coefﬁcient jump-independent approximation of the ...szhang/research/p/2016f.pdf · 724 s. y....

Advances in Applied Mathematics and MechanicsAdv. Appl. Math. Mech., Vol. 8, No. 5, pp. 722-736

DOI: 10.4208/aamm.2015.m931October 2016

Coefficient Jump-Independent Approximation of

the Conforming and Nonconforming Finite

Element Solutions

Shangyou Zhang∗

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716,USA

Received 21 January 2015; Accepted (in revised version) 30 December 2015

Abstract. A counterexample is constructed. It confirms that the error of conformingfinite element solution is proportional to the coefficient jump, when solving interfaceelliptic equations. The Scott-Zhang operator is applied to a nonconforming finite el-ement. It is shown that the nonconforming finite element provides the optimal orderapproximation in interpolation, in L2-projection, and in solving elliptic differential e-quation, independent of the coefficient jump in the elliptic differential equation. Nu-merical tests confirm the theoretical finding.

AMS subject classifications: 65N30, 65D05

Key words: Jump coefficient, finite element, L2 projection, weighted projection, Scott-Zhang op-erator.

1 Introduction

When studying the finite element solutions of elliptic boundary value problems withdiscontinuous coefficients, i.e., interface problems, a useful tool is the weighted L2 pro-jection operator, cf. [5,17,19]. To be specific, we consider a domain Ω which is subdividedinto finitely many, bounded, polygonal subdomains Ωi, i=1,··· , J, in d=2 or 3 space-dimension. On each subdomain Ωi, we are given a positive constant ωi, and we have aquasi-uniform triangulation Th(Ωi) of size h on Ωi, cf. [7], shown by Fig. 1 as an example.Thus each Ωi, and Ω, is Lipschitz. We further assume the subdomain grids are match-ing at the interface so that we can define conforming and nonconforming linear finite

∗Corresponding author.Email: [email protected] (S. Y. Zhang)

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S. Y. Zhang / Adv. Appl. Math. Mech., 8 (2016), pp. 722-736 723

ω1

ω2

ω3

ω4 ω5

Figure 1: Constant weight each subdomain, and a matching grid.

element spaces on the combined grid Th over the domain Ω, [7]:

Vh=

v∈H10 (Ω)∩C(Ω)|v|K ∈P1, ∀K∈Th

, (1.1a)

Vnch =

v|v|K ∈P1, ∀K∈Th, v is continuous at all mid face ∂K and

0 at mid face ∂K∩∂Ω

. (1.1b)

The weighed L2 and semi-H1 inner products are defined by

(u,v)L2ω(Ω)=

J

∑i=1

ωi

∫

Ωi

uvdx, (1.2a)

(u,v)H1ω(Ω)=

J

∑i=1

ωi

∫

Ωi

∇u·∇vdx. (1.2b)

The induced norms are denoted by ‖·‖L2ω

and |·|H1ω

, respectively. The full H1 weighted

norm is ‖·‖2H1

ω= |·|2

H1ω+‖·‖2

L2ω

. The weighted L2 projection Qωh : L2(Ω) 7→Vh is defined by

(Qωh u,v)L2

ω(Ω)=(u,v)L2ω(Ω), ∀v∈Vh. (1.3)

The following important theorem is proved by Bramble and Xu in 1991.

Theorem 1.1 (see [5]). If for all i, the (d−1)-dimensional Lebesgue measure of ∂Ωi∩∂Ω ispositive, then for all u∈H1

0(Ω),

‖u−Qωh u‖L2

ω(Ω)+h|Qωh u|H1

ω(Ω)≤Ch|logh|1/2|u|H1ω(Ω), (1.4)

where C is independent of ωi.

Trying to show the necessity that all subdomains have a part of boundary ∂Ω, andof the log term in the bound, several examples are constructed by Xu in [18]. However,these examples are constructed by some limit argument where no specific function ucan be used in computation to show the sharpness of (1.4). Thus, some people are still





724 S. Y. Zhang / Adv. Appl. Math. Mech., 8 (2016), pp. 722-736

skeptic about the sharpness and the condition for (1.4). In this work, we construct acounterexample, where ∂Ωi∩∂Ω is a set of isolated points, i.e., has a Lebesgue measurezero, violating the condition in Theorem 1.1. Basically, we construct a discontinuous, butH1 function. With this example we show that Bramble-Xu’s Theorem 1.1 can never havethe constant C in the bound independent of coefficient jump, if the boundary assumptionis taken off. In other words, by this example, we show the following standard errorbound is sharp:

‖u−Qωh u‖L2

ω(Ω)+h|Qωh u|H1

ω(Ω)≤Chmax√ωimin√ωi

‖u‖H1ω(Ω).

