Finite element approximation of the Dirichlet problem ... · The finite element approximation of (1.3) requires the construction of finite dimensional spaces approximating HI(O )

Numer. Math. 49, 343-366 (1986) Numerische Mathematik �9 Springer-Verlag 1986

Finite Element Approximation of the Dirichlet Problem Using the Boundary Penalty Method

John W. Barrett and Charles M. Elliott

Department of Mathematics, Imperial College, London, S.W.7., Great Britain

Summary. This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, A u =f, in a region f 2 c N " (n=2 or 3) by the boundary penalty method. If the finite element space defined over D h, a union of elements, has approximation power h ~ in the L 2 norm, then

(i) for f2=D h convex polyhedral, we show that choosing the penalty parameter e = h x with 2 > K yields optimal H 1 and L 2 error bounds if u~HK+I(Q);

(ii) for Of 2 being smooth, an unfitted mesh (f2___D h) and assuming uEHK+2(f2) we improve on the error bounds given by Babuska [1]. As (ii) is not practical we analyse finally a fully practical piecewise linear approximation involving domain perturbation and numerical integration. We show that the choice 2 = 2 yields an optimal H 1 and interior L 2 rate of convergence for the error. A numerical example is presented confirming this analysis.

Subject Classifications." AMS(MOS): 65N30; CR: G1.8.

I. Introduction

Let f2 be a bounded domain in P,," (n=2 or 3) with a Lipschitz boundary 0f2. We assume either t2 is convex polyhedral or 312 is smooth. Let a and c be sufficiently smooth functions satisfying

al>a(x)>ao>O a.e. in f2 (1.1a) and

cl>c(x)>co>O a.e. in Q. (1.1b)

Consider the elliptic boundary value problem of finding u such that

Au= -V. (aVu)+cu=f in f2, (l.2a)

u=g on Of 2, (1.2b)

344 J.W. Barrett and C.M. Elliott

for prescribed data f~L;(O) and g6HZ(f2). The variational form of (1.2) is: find ueHI(O) = {weHI(O): w=g on dO} such that

where

and

a(u,v)=l(v) Vv~H~(O)={weHI(O): w =0 on dO}; (1.3)

a (w, v) - (a F w, V v)s ~ + (c w, v)e, (1.4)

t(v) = (f, v)~ (1.5)

(w,v)~= Swvclx. G

It follows from the assumptions (1.1) that a(', ') is continuous and coercive over HI(O) and hence (1.3) is a well-posed variational problem. From elliptic regularity theory it follows that u~HZ(O).

The finite element approximation of (1.3) requires the construction of finite dimensional spaces approximating HI(O ) and Hi(O ). This is achieved in practice by fitting a mesh to ~2. Either partitioning O into elements if O is polyhedral or approximating O by O h, a union of isoparametric elements, if dO is curved. However, if QO is smooth and the imposed boundary condition is of the Neumann or Robin type; that is, (1.2b) is replaced by

d u a ~ - + c t u = g on dO, (1.6)

dv

where v denotes the outward pointing unit normal on dQ and ~(x)>0; then it is not necessary to fit the elements to the boundary in order to retain the optimal rate of convergence, see Barrett and Elliott [6]. They show that if the finite element space defined over D h, a union of elements, has approximation h K in the L 2 norm and if the region of integration is approximated by (P with dist(O, Oh)<Ch ~ then the optimal rate of convergence for the error in the H a and L 2 norms can be retained whether O h is fitted (Oh--D h) or unfitted (O h c Dh). We note that unfitted meshes have useful practical applications to free and moving boundary problems as at each step only the domain of integration and not the mesh has to be adjusted; see Barrett and Elliott [3, 5] for example.

The need to fit the mesh, Oh-- -- D h, in the case of the finite element approximation of (1.3) is that the Dirichlet boundary condition is imposed essentially on the finite element trial space. To remove this constraint and be able to explore the possibility of using an unfitted mesh, Oh___ D h, when d O is smooth for the Dirichlet problem the boundary condition has to be imposed weakly. One method of imposing the boundary condition weakly is the penalty method, as studied by Babuska [1].

The penalty method considers the penalised problem of finding u, such that

Au~=f in O, (1.7a)

adu~+e-l(u~-g)=O on OO, (1.7b) d v

Finite Element Approximation of the Dirichlet Problem 345

where e, a positive constant, is the penalty parameter . The variat ional form of (1.7) is: find u ~ H ~ ( f 2 ) such that

a~(u~,v)=l~(v) Vv~HI( f2) , (1.8) where

a~(w, v) = a(w, v) + ~- 1 (w, v)o~, (1.9)

l,(v) - l(v)-t- e - 1 (g, v ) o~ (1.10) and

( w , v ) o ~ = ~ wvds . OG

Since ~ > 0 it follows that a~( ' , ' ) is continuous and coercive over Hi(f2) and hence (1.8) is a well-posed variat ional problem.

It is a simple matter , see Theorem 2.1 in the next section, to show that the solutions of (1.3) and (1.8) satisfy

Ilu-u~llo,~_- < Ce Ilufl 2,a, (1.11)

where C is a constant independent of e, and so u~-,u as ~ 0 . As the boundary condit ion (1.7b) is of the Robin type the finite element approx imat ion of (1.8) can be based on an unfitted mesh if ~?f2 is smooth. The impor tan t p rob lem is how to choose the penalty paramete r ~ with respect to h and K so that uh~, the finite element approximat ion of (1.8), converges to u at the opt imal rate as h ~ 0 .

