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 bInstitute 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. Theglobalsuperconvergence 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 dierent coupling strategies, such as the nonconforming constraints, the penalty integrals, and the penalty plus hybrid integrals, where(¿0) is an arbitrarily small number, andhis the maximal boundary length of quasiuniform rectangles ij used. A little eort 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) 1–17. c1999 Elsevier Science B.V. All rights reserved.
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 specic 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)
[3,13–17] is devoted to specic 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 dierence
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 dierence 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,
respec-tively. 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 inS1. Five dierent 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∈H3(S
1), the global superconvergence
rates can be achieved as
kuI −u˜hk1; S1+kuL−u˜Lk1; S2= O(h
2−);
u−22hu˜h 1; S
1+kuL−u˜Lk1; S2= O(h
2−); 0¡
≪1;
whereuI; u˜h and22hu˜h are the solution interpolant, the numerical solution and a posteriori interpolant of ˜uh, respectively. k · k1; S1 is the Sobolev norm in space H
1(S). Here u
L and ˜uL 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 dierent 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
−u=− @
2u
@x2 + @2u @y2
!
=f(x; y); (x; y) in S; (2.1)
u| = 0; (x; y) on ; (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
nite-dierence method is used in S1. For simplicity, the subdomain S1 is again split into small
quasiuni-form rectangles ij only, where ij={(x; y); xi6x6xi+1; yj6y6yj+1}. This connes 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
whereai are the expansion coecients, and i (i=1;2; : : : ;∞) are complete and linearly independent base functions in the sobolev spaceH1(S
2). { i} may be chosen as analytical and singular functions. Then the admissible functions of combinations of the RGM–FEM are written as
v= v−
=v1 in S1;
v+=fL( ˜ai) in S2;
(2.4)
wherev1 is the piecewise bilinear functions on S1; fL(ai) =PLi=1ai i and ˜ai 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 0; (2.5)
we dene another space
H={v|v∈L2(S); v∈H1(S
1) and v∈H1(S2)};
whereH1(S
1) is the Sobolev space. LetVh(⊆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
ˆ
ah(uh; v) =f(v); ∀v∈Vh; (2.6)
where
ˆ
ah(u; v) = Z Z
S1
3u3vds+
Z Z
S2
3u3vds+ ˆD(u; v); (2.7)
f(v) = Z Z
S
fvds; (2.8)
ˆ
D(u; v) =Pc
h c Z
0
(u+−u−
)(v+−v−
) dl− c
Z
0
@u
+
@n +
u−
n
(v+−v− ) dl
−c
Z
0
@v +
@n +
v−
n
(u+−u−
) dl; (2.9)
where @=@n on 0 is the outer normal of @S2, and u−=x|(xi;yj)∈ 0∩(xi+hi;yj)∈S1= (u −
(xi+hi; yj)−
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 dierent parameters in (2.9). [5 –7].
(I) Combination I : = 0 and = 1;
(II) Combination II : = 1 and = 0;
(III) Symmetric Combination : ==1
2;
(IV) Penalty Combination : == 0:
Besides, a direct continuity constraint at the dierence nodes Zk on 0 is also given in [5 –7, 9]
as
v+(Z
k) =v −
where the interface dierence nodes Zk are located just on 0. We then obtain the nonconforming
and Vh(⊂H) is a nite-dimensional collection of functions dened in (2.4) satisfying both (2.10) and (2.2).
The approximate integrals in ˆD(u; v) on 0 are given by using the following integration rules:
Z linear interpolatory functions on 0. For the interior boundary 0 we have
Z
Note that the above integration is also suited for the slant-up boundary 0, while using triangular
elements (see [6,7]). Dene
kvk1= kvk21; S1+kvk21; S21=2; |v|1= (|v|21; S1+|v|21; S2)1=2; (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
kvkh=
where the norms with discrete summation in S1 are assigned on 0 only
kv+−v−k2
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 superconver-gence rates, ku−2
2huhkh= O(h2
−); can be achived by all ve combinations for the quasiuniform
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:
ˆ
uI; L=
u−
I =uI in S
∗
;
ˆ
uI in S;
u+
L =uL in S2;
(3.1)
where uL=PLi=1a˜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 ˆuI in S is also the piecewise bilinear function with the corner values at Pi, dened by
ˆ
uI(Pi) =
u(Pi) if Pi∈ 1=S∗∩S;
u+
L(Pi) if Pi∈ 0;
0; if Pi∈ :
Hence,
ˆ
uI; 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; uN h
whereC is a bounded constant independent ofh; L; u andw; and the normk·k1 is dened in(2:15).