By this explicit counterexample, we put a solid period to the problem on weightedprojection of conforming finite elements. This example points out that it is inappropriateto use conforming finite element methods for large jump-coefficient problems. Whatabout the nonconforming finite element? Yes. With the help of Scott-Zhang interpolationoperator, we show the following theorem that the approximation of weighted projectionis independent of weights for nonconforming finite elements. In addition, we show thatthe nonconforming finite element solves the jump-coefficient elliptic problems optimally,independent of the jump.

Theorem 1.2. For all u∈H10(Ω),

‖u−Qωh,ncu‖L2

ω(Ω)+h|Qωh,ncu|H1

ω(Ω)≤Ch‖u‖H1ω(Ω), (1.5)

where C is independent of ωi, and the weighted L2-projection operator Qωh,nc is defined by

(Qωh,ncu,v)L2

ω(Ω)=(u,v)L2ω(Ω), ∀v∈Vnc

h . (1.6)

Though we prove the jump-independent convergence for the non-conforming finiteelement method only, it would be straightforward to prove the same property for manyother exotic finite elements, in particular, for the mixed finite elements [8,14], for the dis-continuous Galerkin methods [1], for the discontinuous Petrov-Galerkin methods [11],for the hybridizable discontinuous Galerkin methods [10], for the virtual element meth-ods [4], and for the weak Galerkin methods [13, 16].

The rest manuscript is organized as follows. In Section 2, we define a counterexampleshowing the weighted L2 projection does depend on the jump coefficient. In Section 3,we introduce the Scott-Zhang operator and show its jump-independent approximation.Consequently we prove the main theorem. Numerical results are given in Section 4,supporting the new theory.

2 A counterexample

By the standard finite element theory, we show first a weight-dependent approximationof the weighted L2 projection. Then we construct a counterexample, showing this stan-dard approximation property cannot be improved, i.e., the approximation property ofweighted L2 projection in conforming finite element spaces is weight-dependent.






Theorem 2.1. For any u∈H1(Ω),

‖u−Qωh u‖L2

ω(Ω)+h|Qωh u|H1

ω(Ω)≤Chmax√ωimin√ωi

‖u‖H1ω(Ω), (2.1)

where Qωh is the weighted L2 projection in to the conforming finite element space Vh (1.1a), defined

in (1.3).

Proof. Let Ih be the Scott-Zhang interpolation operator (cf. [15] and next section):

Ih : H10(Ω)∩Hǫ+d/2(Ω)→Vh, (2.2a)

Ihu(x¯ i)=

∫

Ki

ψi(x¯)u(x

¯)dx

¯at a vertex x

¯ i of Th. (2.2b)

Here Ki is a (d−1) simplex (an edge in 2D, and a triangle in 3D) with x¯ i as one of its

vertices. For x¯ i ∈∂Ω, Ki is chosen so that Ki⊂∂Ω. The ψi above is the dual basis function

on Ki with respect to the finite element nodal basis φi at x¯ i (restricted on Ki). By the

standard theory [15], without boundary value requirement,

‖u− Ihu‖L2(Ω)≤Ch‖u‖H1(Ω),

|Ihu|H1(Ω)≤C‖u‖H1(Ω).

Thus,

‖u−Qωh u‖L2

ω(Ω)≤‖u− Ihu‖L2ω(Ω)≤max√ωi‖u− Ihu‖L2(Ω)

≤Chmax√ωi‖u‖H1(Ω)≤Chmax√ωimin√ωi

‖u‖H1ω(Ω).