The outline of this paper is as follows: in the next section we define and analyse a finite element approx imat ion to (1.8) in the absence of variat ional crimes; that is f2h=f2___D h and all integrations are per formed exactly. In Subsect. 2.1 we consider the case of c~f2 smooth and so (1.8) requires evaluating integrals over curved domains and hence is not practical. Babuska [1] analysed this approximat ion for Poisson's equation, a = 1 and c=-0, with homo- geneous boundary data, g = 0 . Setting e = h a, 2.>0, Babuska [1] showed that u and u~ satisfied the following error bounds for ~ > 0 arbitrary

Ilu -u~ll 1,~_- < Ch "l-a IlullK,~ (1.12a) and

Ilu-u~llo,a < Ch u~ Ilurlg,~, (1.12b) where

m = max [1, �89 + 1)], (1.13 a)

p = (K - m) / (K - 1), (1.13 b)

g~ = min [2., K - 1, g -�89 + 1), K + �89 - 3)] (1.14 a)

(1.14b) and

Po = m i n [ 2 , P +12 ,P l +�89 +P] .

Thus for piecewise linears, K = 2 , the only choice of 2 which yields an opt imal H* error estimate, / q = l , is 2 = 1 and this leads to a subopt imal L 2 error estimate, # o = 1 . For K > 3 there is no choice of 2 which yields an opt imal H 1 estimate.


Assuming more smoothness on u, ueHr+z(f2), we show in Subsect. 2.1 that these bounds can be improved so that (1.12) holds with 6=0 , # 1 = # * and Po =#~ where

p* = min [2, K - 1, K --�89 (1.15 a) and

p* = min [2,/~* + 1, p* +~2, p l l * - � 8 9 (1.15 b)

Thus with K = 2 an optimal H a error bound is obtained if 2 is such that 1 < 2 < 2 which leads to #~=min[2 , 1+�89189 Hence the best choice of 2 is ~ leading to , _5 /~0-3- This is an improvement over (1.14), but of course still not optimal. For K =3 the choice 2 = 2 yields an optimal H a error bound.

In Subsect. 2.2 we consider the case of f2 being convex polyhedral. We analyse a finite element approximation to (1.8) in the absence of variational crimes assuming f2 h=- f2 =- D h. This is a less interesting case as in practice it is straightforward to impose the boundary condition essentially using a finite element approximation of (1.3) as opposed to using the penalty method. How- ever, Zhong-Ci Shi [14] has analysed this case to explain the numerical results of Utku and Carey [13]. Zhong-Ci Shi [14] shows that for piecewise linears

choosing 2 = 1 one obtains an optimal H a error bound if ~veH~(Of2). We show

for K ~ 2 that if usHK+a(O) then choosing 2 > K one obtains optimal H a and L 2 error bounds.

In Sect. 3 we return to the more important case of 0f2 smooth and analyse a fully practical piecewise linear finite element approximation of (1.8) on an unfitted mesh involving domain perturbation and numerical integration. We show that with 2 = 2 the results (1.12) and (1.15) remain valid. Moreover, although the global L 2 error bound is only O(h ~) we obtain an optimal order interior L 2 error bound over a domain f2 o c c f21 c c f2 h. Finally, in Sect. 4 we report on some numerical calculations with piecewise linear elements which confirm the error bounds derived above. The results show the superiority of choosing 2 = 2 as opposed to 2 = 1; which is generally quoted in the literature, see Utku and Carey [13] and King [8] for example, not only asymptotically but also for finite h.

We end this section by stating the notation we shall adopt throughout this paper. Given m~N and a bounded domain G in •",

W"P(G) = {w~LP(G): D~weLP(G) for all Iq] <m} and

Hm(G)_= W",2 (C)

denote, for l < p < m , the standard Sobolev spaces, where we use the multi- index notation for the derivatives. We use the following norms and semi-norms on functions w defined by

Ilwll,,,p,~=[ Y, D"w p 1alp 0 , ~ . ~ , I l w l l m , ~ - I l w l l . . . 2 , G ,

Iwt.. ,~,~--[ Y~ D"w" 1~/,, 0,~,G- , IWI..,G--IwI..,2,G I,t[=m

Finite Element Approx ima t ion of the Dirichlet P rob lem 347

and

I w Io.p,~-- [jlwlPdx] 1/~ G

for m e n and l < p < ~ with the standard modification for p = ~ . For dG a section of the boundary of G we define wm'p(dG) and II.ll...p,~G as above with G replaced by dG if aG is of class cg,,,1, see Kufner, John and Fucik ([10],

N p. 305). If dG is only piecewise cs that is, 0 G = U ~G with ~G of class ~,,,, 1 ; we define i= 1

P (1.16) Ilwll,.,p,0G-- IIWlIm,~,0,G i=1

The measure of a domain G is denoted by re(G). Throughout C denotes a positive constant independent of h and the penalty parameter e whose value may change in different relations. We require the trace inequalities: for OG of class cs t

IlOnwllo,~G< C Itwlll,l+ 1.~ VweHInI+I(G), (1.17a)

and for ~G of class ego,1 and piecewise ogre-l,1 this implies that for m e n

Ilwll,._l,e~<CllwN,.,G VweH"(G) (1.17b) and

~w <=CIIWlIm+I,G VweHm+I(G) �9 (1.17c) m,- 1,~G

We require also the fractional Sobolev space Hm-~(aG), meN, for 0G of class ~m--1,1 with norm defined by

Ilwll.,-~.~G--

The following trace inequalities hold

inf IlvtI~,G. v~Hm(G)

v ~- w o n ~G

Ilwllm_ ~,~G ~ IIwlIm, G VweHm(G) (1.18 a) and for m > 2

~-v w < C IlWllm,~ Vwenm(G) �9 (1.18b)

In (1.17) and (1.18) above v is the outward pointing unit normal to 0G and C is a constant independent of w; see Kufner, John and Fucik [10].

Finally, we require the Friedrichs' inequality for ~3 G of class fro, 1

iiwll~,~<C{iwl~,G+ 2 Ilwllo.0G} VweHI(G), (1.19)

where C is a constant independent of w.

2. Error Bounds in the Absence of Variational Crimes

Before discussing the finite element approximation of the penalised problem (1.8) we first show that u ~ u as e~0.