Proof. For the true solution u, we have
I(u−uN
Desired results (3.4) are obtained from (3.5) – (3.7) and the Poincare–Friedricks inequality
kwk16C|w|1
with a bounded constant independent of w. This completes the proof of Theorem 3.1.
Lemma 3.1. Let the rectangles ij be quasiuniform; then for w∈Vh
Z Z
S
3(uI −uˆI)3wds
6Ch−(1=2)kRLk0; 0kwk1: (3.8)
Proof. From the Schwarz inequality,
Z Z
S
3(uI −uˆI)3wds
6|uI −uˆI|1; S|w|1: (3.9)
Denoting =uI −uˆI, we obtain from the inverse estimates
|uI −uˆI|1; S=||1; S6Ch −1
kk0; S: (3.10)
Note that is the piecewise bilinear function on S and the fact that (Zi) = 0 for all element nodes Zi6∈ 0 in S, the maximal values of|| along any vertical and horizontal lines in S are just located on 0, i.e.,
kk2 0; S =
Z
S
2ds6ChZ
0
2dl=Chku
I −uˆIk20; 0: (3.11)
Moreover, since
uI −uˆI =uI −uˆL= ˆRL on 0:
Since ˆRL is the piecewise linear interpolant of the remainder RL(=u−uL) on 0, we obtain from the
triangle inequality
kuI −uˆIk0; 0 =kRˆLk0; 0
6kRˆLk0; 0+h kR
L−RˆLk;0
6kRLk0; 0+Ch
kRLk; 06CkRLk0; 0; (3.12)
where ¿0 and →0. Hence, we have from (3.10) and (3.11)
|uI −uˆI|1; S=||1; S6Ch
−1h1=2kR
Lk0; 0=Ch −1=2kR
Lk0; 0: (3.13)
Combining (3.9) and (3.13) yields (3.8). This completes the proof of Lemma 3.1.
Lemma 3.2. Let u∈H3(S
1); then
Z Z
S1
3(u−uI)3wds
6Ch2|u|3; S1kwk1; ∀w∈Vh; (3.14)
where |u|3; S1 is the Sobolev seminorm over the space H
3(S 1).
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
u(2)I =
2
X
i+j=0
ijxiyj=uI +20x2+02y2; (3.15)
where ij are coecients. By virtue of the following equality (shown later): Z Z
ij
3(u−uI)3wds=
Z Z
ij
we obtain This is desired result (3.14).
Now we show (3.16). Denote
ˆ ={(x; y);−1
Then by the linear transformations
ˆ
x=−1
2 + (x−xi)=hi; yˆ=− 1
2 + (y−yj)=kj; (3.17)
the integrals on ij Z Z
where ˆ20 and ˆ02 are also constant. In the last equality of (3.18), we have used the following
equation for any bilinear functions ˆw on ˆ : Z Z
By the inverse transformtion of (3.17), Z Z
Combining (3.18) and (3.20) yields Z Z
Similarly, we have Z Z
Lemma 3.3. Assume u∈H2(S) and that there exists a bounded power (¿0) independent of
l; 0 is the Sobolev seminorm over the space H l(
Proof. From the Schwarz inequality, we have
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−
Desired results (3.24) follows from (3.25) and (3.26).
Now we provide an important theorem.
Theorem 3.2. Let u∈H3(S
Proof. We have
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 betweenuN
h and ˆuh; Lonly, but not the true global errors be-tween uN
h and u. The key idea in raising convergence rates is to constitute an a posteriori interpolant,
puNh, based on uNh. The global errors ku−puNhk1 can attain the same global superconvergence.
The solution of (2.11) is denoted as
uNh =
Based on Lin’s techniques [10] for the FEM solutions (uN h)
−
in S1, the bi-quadratic interpolant, 2
2h(uNh)
−, can be formed in 2
is made only in S1. Denote
pv= 2
2hv −
in S1; ∀v∈Vh;
v+ 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−puNhk16C1; (3.31)
where 1 is given in Theorem 3:2.