For the weighted semi-H1 norm, we need an inverse inequality [7].

|Qωh u|2H1

ω(Ω)=∑i

ωi|Qωh u|2H1(Ωi)

≤∑i

ωi2(|Ihu−Qω

h u|2H1(Ωi)+|Ihu|2H1(Ωi)

)

≤C∑i

ωi

(h−22[‖Ihu−u‖2

L2(Ωi)+‖u−Qω

h u‖2L2(Ωi)

]+‖u‖2H1(Ωi)

)

=C(

h−2‖Ihu−u‖2L2

ω(Ω)+h−2‖u−Qωh ‖2

L2ω(Ω)+‖u‖2

H1ω(Ωi)

)

≤Cmaxωiminωi

‖u‖2H1

ω(Ω).

Thus, we complete the proof.






0

4

2

1

6

7

8

9

0

4

2

1

6

7

8

@@

@@

@@3

5

Figure 2: f2 is defined on tetrahedron K0126=K0135∪K1356∪K1623, cf. (2.5).

We show next a counterexample. We define a ”reference” function on the referencetetrahedron K=x+y+z≤1|x,y,z≥0:

f1(x,y,z)=

0, if x=y= z=0,z

x+y+z, elsewhere on K.

(2.3)

The function f1(x,y,z) is continuous everywhere except at (0,0,0). At point (0,0,0), thedirectional limit of f1(x,y,z) is between 0 and 1, depending on the direction. In particular,f1(x,y,z)=1 on the vertical axes, but it jumps to zero at the origin. Actually, the functionf1 is very smooth in terms of Sobolev spaces. It is in H3/2−ǫ(K). It can be shown underthe spherical coordinates that | f1|H3/2(K)=∞. For our need, we verify it is in H1(K), viaspherical coordinates:

| f1|2H1(K)=

∫ π/2

0

∫ π/2

0

∫ 1/(cosθ+sinθ(cosα+sinα))

0|∇ f1|2ρ2sinθdρdθdα=

5

12. (2.4)

Next, we glue two f1 functions and a constant function on 1/6 of unit cube, i.e., on atetrahedron K0126=x≥0, z≥x, x≥y, z≤1. K0126 is shown in Fig. 2 as tetrahedron 0126.For convenience, we let Fijkl be the affine mapping (standard finite element reference

mapping) from K to Kijkl : F(0,0,0)=x¯ i, F(1,0,0)=x

¯ j, F(0,1,0)=x¯ k, F(0,0,1)=x

¯ l. Note thatf1 has a constant value 0 on one face triangle (z=0) while having a value 1 on one edge(x= 0, y= 0), on the reference tetrahedron K. Here we need one such function, but alsoanother function which has a value 1 on one face triangle but 0 on an edge, opposite tof1. We define f2 on K0126, cf. Fig. 2,

f2(x,y,z)=

1 on K0135,

1− f1(F−11356(x,y,z)) on K1356,

f1(F−11623(x,y,z)) on K1623.

(2.5)






Note that on the face triangle x¯1x

¯3x¯6, the value of f2|K1623

is constant, the ratio of distances,|x¯x¯6|/|x¯3x

¯6|, on each radial line connecting x¯1 and x

¯for some x

¯on the line segment x

¯3x¯6.

Thus, 1−[

f2|K1623

]x¯ 1x¯ 3x¯ 6

would have a constant value too, |x¯x¯3|/|x¯3x

¯6|, on each radial line

connecting x¯1 and x

¯. This is the key point of the counterexample. Later, we have another

face-value matching, but that is a matching of two linear functions. f2 is continuouseverywhere except at one point, (0,0,1). Its semi-H1 norm can be computed by affinemapping and (2.4),

∫

K1623∪K1635

|∇ f2|2dx¯=

2

3+

13

72. (2.6)

By two reflections, we glue three copies of function f2, to get a function on 1/2 of a cube:

f3(x,y,z)=

f2(x,y,z) on K0126,

f2(F0126(F−10426(x,y,z))) on K0426,

f2(F0126(F−10486(x,y,z))) on K0486.

(2.7)

Here in (2.7), the affine mapping Fijkl is no longer the finite element reference mapping aswe allow ”negative” volume (negative Jacobian). This is to form a mirror image for thepiecewise function. We compute its norm, cf. Fig. 2,

| f3|2H1(K1623∪K1635∪K4263∪K4536∪K4,10,5,6∪K4,6,8,10)

=2

3+

13

72+

3

4+

13

72+

13

72+

3

4=

65

24.