348 J.W. Barre t t and C.M. Elliott

Theorem 2.1. 7he solutions u and u~ of (1.3) and (1.8) satisfy:

I l u - u ~ l l o , ~ _ - < C~ Itull 2,~.

Proof From (1.3) and (l.8) it follows that

a~(u-u~,v)=<a~v,V>o ~ Vv~Hl(f2)

and so choosing v=u-u~ we obtain

~-1 ilu_u~ii2,ao <a~(u_u~,u_u)

From the trace inequalities (1.17) it follows that

[In -u~l l o,0~ < C~ Ilull 2,o' Observe that

I lu -u~ l lo ,~= sup

For any r/~L2(f2) define z such that

(2.1)

(2.2)

(2.3)

(2.4)

and T= {z~T*: zest24:0}.

Az=q in f2 z = 0 on ~12. (2.5)

It follows from elliptic regularity for either ~ f2 smooth or f2 convex polyhedral that

[Izll 2,~< C Ii~tll o,~, (2.6)

Note that as z~H~(f2) we have

[(u-u~,Az)~l= a(u-u~,z)-Iu-u~,a~z ~ Ov / o~l

__< C~ Ilull z,~ llzH z,~, (2.7)

where we have noted (2.2) and applied the bound (2.3) and the trace inequality (1.17c). Combining (2.4), (2.5), (2.6) and (2.7) yields the desired result (2.1). []

Let D* be a bounded domain in R" containing f2 such that b *= U ~, ~ T*

where T* is a collection of disjoint open regular simplicial elements z, each with maximum diameter not exceeding h. We have in general as f2 is not a union of elements that

f2~_D~_D*(h)cR", where

b = [..)~ ~ T

(2.8a)

(2.8b)

(2.8c)


In addition, we set

B= {zET: m(t-c~ ~ I2) 4=0 in R"-1}. (2.8 d)

Associated with T is a finite dimensional subspace S h of C~ depending upon an integer K > 2 . Let 7rh: C~ h denote the interpolation operator. Sometimes we shall require a generalised interpolation operator 7r~: L2(D)~S ~, see Clement 1-7] for example. We assume that the following approximation properties hold:

(AI) For integers K' and m satisfying K'>_m>O, K > K ' > 2 and for p~[2, ~ 3

K ' - - m Iw-rrhwl,,,p,~<Clh IwlK,,~,~

Vw~WK"P(z), Vz~T, (2.9 a)

and for integers K' and m satisfying K ' > m > 0 and K > K ' > 0

I w - rc~,wl,,,,~< C2h~'-" lwlr, ~

Vw~HK' (z), Vz~T; (2.9b)

where C a and C 2 are constants independent of h and w. This assumption is satisfied by the standard piecewise polynomial spaces S h used in practice.

As the mesh is unfitted, in order to define the interpolate (generalised interpolate) in S h of a given function wEH2(f2) (L2(f2)) it is necessary to extend the domain of definition of w. It is convenient to introduce a domain f ) c R" with a smooth boundary such that

12~_D~_(2 Vh<h o. (2.10)

For all integers s > 0 there exists an extension operator E: H~(f2)---,H~((2) such that

E w = w onI2 (2.11a) and

llEw[l~,~ < C Ilwlls,~, (2.11 b)

where C is independent of w (see Kufner, John and Fucik [10]). A finite element approximation to (1.8) is: find u ~ S h such that

a~(u h, Z) = I~(Z) Vz~S h �9 (2.12)

This approximation has been analysed by Babuska [1] for Ot2 smooth and assuming u~HX(g2). As (2.12) requires the exact evaluation of integrals over curved domains it is not a practical finite element approximation. In Sub- sect. 2.1 we also analyse this approximation in order to see what one can expect from the penalty method in an ideal situation. Assuming more smoothness on u, u~HK+2(O), we are able to improve on the rates of convergence in the H a and L 2 norms given by Babuska [1]. In Sect. 3 we see that these rates are retained by a fully practical piecewise linear finite element approximation of (1.8) involving domain perturbation and numerical integration. In Sub-

350 J.W. Barre t t and C.M. El l io t t

sect. 2.2 we analyse the approximation (2.12) for the case of f2 being convex polyhedral with t) = D.

In the results that follow we adopt the notation 11"[12=a( ~ Lemmas 2.1 and 2.2 below derive abstract H ~ and L 2 error bounds for the Approximation (2.12).

Lemma 2.1. 7he solutions u and u~ of (1.3) and (2.12) satisfy:

-u~lla +~- u - e a - - - u ~ 1) 0, OD

< inf Ilu--Zlla~+~ -~ - ea6v z~S h

Proof. From (1.2) we have that

(2.13)

a(u ,v)=l(v)+<a~v,Vlo ~ Vv~H~(O).

Choosing v = z and subtracting (2.12) we obtain

(u c~u h \ a(U --Uh, Z ) + 8 - 1

Therefore for all x~S h it follows that

(2.14)

Vz~S h. (2.15)

h 2 e - ' u ~u hll z Ilu - u.tl. + - e a - - - u ~ ,

~ O,~D

"/o = a ( u - u ~ , u - z / + ~ -~ u - ~ - u ~ , u - ~ o ~ - z ~

<= [llu-u~ll~ +llu-zll:.]+- T u-ea---u~ov o.oo - ~ a ~ v - z o.o~

Hence the desired result (2.13) holds. []

Lemma2.2. Assuming (A1) holds the solutions u satisfy:

Ilu-u~llo,o_- < C[h flu -u~lla + Ilu-u~ll o , j

where ~ - E z.