Proof. By noting the interpolation operation p in (3.30) we have
ku−puNhk216ku−puˆh; Lk21+kp( ˆuh;L−uNh)k21
=ku−22huIk21; S1+k
2
2h( ˆuI−uI)k21; S∪S∗
+kp( ˆuh;L−uNh)k
2
1; S1+kRLk
2
1; 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
|22hv|l; ij6C|v|l; ij; to lead to
k2
2hvkl; S16Ckvkl; S1; ∀v∈Vh: (3.33)
Also from Theorem 3.2,
kp( ˆuh; L−uNh)k1; S1+kRLk1; S26C(kuˆh; L−u N
hk1; S1+kRLk1; S2)
6Ckuˆh; L−uNhk16C1: (3.34)
Let v= ˆuI −uI. We then have from (3.13)
k22h( ˆuI −uI)k1; S∪S∗
6CkuˆI −uIk1; S∪S∗ =CkuˆI −uIk1; S6Ch −(1=2)
kRLk0; 0: (3.35)
Since 2
2huI is the bi-quadratic interpolant of u, then we have
ku−22huIk1; S16Ch
2
|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(h
2);
kRLk0; 0= O(h
Then
Corollary 3.1 displays the global superconvergence in the entire solution domain S. In particular in S1.
ku−22huNhk1; S1= O(h
2−):
In fact, the a posteriori quadratic interpolation 2
2h on the solution uNh costs a little more compu-tation. 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 uP
h by (2.6) as == 0. We rst
give a basic theorem in the norm k·kh dened in (2.16).
Theorem 4.1. There exist the error bounds between the solution uP
h and uˆI; L dened in (3:1);
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:
Then for w=uP
The penalty term
ˆ
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 dierence is that the space Vh does not oer the explicit relation between w+ and w− along 0. Hence the bounds of
the last term in (4.1) should be estimated in dierent manner from the above. We can prove the following lemma.
6C
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 uP h
Moreover for the a posteriori interpolant puPh; we also have the following theorem.
Theorem 4.3. Let all conditions in Theorem 3:1 hold. Then
u−puPh
h6C2: (4.11)
Proof. We have
Since interpolation rule (3.30) gives
(2
This leads to
ku−puI; Lk0; 0= 0: (4.14)
Also,
kp( ˆuI; L−uPh)k0; 0=kuˆI; L−u P
hk0; 0: (4.15)
We then have
ku−puˆI; Lkh6C
From (3.35) and (3.36)
ku−puˆI; Lkh6C{h2|u|3; S1+h −1=2
On the other hand, we obtain from (4.15) and Theorem 4.2
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−pu
The penalty combination leads to the same global convergence rates as the nonconforming com-bination does because
ku−pu
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=puph at the element nodes Zi∈ 0 have the convergence rates
where N is the number of element nodes on 0.
Proof. Denote rst the trapezoidal rule
Since the rectangles are quasiuniform, we obtain from the Schwarz inequality
from Corollary 4.1. This completes the proof of Corollary 4.2.
Corollary 4.2 displays an interesting fact that the average jumps between (puph)+ and (puph) −
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 com-plicated, 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 dened as
|v|1; S
Fig. 3. The rectangle ij. Lemma 5.1. Let
u∈C3(S1) (5.3)
and assume the numerical solutions uh∈Vh have the errors
|u−uh|1; S1= O(h2
−); 0
6 ¡1: (5.4)
Then the global errors between uh and the solution interpolant uI have also the error bounds
|uI −uh|1; S1= O(h
2−): (5.5)
Proof. For the piecewise bilinear functions, v∈Vh, we have
|v|21; S1=X ij
Z Z
ij
(v2x+v2y) ds;
where Z Z
ij
v2
xds=
hikj 3 [v
2
x(A) +v
2
x(B) +vx(A)vx(B)]; Z Z
ij
v2yds=hikj 3 [v
2
y(C) +v
2
y(D) +vy(C)vy(D)]:
Since x2+y2+xy63 2(x
2+y2), we obtain for v∈V
h, Z Z
ij
v2xds6hikj
2 [v
2
x(A) +v
2
x(B)] = d Z Z
ij
v2xds:
Hence
|v|1; S16|v|1; S1; ∀v∈Vh: (5.6)
We then have
|uI −uh|1; S6|uI −uh|1; S1=|u−uh|1; S1+|u−uI|1; S1: (5.7) Also,
|u−uI|1; S1=Ch2|u|3;∞; S1= O(h
2); (5.8)
Fig. 4. Partition of Motz’s problem with MS = 2. Lemma 5.2. Let (5:3) and
|uI −uh|1; S1= O(h
2−); 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; S16
√
3|uI −uh|1; S1:
Then
|u−uh|1; S16|u−uI|1; S1+|uI −uh|1; S16O(h2) +
√
3|uI −uh|1; S1
=O(h2) + O(h2−) = O(h2−): (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 super-convergence 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-dierence 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∈H3(S
1) in Theorem 3.2 is weaker than (5.3).