We will reflect f3 once to define our counterexample u on one cube C04871269 = [0,1]3,shown in Fig. 2:

f4=

f3(x,y,z) on x

¯0x¯1x

¯2x¯4x

¯6x¯8,

f3(y,x,z) on x¯0x

¯1x¯9x

¯7x¯6x

¯8.

Finally, we define u on a big cube Ω=[−1,1]3:

u(x,y,z)=

f4(x,y,z) on 0≤ x,y,z≤1,f4(−x,y,z) on 0≤−x,y,z≤1,f4(−x,−y,z) on 0≤−x,−y,z≤1,f4(x,−y,z) on 0≤ x,−y,z≤1,f4(x,y,−z) on 0≤ x,y,−z≤1,f4(−x,y,−z) on 0≤−x,y,z≤1,f4(−x,−y,−z) on 0≤−x,−y,−z≤1,f4(x,−y,−z) on 0≤ x,−y,−z≤1.

(2.8)

The constructed function u(x,y,z) is continuous everywhere except at six mid-face points,(±1,0,0), (0,±1,0) and (0,0,±1). The function is depicted in Fig. 3, where u=1 at purple






xy(-1.0,-1.0)

f=[ 0.0000, 1.0000]z=-0.992

xy( 1.0, 1.0)

f=[ 0.0000, 1.0000]z=-0.510

f=[ 0.0000, 1.0000]z= 0.001

f=[ 0.0000, 1.0000]z= 0.510

f=[ 0.0000, 1.0000]z= 0.992

Figure 3: An H10((−1,1)3) function, jumps at 6 boundary points, cf. (2.8).

colored region, and u=0 at red colored region (only the boundary of the cube). By (2.4),via affine mappings, it follows that

u(x,y,z)∈H10((−1,1)3),

with

|u|H1(Ω)=√

16|u|2H1(x

¯ 0x¯ 1x¯ 2x¯ 4x¯ 6x¯ 8)=

1

3

√390≈6.6833.

This value is also verified by our numerical integration, see the data in column 4 of Table1, in column 4 of Table 2, and in column 6 of Table 4. In fact, the weighted semi-H1 normof u is independent of the jump of coefficients, see (2.10) below, as we purposely madethe semi-H1 norm zero in the region of large weight.

We define a weight function of two constant weights on the domain:

ωǫ0(x,y,z)=

ω1=ǫ−1

0 , if |x|+|y|+|z|≤1,

ω2=1, elsewhere on [−1,1]3.(2.9)

Note that, u≡1 in the diamond region |x|+|y|+|z|≤1 where the weight could be big,ǫ−1

0 . Thus, |u|H1|x|+|y|+|z|≤1=0 and

|u|H1ωǫ0

(Ω)= |u|H1(Ω)=1

3

√390, ∀ǫ0>0 in (2.9). (2.10)






Table 1: The error and order of convergence, for conforming elements.

‖u−Qωh u‖L2

ωhn |Qω

h u|H1ω

# CG # dof

ǫ0=1 in (2.9)3 0.5507 6.2114 19 1154 0.2017 1.4 6.4570 43 6455 0.0720 1.5 6.5919 52 42336 0.0258 1.5 6.6486 53 304817 0.0092 1.5 6.6712 50 230945

ǫ0=10−12 in (2.9)3 235702.2604 1865872.7642 30 1154 85625.7102 1.5 1230494.9355 977 6455 30269.8888 1.5 869303.8840 2481 42336 10702.0200 1.5 614691.3227 3217 304817 3783.7355 1.5 434652.4027 3367 230945

The first level triangulation on Ω is shown in Fig. 4. Then the standard multigridrefinement [20] is applied recursively to define the higher level grids. The second lev-el grid is also shown in Fig. 4. We note that there is no interior node on the first twolevel grids for the conforming finite element functions. So the meaningful computationstarts from the level 3 grid. To verify Theorem 2.1, we select two ǫ0, 1 and 10−12, in theweight ω(x,y,z) (2.9). When ǫ0 =1, the weight ω(x,y,z) is identically 1 on the whole do-main Ω. The approximation is standard, shown in Table 1. Note that u ∈ H3/2−ǫ(Ω),the L2 convergence order is 1.5. More details on the computation will be explainedin Section 4, comparing to that of nonconforming finite element. To confirm the ratio,

max√ωi/min√ωi=√

ǫ0−1=106, in Theorem 2.1, we list the ratio of errors below,

‖u−Qω10−6

h u‖L2ω

10−6

‖u−Qω1

h u‖L2ω1

min√ωimax√ωi

=0.428, 0.424, 0.420, 0.414, 0.411,

when h=1/4, 1/8, 1/16, 1/32, 1/64, respectively. That is, the constant C in Theorem 2.1is roughly 0.4 (C depends also on shape regularity of grids).