Proof. As

h and u s of (1.3) and (2.12)

sup u au ,,11 - e a - - - u ~ i (2.16) ,~H~o) ,~o~ I_ Ilzll2,o J ~v o,oo'

U h I(u- ~,n)ol Iju-u~flo,o= sup r/~L2 (~Q) Ilrtll0,o

Finite Element Approximation of the Dirichlet Problem

defining z by

[(u - u ~ , A z ) a [

351

(2.5) it follows from (2.15) that for all z ~ S h

h OZ = ~

/u eu " \ / h az \ I a ( u - -uh , z - - z ) + ~ - 1 \

< Ilu --uhlla Ilz-;tlla + h ~Z IlU--U~I[o,o~ a~V v 0 ,~

Choosing )~ = nh~, the desired result (2.16) follows from (2.9), (2.11 b), (1.17 c) and (2.6). [ ]

F rom the abstract error bounds (2.13) and (2.16) we see the importance of the approximat ion property of S h on Of 2. F rom the proof of the trace theorem in Necas [11] p. 15 it follows that V 6 > 0 and VweHa(f2),

qlwll2 a < = a l l w l t ~ , ~ + f a - , 2 Ilw[Ix,a, (2.17)

where C is independent of 6 and w. Applying this bound with 6 = h -~ it follows from (A1) that for 1 < r < K and for all w~H"(D) that

I lw-~gwlt2,0a<h -~ I I w - ~ w l l ~ , o + Ch I Iw-~wl l~ ,o

< C h 2 , - 1 2 (2.18) = Iwl~,D.

However, if we assume more smoothness on w it follows from (A1) that for O < r < K and for all w e H ' + 2 ( D )

11 w - ~h wll 0,~a --< C II w - r~hwll o, ~ , ~

< C Ilw -nhwll o, ~,a

<= C h" l w[ . . . . D < C h" llwll,+ z,o" (2.19)

The application of the bound (2.19) instead of (2.18) leads to the improvement in the rate of convergence over that given by Babuska [1].

2.1. c3 t2 Smooth

We make the following regularity assumptions on the data:

(A2) Of 2 is of class cg~+l,1, f~HK(K2), g~H~+~(0f2); and a, c~CK+I(f2). It follows that the solution u of (1.3) is such that u ~ H K+ 2(f2).

In order to obtain H 1 and L z error bounds we require the following approximat ion result.

352 J.W. Barret t and C.M. El l iot t

Lemma 2.3, Assume (At) and (A2) hold. Setting e=h ~, ).>0, it follows that:

-1 ~u 2 inf 000] fCh2"l ]lull2'~ (2.20)

<=~Ch 2"~ Ilull~+2,o' where

#~ = min[2, K - 1 , K - � 8 9 1), K + �89 3)] (2.21 a) and

#* = min [-J., K - 1, K - �89 (2.21 b)

Proof. If u~H~(f2) then ~ H ' - ~ ( ~ 3 f 2 ) for 2 < r < K + 2 . Hence there exists a

~u harmonic function v~H'-l(f2) such that v=a~v on dr2 and from elliptic

regularity theory and the trace inequality (1.18b) it follows that

llvll~_ 1 , ~ - < _ C ~ r ~ ,_~,~ _<_ C Ilull,,o. (2.22)

Choose z = n g f t - e n ~ in (2.20), where ~=Eu and ~=Ev. The approximation assumption (A1), (2.11 b) and (2.22) yield

<= C[hZ(K -1)[fil2,D+g2 ~ 2 Ilvlll,v] <=C[h2(K-1)+h2~. ] 2 IluNK,o. (2.23)

The approximation result (2.18), (2.11 b) and (2.22) yield

u ~ u II 2

< 2 I-Ilu- ~f, tTII ~,0o + ~ 2 I Iv-z~<l 2 O,d f/']

< CI -has- x [t~12,0 + h2z+2r-3 I~1~- 1,v]

<= C[h2X - i + h2a+ 2K- 3"] tlullK,o.2 (2.24)

Combining (2.23) and (2.24) yields the desired result (2.20) with (2.21 a). Assuming u~H r+ 2(0) and choosing Z = 7~h/~ - - g 7Zh ~ in (2.20), it follows that

[lu_nn~+enh~ll2<C[h2(X-ll+hZa] 2 IlulIK+ x,o (2.25)

and from (2.19), (2.11b) and (2.22) that

~U 7~h~l"~-e~h~ 2 ~-~C[ h2K+h2;~+EK-2] 2 - Itull~+2,o. (2.26) u - - 8 0 " ~ 0,dl2

Combining (2.26) with (2.25) yields the desired result (2.20) with (2.21b). []


Combining the above lemmas we obtain the H 1 and L z error bounds.

Theorem 2.2. Assume (A1) and (A2) hold. Setting e=h ~, 2 > 0 , the solutions u and uh~ of (1.3) and (2.12) satisfy:

~ Ch ~2 IlulIK,~ (2.27)

and for i= 0 and 1

" ~Ch u' IlullK,a ; (2.28) lu -un~li.a < [Ch"*/lullg+ 2.~

where #1 and #* are as defined in (2.21),

/~*) = m i n [2, p~*)+�89 (2.29) and

/t~o *) = min [#~*~ + l,/~*) + �89 - 2), #~*)]. (2.30)

Proof. From (2.13) and (2.20) we obtain

u - ~ a ~ v v -u~ o,oa ~Ch "'+~lrutrK,a (2.31) 0u h < [ Chu~+�89 I[U[[g+ 2,a"

As

_-< u ~^u h ~u [lu--uhHo,~a --gaov--U ~ +e a ~ v , (2.32)

0,(312 O,0D

applying the trace inequality (1.17c) and the result (2.31) we obtain the desired results (2.27) and (2.29).

