching between
L + 1 (the total number of basis functions used) and MS given in [4], we will choose
MS = 2 and L + 1 = 4;
MS = 3; 4 and L + 1 = 5;
MS = 6; 8 and L + 1 = 6:
(5.14)
Numerical solutions are conducted by the nonconforming combinations of RGM–FEM; and their
error norms and the coecients are provided in Tables 1 and 2. “Con.” in Table 1 denotes the
condition number of the associated matrix resulting from the nonconforming combination, and other
error norms are de ned by
kk0; S =
Z Z
S
2
2
ds
1=2
;
||1 = (||1; S1 + ||21; S1 )1=2 ;
max = max ||;
S
2
kk1 = (kk1; S1 + kk21; S1 )1=2 ;
||1 = (|uh − uI |21; S1 + |uL − u˜ L |21; S2 )1=2 ;
kk1 = (kuh − uI k21; S1 + kuL − u˜ L k21; S2 )1=2 ;
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
Table 2
Approximate coecients by nonconforming combination of RGM–FEM
Coe s.
D˜ 3
D˜ 4
D˜ 5
–
–
–
13.759
−23:291
–
–
86.998
15.677
−15:336
2.936
–
400.847
87.278
16.350
−12:169
2.199
–
D˜ 2
D˜ 0
D˜ 1
403.724
81.257
2.670
399.364
86.172
400.593
MS = 1
L+1=3
MS = 2
L+1=4
MS = 3
L+1=5
MS = 4
L+1=5
MS = 6
L+1=6
MS = 8
L+1=6
401.023
87.585
16.773
−9:898
1.743
1.080
401.085
87.616
16.962
−9:097
1.601
0.740
True
401.162
87.656
17.238
−8:071
1.440
0.331
where = u − uhN and u˜ h = u˜ L in S2 . It is easy to see from the data in Tables 1 and 2 that
kkh = O(h);
||1 = O(h2− );
kk0; S = O(h2 );
max = O(h2− );
kk1 = O(h2− );
(5.15)
(5.16)
2
uh k1 = O(h2− );
ku − 2h
(5.17)
||1 = O(h2− );
(5.18)
kk1 = O(h2− );
|D0 − D˜ 0 | = O(h2 );
(5.19)
where D0 and D˜ 0 are the true and approximate coecients, respectively. In Table 1, the data for
2
ku − 2h
uh k1 are not available when using odd MS = 1; 3. Note that Eqs. (5.15) and (5.19) coincide
with the optimal results in [4]; Eqs. (5.16), (5.17) and (5:18) verify perfectly the analysis in Section 3
and this section, respectively.
Finally let us make a few remarks.
1. The rectangular elements are discussed on this paper. In fact, when the uniform right triangle
elements with the interior angle =4 are chosen, only the global superconvergence rates O(h3=2 ) can
be gained, based on the analysis on this paper and Lin and Yan’s [12]. Once the solution domain
is not very complicated, the rectangles and such simple triangles can be employed. Both global and
nodal superconvergence can be achieved simultaneously. Note that for the global superconvergence,
the a posteriori interpolant on the numerical solution costs a little more computation e ort.
2. The nonconforming combination will deal with the unknown constraints (2.10). Instead, we
may solicit other combinations. Corollary 4.2 implies that when is large, the penalty combination
approaches, indeed, the nonconforming combination. In fact, such a large weight technique has
already been applied in engineering computations. Among ve combinations, the nonconforming
combination is still basic.
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Z.-C. Li et al. / Journal of Computational and Applied Mathematics 106 (1999) 325–344
3. The di erent coupling strategies are studied in [7,9] for Ritz–Galerkin–FEM and in [5,6] for
Ritz–Galerkin–FDM. The optimal convergence O(h) and the nodal superconvergence O(h2− ) have
been proven. In this paper the global superconvergence O(h2− ) has been exploited together. Note
that the high superconvergence of combinations yields the higher accuracy of both uh in S1 and the
coecients ai of the solution in S2 . The leading coecient a1 is, in fact, the stress intensity factor
at the singularity, which has important application in fracture mechanics.
Acknowledgements
The authors are grateful to the referee for his=her valuable comments and careful reading.
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