In this counterexample, the weighted semi-H1 norm is independent of the coefficientjump. Thus, on a fixed grid, as the conforming finite element is continuous that its semi-H1 norm on the inner diamond region is nonzero, bounded below by some constantdepending on the grid. Thus the conforming finite element projection, interpolation,or PDE solution would have its weighted semi-H1 norm proportional to the 1/

√ǫ0, as

shown in Theorem 2.1.

3 The Scott-Zhang operator on nonconforming elements

We define first the Scott-Zhang interpolation for the nonconforming finite element. Thekey is that the interpolation is subdomain independent, i.e., jump-independent. Then,






with the standard theory on the Scott-Zhang interpolation, we will show the L2 weightedprojection is of optimal order, independent of jump coefficients. Also, we show thatthe nonconforming finite element solution is of optimal order, jump-independent, whensolving elliptic partial differential equations with large jump coefficients.

Because the nodal degree of freedom of P1 nonconforming finite elements (1.1b) inat the barycenter of a (d−1) dimensional simplex, the Scott-Zhang operator is defineduniquely without any face-selection

Inch : H1

0(Ω)→Vnch ,

(Inch u)(x

¯ i)=1

|Ki|∫

Ki

u(x¯)dx

¯at a barycenter x

¯ i of Ki⊂K∈Th.

The Scott-Zhang operator preserves the image function that

Inch vh =vh, ∀vh ∈Vnc

h .

Because the value of Inch u(x

¯ i) is defined by the value of u on the common interface on thetwo sides of Ki, it follows that

Inch = Inc

h |Ω1⊕···⊕ Inc

h |ΩJ.

This cannot be done for continuous finite element functions. In this section, to avoidintroducing discrete norms and inner products, we simply extend the conventional nota-tions to nonconforming finite elements, for example,

|vh|2H1(Ω) := ∑K∈Th

|vh|2H1(K), ∀vh ∈Vnch .

Lemma 3.1. For any u∈H10(Ω), it holds that

|Inch u|H1

ω(Ω)≤C‖u‖H1ω(Ω),

‖Inch u−u‖L2

ω(Ω)≤Ch‖u‖H1ω(Ω),

where C are independent of the jump ωi.

Proof. On one subdomain, by trace theorem, as in [15], we have

|Inch u|H1(Ωi)

≤C‖u‖H1(Ωi),

‖Inch u−u‖L2(Ωi)

≤Ch‖u‖H1(Ωi).

Thus, summing up all subdomains,

|Inch u|2H1

ω(Ω)=∑i

ωi|Inch u|2H1(Ωi)

≤C∑i

ωi‖u‖2H1(Ωi)

=C‖u‖2H1

ω(Ω).

The L2 estimate is proved similarly.






We are ready to prove the main theorem, Theorem 1.2.

Proof of Theorem 1.2. The proof is standard (cf. [15]) except that it involves weights. Letu∈H1

0(Ω). Let Qωh,ncu be defined in (1.6). By Lemma 3.1,

‖u−Qωh,ncu‖2

L2ω(Ω)≤‖u− Inc

h u‖2L2

ω(Ω)≤Ch2∑i

ωi‖u‖2H1(Ωi)

=Ch2‖u‖2H1

ω(Ω).

Next, for semi-H1ω norm, we have, by the inverse inequality,

|Qωh,ncu|2H1

ω(Ω)=∑i

ωi|Qωh,ncu|2H1(Ωi)

≤C∑i

ωi|Qωh,ncu− Inc

h u|2H1(Ωi)+|Inc

h u|2H1(Ωi)

≤C∑i

ωih−2‖Qω

h,ncu− Inch u‖2

L2(Ωi)+|Inc

h u|2H1(Ωi)

≤Ch−2(‖Qωh,ncu−u‖2

L2ω(Ω)+‖u− Inc

h u‖2L2

ω(Ω))+C|Inch u|2H1

ω(Ω)

≤C‖u‖2H1

ω(Ω).