As ['[1.o----C ]l'll~ the H 1 error bound (2.28) follows directly from (2.13) and (2.20). The L 2 error bound (2.28) follows directly from (2.16), (2.18), (2.11b), (2.13), (2.20), (2.27) and (2.31). [ ]

The value of #1 is exactly that obtained by Babuska [1]. The value of #o is a slight improvement to that given by Babuska [1], see (l.14). Clearly there is a substantial improvement in the rate of convergence by assuming more smoothness on u and this is usually not a restriction in practice. As stated previously an optimal H 1 error bound is now obtained for piecewise linears, K =2, by choosing ). such that 1 _<2 < 2, but this does not lead to an optimal L 2 error bound. For K = 3 the choice 2 = 2 yields an optimal H 1 error bound, but once again a suboptimal L 2 bound. For K => 4 no choice of 2 leads even to an optimal H 1 error bound. Therefore in practice the penalty formulat ion for dr2 smooth and f2~_D should only be used for low order finite element spaces. In Sect. 3 we analyse a fully practical piecewise linear approximat ion to (1.8) and show that the bounds (2.27) and (2.28) remain valid if 2 is chosen to be 2. Moreover, al though the global L 2 error bound is only O(h ~) we obtain an optimal order interior L 2 error bound over a domain O o c ~ f21 c c f2 h.

Before discussing the case of f2 being a convex polyhedral we end this subsection by mentioning the extrapolat ion method of King [8]. Let uh~ solve

a,(u~, Z) = l,(z) VZe Sh (2.33)


where e=2-JTh, j = 0 , 1 , 2 .... and define

V2- Jh ~uh2- J.eh

where h and 7 are fixed. Using Richardson's extrapolation with respect to j,

u~ x) = 2vh/2 - v h (2.34 a)

2Ju(j - 1) __U(j-- 1) u~j) - - hi2 2j_ 1 j__>2; (2.34b)

one obtains the following error bounds for Poisson's equation,

Ilu -u~K-2)III,~=< Ch K-1 IIUI/K,~ (2.35 a)

hlu--Uth ~- X)ll 0m < Ch ~ Ilull~+~+~,~ (2.35b)

where 6 > 0 is arbitrary. Once again this analysis is for smooth 0 0 and in that case (2.33) is an impractical method. Numerical results for this approach when f2 = D is a polygon are given in King and Serbin [-9].

2.2. f2 Convex Polyhedral

Zhong-Ci Shi [14] analysed the apl2rox_imation (2.12) for Poisson's equation assuming that f2 is polygonal and I2=D. As stated previously, this is a less interesting case as now the Dirichlet boundary condition can be imposed essentially in practice using a standard finite element approximation of (1.3). However, this analysis was undertaken to explain the numerical results of Utku and Carey [13]. They solved the problem

- V 2 u = 0 in f2= [0,1] x [0,1] u = g on 0f2, (2.36)

where g was chosen so that the solution u = x + y - 2 x y . They found in practice using piecewise linear elements with 2 = 1 that the approximation (2.12) converged at the optimal rate in H t, although their suboptimal analysis predicted O(h�89 Zhong-Ci S hi [14] showed that for f 2 - D polygonal, e = h ~ and

0u �89 assuming that ~vv~H (Of2) the solutions of (1.3) and (2.12) for K = 2 satisfy

Ilu-u~[ll a < C h ul [[lull2,,+ 8 u 2 ]~, (2.37) ' = OV �89

where #1 = rain [2, 1,�89 -2) , �89 + 1)]. (2.38)

The proof of (2.37) and (2.38) follow the same lines as for 0/2 smooth, except

that in the proof of Lemma 2.3 u~H2(f2) does not imply that ~vU~H~(0f2) so

one needs to make it an assumption. Hence the value of #1 agrees exactly with


the analysis of Babuska [1] for 0t2 smooth, (2.28) and (2.21a) with K =2 . du

Clearly the result (2.37) with (2.38) applies to the problem (2.36) as ~vv~H 2 (d t2).

We now present an alternative analysis. For f2=D polyhedral we define dS h where S h = S h • d S h and

Sho = {z~Sh: ; (=0 on 0f2}. (2.39)

We introduce Phg, the LZ(df2) projection of the boundary data g onto dS h, defined by: Phg~OS h such that

(g --Phg, Z)e~ =0 VZ ~dSh (2.40) and set

sh-- {Z ~Sh: Z =Phg on dr2}. (2.41)

Then a finite element approximation of (1.3) is: find uh6S h such that

a( uh, Z) = I(Z) VzeSho �9 (2.42)

As f 2 - D if we let e-~0 for fixed h, uh~u h. It is well known that uh--~U at an optimal rate in the H 1 and L 2 norms and so we would expect the convergence

n to u for fixed K to tend to the optimal rate as ,~ increases. This is not rate of u~ reflected in the analysis of Zhong-Ci Shi [14] due to the presence of the term �89 in (2.38). The reason for this is the analysis of Zhong-Ci Shi [14] does not make use of the fact that f2 -D. This we do below.

Firstly, we require some assumptions. Note that the interpolation operator

Xh induces an interpolation operator on C~ into dS h, which for notational convenience is still denoted by xn. We assume the following approximation property holds:

(A3) For an integer K' with K>K'>_2

Ilw -nhWllo,0o, < Cah K' llwll~, 0~

VweHr ' (dt2,) n C~ VzeB, (2.43)

where Ot2~=df2c~ and C a is a constant independent of h and w. In addition we assume the inverse inequalities:

(A4) For all z~S h IIZIll,,<C4 h-1 IlXllo,, VzeT (2.44a)

and Ilxll o,o~,~ < C5 h-<~-1~/2 IlXllo,~ VzeB, (2.44b)

where C 4 and C 5 are constants independent of h and Z. Once again these assumptions are satisfied by the standard piecewise polynomial spaces S h.