In this section, numerical experiments are also carried out to conrm the global superconvergence O(h2−) made in Section 3 by the nonconforming combination. Other kinds of combinations can also be veried by numerical experiments. Let us consider the typical Motz problem (see Fig. 4):
u= @
2u
@x2 + @2u
@y2 = 0 inS; (5.10)
u|x¡0∧y=0= 0; u|x=1= 500; (5.11)
@u @y
y=1
= @u
@y
x¿0∧y=0
=@u
@x
x=−1
Table 1
Errors norms and condition number of nonconforming combination of RGM–FEM
Divisions Max kk0; S kkh ||1 kk1 ||1 kk1 ku−22huhk1 Con. MS = 1
L+ 1 = 3 8.10 3.77 42.8 15.6 15.9 12.6 13.0 – 8
MS = 2
L+ 1 = 4 3.23 1.29 22.4 9.66 9.75 8.72 8.82 6.90 33
MS = 3
L+ 1 = 5 1.33 0.559 14.3 4.19 4.22 3.93 3.97 – 82
MS = 4
L+ 1 = 5 0.779 0.315 10.6 2.33 2.35 2.22 2.23 1.83 163
MS = 6
L+ 1 = 6 0.358 0.141 6.97 1.04 1.05 1.01 1.02 0.834 462
MS = 8
L+ 1 = 6 0.214 0.0794 5.20 0.583 0.587 0.567 0.572 0.470 1000
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 (−126x612;
06y61
2). Also the subdomain S1 is again split into uniform rectangular elements shown in Fig. 4.
The admissible functions are chosen as
v=
v− =v1;
v+ = L X
l=0
˜
Dlrl+(1=2)cos(l+12);
(5.13)
where ˜Dl are unknown coecients, and (r; ) are the polar coordinates with origin (0,0).
Let MS denote the dierence 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 dened by
kk0; S= Z Z
S
2ds
1=2
; max = max
S ||;
||1= (||21; S
1+||
2 1; S1)
1=2;
kk1= (kk21; S
1+kk
2 1; S1)
1=2;
||1= (|uh−uI|21; S1+|uL−u˜L|
2 1; S2)
1=2;
kk1= (kuh−uIk21; S1+kuL−u˜Lk
2 1; S2)
Table 2
Approximate coecients by nonconforming combination of RGM–FEM
Coes. D˜0 D˜1 D˜2 D˜3 D˜4 D˜5
MS = 1
L+ 1 = 3 403.724 81.257 2.670 – – –
MS = 2
L+ 1 = 4 399.364 86.172 13.759 −23:291 – –
MS = 3
L+ 1 = 5 400.593 86.998 15.677 −15:336 2.936 –
MS = 4
L+ 1 = 5 400.847 87.278 16.350 −12:169 2.199 –
MS = 6
L+ 1 = 6 401.023 87.585 16.773 −9:898 1.743 1.080
MS = 8
L+ 1 = 6 401.085 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−uN
h and ˜uh= ˜uL in S2. It is easy to see from the data in Tables 1 and 2 that
kkh= O(h); kk0; S= O(h2); max = O(h2
−); (5.15)
||1= O(h2
−);
kk1= O(h2
−); (5.16)
ku−22huhk1= O(h2−); (5.17)
||1= O(h2−);
kk1= O(h2−); (5.18)
|D0−D˜0|= O(h2); (5.19)
where D0 and ˜D0 are the true and approximate coecients, respectively. In Table 1, the data for ku−2
2huhk1 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 eort.
3. The dierent 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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