So, we complete the proof.

Finally, we study the nonconforming finite element solution for jump-coefficient el-liptic problems. The weighted Laplace equation is:

−∇·(ω(x¯)∇u)= f in Ω,

u=0 on ∂Ω.

The weak variational form is: Find u∈H10(Ω) such that

(u,v)H1ω(Ω)=( f ,v)L2(Ω), ∀v∈H1

0(Ω).

We seek the nonconforming finite element solution: Find uh∈Vnch such that

(uh,vh)H1ω(Ω)=( f ,vh)L2(Ω), ∀vh ∈Vnc

h . (3.1)

Theorem 3.1. The nonconforming finite element solution in (3.1) approximates the true solutionin the optimal order, independent of the jump coefficients ωi, i.e.,

|u−uh|H1ω(Ω)≤Chβ−1‖u‖

Hβω(Ω)

,

where β>1 is determined by the regularity of the true solution u.

Proof. By Lemma 3.1, the proof is standard, cf. [6], for example. First, by the secondStrang lemma, cf. [9],

|u−uh|H1ω(Ω)≤C inf

wh∈Vnch

|u−wh|H1ω(Ω)+C sup

wh∈Vnch

(u,wh)H1ω(Ω)−( f ,wh)L2(Ω)

|wh|H1ω(Ω)

.






The first term is the approximation error, bounded by the interpolation error, by Lemma3.1,

|u−wh|2H1ω(Ω)≤

∣∣u− Inch u

∣∣2H1

ω(Ω)

=∑i

ωi ∑K∈Th∩Ωi

∣∣∣u− Inch |Ωi

u∣∣∣2

H1(K)

≤∑i

ωi ∑K∈Th∩Ωi

C infp1∈P1(SK)

∣∣∣Inch |Ωi

(u−p1)∣∣∣2

H1(K)

≤∑i

ωi ∑K∈Th∩Ωi

Ch2β−2|u|2Hβ(K)

=Ch2β−2|u|2H

βω(Ω)

,

where SK is the union of triangles touching triangle K. We note that SK can cross severalregions Ωi. But as u∈H1(Ω), the existence of p1 is guaranteed.

The second term is the consistent error, cf. [6].

(u,wh)H1ω(Ω)−( f ,wh)L2(Ω)=∑

i

ωi ∑K∈Th∩Ωi

∫

∂K

∂u

∂n¯

whds= ∑e∈T e

h

∫

eωi

∂u

∂n¯ e

[wh]ds,

where T eh is the set of edges in the triangulation Th, [wh] is the jump of wh on the edge of

e, and ωi∂u

∂n¯ e

is continuous on the two sides of e. We introduce two notations,

f =∫

e

f ds

|e| and f =∫

K

f dx¯|K| .

Then, we continue to estimate the second term,

∣∣∣ ∑e∈T e

h

∫

eωi

∂u

∂n¯ e

[wh]ds∣∣∣

=∣∣∣ ∑

e∈T eh

∫

e

(ωi

∂u

∂n¯ e

−ωi∂u

∂n¯ e

)(wh|+e −wh|Ke+

−wh|−e +wh|Ke−

)ds∣∣∣

≤(∑

i∑

K∈Th∩Ωi

∑e⊂∂K

1

ωi

∫

e

(ωi

∂u

∂n¯ e

−ωi∂u

∂n¯ e

)2ds)1/2

·(∑

i∑

K∈Th∩Ωi

∑e⊂∂K

ωi

∫

e2(wh−wh)

2ds)1/2

,

noticing that on edge e, wh|Ke+and wh|Ke− are two constants, averaging wh on the plus

and the minus side triangle of edge e, respectively,






∣∣∣ ∑e∈T e

h

∫

eωi

∂u

∂n¯ e

[wh]ds∣∣∣

≤C(∑

i

ωi ∑K∈Th∩Ωi

∑e⊂∂K

∫

e

( ∂u

∂n¯ e

− ∂Ihu

∂n¯ e

)2ds)1/2

·(∑

i

ωi ∑K∈Th∩Ωi

(1

h‖wh−wh‖2

L2(K)+h|wh|2H1(K)