Lemma 2.4. Assume f2 =D polyhedral (A1) and (A4) hold. The solutions u and u~ of (1.3) and (2.12) satisfy:

Ilu-uh~ll~ < Ilu-rchull~ + Ch-~['ltu--rchullo,o~+ Ilu-u~ll o , j . (2.45)

356 J.W. Barrett and C.M. Ellion

Proof. F r o m (2.12) and (1.3) we obtain for all z~S~ that

a(@ X) = G(u~, Z) = l~(z) = I(Z) = a(u, ~). (2.46)

Let rh~3s h be such that rh=~hu--u~ on 3 0 and so from (2.46) it follows that

h 2 IlU--Uella =a(u--GU--~hU+1" h)

=</Fu -uh~l[~ [l[u-~hull.+ II ?11 j . (2.47)

Let B* denote those elements z~T with a node on ~2. It follows from the inverse inequalities (2.44) that

Itrhll~,~ = • ]lrhll~,~ <= Ch-2 2 ]trh]lg,~

Chn-2 2 h 2 Ilr IIo . . . . f e B *

<=Ch.-2~ h 2

< _ _ C h - l Z a 2 ]Jr [] o,oa~ "c~B

=Ch-1 h 2 t[r ]lo,0a. (2.48)

As ][-[[,-<_ C [[" t[1 the desired result (2.45) follows from (2.47) and (2.48). []

Theorem 2.3. Assume Q=D convex polyhedral (A 1), (A3) and (A4) hold. Setting e = h ~, 2 > 0 , the solutions u and u~ of (1.3) and (2.12) satisfy:

~C h ~i~ t1 u {1K,a (2.49) ]IU--U~[IO,OO < [Ch~ ~ ]lujlK + l, ,

and for i= 0 and 1

I C ha' [Jul[ K,a (2.50) lu-u)l i ,~ < ( C h~, IlulIK+ t,~'

where /i 2 = min [2, K - �89 K - 1 + 1 2 ] , (2.51 a)

fi* = min [2, K, K - 1 + 1 2 ] , (2.51 b)

#J~l*) = min [K - 1,/~*) - �89 (2.52) and

/7~o*) = min [#~*) + 1,1~*)]; (2.53)

assuming u is sufficiently regular.

Proof. F r o m (2.13) it follows that

rll ou .11 ~ ~ 0u~ ] Ilu-u~ll~,or~<2 [ u-~o~-u~ o,o+~ O-?llo,o~J

aOU_u 2 . (2.54) <x~s~infE2~jlu-zll2+4llu-zll2~ II 3Vllo,o~


Choosing )~=r~hu, applying the approximation results (2.9a), (2.18) and the trace inequality (1.17c) yields the desired result (2.49) with (2.5la). Applying (2.43) and the trace inequality (1.17b) instead of (2.18) yields the desired result (2.49) with (2.51 b).

The result (2.50) with (2.52) follows from (2.45), (2.9a), (2.18) and (2.49). The result (2.50) with (2.53) follows from (2.16), (2.49) and (2.52). []

Hence it follows for a given K>2 , if usH~:(O) choosing 2 > K - � 8 9 yields an optimal H 1 error bound and if ueH~:+~(f2) choosing 2 > K yields optimal H a and L z error bounds.

3. Error Bounds in the Presence of Variational Crimes

As stated previously the approximation (2.12) is not practical for ~?O smooth because it requires the exact evaluation of integrals over curved domains. We now study a fully practical approximation using piecewise linears, K = 2, under the assumption (A2). The domain f2 and its boundary ~3 I2 are approximated by O h and ~12 h. We make the following assumption on this boundary approximation: (A5) dist(O, Oh)<= Ch 2, (3.1)

which is achieved by choosing OO h in each element f eB to be the linear polynomial interpolating ~f2. The interpolation points being where ~2 crosses either the elements sides (n=2) or edges (n= 3) yielding a polyhedral domain O h"

We now have

where

and

(3.2a)

~h=+~, f , (3.2b)

Th-- {z~T*: r c~f2h+0} (3.2c)

B h= {zsTh: m(r Of 2 h) +0 in IR"- 1}. (3.2 d)

We require the approximation property (2.9) to hold now for all z ~ T h and f) in (2.10) to be such that

Ohc. Dh~___~-2 Vh<h 0 and O~f2. (3.3)

It follows immediately from (3.3) and the proof of the trace theorem in Necas ([11], p. 15) that the constants C are independent of h for the trace inequalities (1.17) with G-=O h. Similarly (1.19) and (2.18) hold with dI2 and 12 replaced by ~ O h and f2' with C independent h and so for 1 < r < 2

tlw-~,wH0,0m < Ch'-�89 (3.4a)

Assuming more smoothness, as in (2.19), we have for 0 < r < 2

IIw --TthWll o,oeh < Ch" II wll,+ 2,o~. (3.4b)

358 J .W. Ba r r e t t a n d C.M. El l iot t

In addition there exists an extension operator Eh: ns(f2h)--*ns(~2), in analogue to (2.11), for all integers s > 0 such that

and Ehw=w on O h (3.5a)

IIEhwJl~,~ C I/wlls,~ (3.5 b)

where C is independent of w and h as it depends only on the Lipschitz constants of ~O h, which are clearly independent of h, by its construction, see Babuska and Aziz ([2], p. 30).

Because, in general, f2hSf2 it is necessary to extend the data. We assume that f, g, a and c are restrictions to f2 of feH2(t)) , ~eH4(f)), # and c e C 3 ( ~ ) )

and in addition that ~>6(x)>~o>O Vx~(2,

We define

and

~ >~(x)>?o>O Vxe(2.

.~ w= -V. (~V w)+ ~w

fib(w, v) - (# V w, V v).. + (~ w, v)~.

where v h is the outward pointing unit normal to O f2 h. We define also

and

a~(w, v) = a"(w, v) + ~- 1 (w , v ) o . . ,

i~(v) = (f, % .

~(v) = "0(~) + ~- ~ (~., ~ )o . .