))1/2

≤C(∑

i

ωi ∑K∈Th∩Ωi

(1

h‖∇(u− Ihu)‖2

L2(K)+h2β−3|∇(u− Ihu)|2Hβ−1(K)

))1/2

·h1/2|wh|H1ω(Ω)

≤C(∑

i

ωih2β−3|u|2Hβ(Ωi)

)1/2·h1/2|wh|H1

ω(Ω)

=Chβ−1|u|H

βω(Ωi)

|wh|H1ω(Ω).

Combining the two estimates, the theorem is proved.

4 Numerical tests

We numerically solve the weighted L2 projection problems (1.3) and (1.6), for u definedin (2.8) and ω(x,y,z) defined in (2.9). The initial grid is show in Fig. 4. The standardmultigrid refinement is performed to generate high level grids, [20].

In the first test, we use the conforming element Vh in (1.1a). The error and the orderof convergence is listed in Table 1. The resulting linear system is solved by the conjugategradient method. It is interesting to notice that the number of CG iterations for the roughcoefficient is about 60 times of that for constant 1 coefficient. It is apparent the weighted

Figure 4: The initial grid and the second level grid.






Table 2: The error and order of convergence, for nonconforming elements.

‖u−Qωh,ncu‖L2

ωhn |Qω

h,ncu|H1ω

# CG # dof

ǫ0=1 in (2.9)2 0.4827 1.2 4.8000 27 1043 0.1695 1.5 5.5932 34 7364 0.0599 1.5 6.1224 35 55045 0.0215 1.5 6.3954 33 424966 0.0077 1.5 6.5347 31 3338247 0.0028 1.5 6.6064 29 2646016

ǫ0=10−12 in (2.9)2 0.4929 18.7 4.8575 120 1043 0.1772 1.5 5.5887 562 7364 0.0789 1.2 6.1207 844 55045 0.0556 0.5 6.3944 879 424966 0.0519 0.1 6.5342 860 3338247 0.0513 0.0 6.6061 826 2646016

Table 3: The error for nonconforming elements.

‖u− Inch u‖L2

ωhn ‖u−Qω

h,ncu‖L2ω

hn |u−Qωh,ncu|H1

ωhn

ǫ0=10−6 in (2.9)2 0.6271 1.0 0.4903 8.8 2.7376 0.63 0.2937 1.1 0.1696 1.5 2.2915 0.34 0.1159 1.3 0.0600 1.5 1.6410 0.55 0.0443 1.4 0.0215 1.5 1.1974 0.56 0.0167 1.4 0.0077 1.5 0.8812 0.47 0.0063 1.4 0.0028 1.5 0.6494 0.4

L2 error deteriorates when ǫ0 →0, at order 1/√

ǫ0.

In the second test, we use the nonconforming element Vnch defined in (1.1b). On grid-

s shown in Fig. 4, we have internal degree of freedom for Vnch functions on level 2 and

higher. We note that the arithmetic steps are about the same for both conforming andnonconforming methods on a same grid, though the dimVnc

h ≈7dimVh. In fact, the con-dition number for the nonconforming element is even better, as seen from the number ofCG iterations that the number for nonconforming element is about 1/4 of that for con-forming element. We listed the weighted L2 error and the order of convergence in Table2. Also in the fourth column, we list the H1

ω norm of uh. Table 2 confirms the estimate inTheorem 1.2. In particular, we can see in the 4th column of Table 2 that the |Qω

h,ncu|H1ω

istruly independent of ωi, as the two sets of data are nearly identical. However, it seemsthat the computation is beyond computer accuracy that the ‖u−Qω

h,ncu‖L2ω

could not as-

sume the convergence order, 1.5 when ǫ0=10−12. We can compare the data in the secondcolumn of Table 2 with that 4, supposedly ‖u−Qω

h,ncu‖L2ω≤‖u− Inc

h u‖L2ω

. For a computer

accuracy checking, we let ǫ0=10−6 in Table 3. We also increased the stop accuracy for theCG iteration in Table 3. Now, in Table 3, the ‖u−Qω

h,ncu‖L2ω

converges at the correct order,






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