It follows from (3.2a) that for all w, veHl(f2 h)

I~h(w, V)I ~ C Ilwl[ i,~. Ilvll x,.h,

(3.6)

(3.7a)

(3.7b)

(3.8a)

(3.8b)

(3.8c)

(3.9)

where t is a simplex in 1t" and V is a simplex in R . - 1 being the boundary face of a tcg-~n~. We now choose integration rules over t and V. With ~,,~.+1 l ~ i l i = 1

and t2hc~z=Ut VzeT h (3.11a)

Ot2hc~=V VzeB h, (3.11b)

where C is independent of h and e. A fully practical piecewise linear finite element approximation to (1.8) is:

find u~eS h such that h h h a,(u~,z)=l,(x) Vz~S h, (3.10)

where a~(.,.) and h l,(.) are approximations to ~ ( . , . ) and ~(.). As the mesh is unfitted we have to approximate integrals over the sub-

regions f2hnz and 0f2hc~f for those elements zEB h. The subregion f2hnz is either a simplex or a union of 2 (3) simplices for n =2 (3) so that


being the vertices of t and {a~}~'= 1 the vertices of ~ we define quadrature rules It(v ) and lr(v) approximating the integrals ~v(x)dx and ~v(x)dx by

t y

and

mit~ .+ 1 (3.12a)

[ ~ - ~ (v(a~) + v({(a 1 + a2) ) + v(a2) ) 4

I~(v) = / + a;) [ 3 i , j

if n = 2

if n=3.

(3.12b)

We denote by (.,.)h and ( . , . )h the approximations of (',')nh and ( ' , ' ) 0~ , by the quadrature formula (3.12a) and (3.12b), respectively. Then we set

a h(., .)_= (~17., V.)h + (y., .)h + e- ~ ( . , . )h (3.13 a) and

l h(.)_ (f, .)h + e- 1 (~,.)h. (3.13 b)

Lemma 3.1. With ah~( ", ") and h l~(') defined by (3.13) it follows that (i) a~(z,Z)~ Ca~(Z,Z) VzeS h,

(ii) for all weH2(~) and for all w h, z e S h' (3.14)

t ' , , a h w h " ~ ~htWhZ)-- ~( ,ZJI

< Ch{h IIwll 2.~h+ IIw-whll x. ,4 Ilzll a.~h (3.15 a) and

[~(z)-l~(z)l <ChZ{l l f l l2 ,~h[Iz l l , ,nh+e -1 II~lI 3,~ Ilzll 0,0,~,}. (3.15b)

Proof As (3.12b) integrates quadratics exactly it follows that

(w h, Z) h = (w h, Z)~ah Vw h, zeS h . (3.16)

Therefore the result (3.14) is a direct consequence of Lemma 5.10 in Barrett and Elliott [6]. Likewise the results (3.15) are a direct consequence of Lem- mas 5.8 and 5.9 in Barrett and Elliott [6] and the trace inequality (1.17b). []

Remark 3.1. The result (3.14) implies the well-posedness of the approximation (3.10). We note that the sampling points of (-,.)h lie in l] so that the approximation u~ h is independent of the extensions f, 8 and ?. []

In the results that follow we adopt the notation

II [l#h--ah(', ") and I1" 2--~h Ila#----a,(', ").

Since the approximation u~ is independent of the extension f, f may be chosen for the convenience of the analysis. A suitable choice is

f=At~, (3.17)


where fi=Eu. Clearly f = A u = f in f2, which implies, together with (A2) and (2.11) that

llfll2,~, = [//iffll2m, < C It~11,.~, _-< C Ilull4.•. (3.18)

The following lemma derives abstract H 1 and H-' , r>O, error estimates. The negative norm estimate is required later in obtaining an interior L 2 error bound.

Lemma 3.2. Let (A2) hold. The solutions u and u~ of (1.3) and (3.10) satisfy:

~h h ~.. la~(~, x)-a~(~, Z)]] jl~7-u~l]~,<C inf II~-~rla~+sup ~ /

~ S " z~S" rlZIla~ J

+ s u p II~(z) h --I,(z)I (3.19)

and for r > 0

I [ ~ - u ~ l l - , , o , < C{ l l~ - u h l l 0 ,O,_~ + ITS-- U~It 0,00"

+ sup II z l1 ;-+12m l-ll ~ -rch zll Lo, flu -u~ll 1,o, zeH~+ 2(12) nH~(f2)

+ ~ - a l l z h ~ l l o , ~ , l l ~ 7 h - , h ~ h h -u~ I{ o.on" + la,(u,, ~nz) -a,(u,, ~Zh ~) I + l ~(~hS) --lh(r~, ~)l + l ~",(~, ~rn2)- ~(rc, ~)I]}, (3.20)

where f~=Eu and ~=Ez.

Proof Evidently the inequality

II~--uhlla, <11~-~11~+11~ h -u~ll~ (3.21) e ~

holds for all ~eS h. Setting Z = u ~ - ~ it follows from (3.14) that

2 < C - ~ a h IIzlta~ = ~(Z, z) (3.22) and

h ~ h ~ h [a~ (r Z) - a~ (~, Z)] a~(z,z)=a,(u_~,Z)+ "~

h + [l~ (Z) - ~ (~0] + [~ (Z) -fih (~, ~0]. (3.23)

Combining (3.21), (3.22) and (3.23) yields the desired result (3.19). We have for r > 0

II~-u~ll_~.~,= sup (3.24)

Extend r / to ~ by setting f /=Ehr / and define z such that

Az=~ in g2 z = 0 on dO. (3.25)

F rom (A2) and elliptic regularity theory and (3.5 b) it follows that

Ilzll.§ C I1~11.., <__ C II~t L. , , . (3.26)


It follows from (3.7b) and (3.10) that

--u~, r/

=a~(~_u~,~_~e)_/~\ ~ . a~ _~ ~ \ -~ ,~v ~§ ~ / ~

h h ~ ~h h + [a~ (u~, ~z h z) - a~ (u~, ~z h ~)] + [~ (Tz h ~) - l~ (re h ~)]

+[a~(~,~h~)--~(~h%)]+(fi--u~,~l--74~)n~_n. (3.27)

Bounding the right hand side of (3.27) using (3.9), noting that

IIA~llo,~_~ < CIl~ll2,~._- < CIIzll2,~ (3.28)

and applying the trace inequality (1.17b) the desired result (3.20) follows from combining this bound with (3.24) and (3.26). []

The following estimates are required in obtaining the error bounds.

Lemma 3.3. Assuming (A5) we have for all w~Hi(l)):

I l w l l o , ~ _ ~ - < _ C{h2lwll,o~_s~+ h II wll o,~n~} (3.29a) and

IIw[I o,~- n.--< C{ h2 IWll,~-~ + h II wll o,0~1. (3.29 b)

Proof The proof of (3.29a) is given in Lemma 3.2 in Barrett and Elliott I-4]. The proof of (3.29b) follows in a similar manner. []

Lemma 3.4. Assume (A5) holds. I f weH2(f)) and w =0 on OO then

Ilwll o,on. < Ch2 IIw" 2,~. (3.30)

Proof For each element z~B h there exists a local co-ordinate system (X,, 11,) such that X,~A~ and Y~P,, where A, is either an interval (n =2) or a triangle (n =3). The surface dDh--t3f2h~ is locally described by Y~=t~(X,). The surface 0f2 is locally described by Y~--~k,(X,). Assuming (A5) it follows that

I1~',- 0~11 o,o~,~. < Ch2 VzeBh, (3.31)

as ~ is sufficiently smooth by (A2). Adopting the notation w = w(X, tk~(X)) and w(h)= w(X, ~kh~(X)) we have

.~eB h

and Ilwl120~ = S (w(h))2 ( 1 + [ 17 ~b~ 12)~ dX~

d~

= I [w(h)-w]Z( 1 § Iv~bh[2) �89 A~

[ wl: v0,1) (O,-r $ LoY, J dY~dX,. < [ l ( l + l h 2 � 8 9 h

A, O(x)


Therefore we obtain

Ilwll o~,~,~ < Ch2[lwl~,,~_~+lwl~,~,_~], (3.32)

applying the bound (3.31) above. From the trace inequality (1.17b) and (3.29) we have for all w~HI(~'~) that

II w [I o ,~h- ~ -<_ Ch II w II 1,~h (3.33 a) and

Ilwllo,~_~._- < Chllwllx,~, (3.33b)

which imply for all weH2(f)) that

2 2 , ( 2 Iwla,~h_~ + Iwl~,~_ ~. = Ch2 If wll 2,~. (3.34)

Combining (3.32) and (3.34) yields the desired result (3.30). []

Combining the above lemmas we obtain the H 1 error bound.

Theorem 3.1. Let the assumptions (A1), (A2) and (A5) hold. Then the solutions u and uh~ of (1.3) and (3.10) satisfy:

tl~7-u~ll ~,o. < C{[ h§189 Ilull4,a §189 II~ll a,~.} (3.35) and

t1 u - u~[I o,aah < C {[e * h + h 2 § 13"] I[U II 4,~ + h2 Ilgll 3,~}, (3.36)

where ~ = E u.

Proof. Choosing r in (3.19) together with (3.15) and (3.18) yields

Ila -u~l[~ < C/lift -rtnfilla.+ h 2 ILutl4 O§ g- ~r h2 ll~ll 3,a~

§ sup z; - (x)l / (3.37) �9 ~ ~ J

From (3.7), (3.8) and (3.17) it follows that

I ~ ( ~ , x ) - ~ ( x ) l = ~ - \~-~vh+~-~,xlo~ [00 ] _-< C o,o~ +~ -x 11~7-~11o,0~. IlZllo,0~h

<C[( l +~-Xh2)llull2,~+~-Xh2{l~ll2,~,]llzllo,o~, (3.38)

where we have applied the trace inequality (1.17c), (2.11), the bound (3.30) as u =g on ~t2 and noted that Ilzllo,0~._<_ C ~ Ilzll~.

From (A 1), (3.4b) and (2.11) we obtain

I1~ - r~h~ll~ < C [h 2 § e - 1 ha]�89 Ilull~,~. (3.39)

From (1.19) it follows that I1" I Ix ,~< C I1"11~ and so combining (3.37), (3.38) and (3.39) yields the desired results (3.35) and (3.36). []


We see from (3.35) and (3.36) that optimal rates of confergence are obtained if and only if e = O(h2).

Theorem 3.2. Let the assumptions (A1), (A2) and (A5) hold and e=h 2. Then the solutions u and uh~ of (1.3) and (3.10) satisfy:

I1~ -u~llo,oh < Ch~[llull,,o+ I1~/I 3,o~] (3.40) and

Ilft-uh~ll-2.o,<fh2[llul14,o+ Ilgll 3,o~], (3.41) where ~ = Eu.

Proof. From (3.20) we see that an important term to estimate is Ilrch~llo,0o~. If r = 0 , so zeH2(f2)c~H~(f2), then from (3.4a) we have

II r~h ~110,0r~ < Ch~ II zll 2,0, (3.42 a)

where ~=Ez. However if r = 2 , so zeH4(t2)nH~(f2), then from (3.4b) we have

I1 rCh~[[ O,0~h < Ch2 Iiz114,o- (3.42b)

Therefore combining (3.20) with r = 0(2), (3.29a), (3.35), (3.36), (A1), (3.42a(b)), (3.15) and (3.38) with )~=gh2 yields the desired results (3.40) and (3.41). [ ]

Although we have only been able to prove that the approximat ion (3.10) with e=h 2 converges at the rate O(h ~) globally in L 2, we now show that it converges at the optimal rate in L 2 o v e r a domain t2 o c o f 21 c c t-~ using the techniques of Nitsche and Schatz [12] for obtaining interior estimates.

L emma 3.5. Let the assumptions (A1), (A2) and (A5) hold. Then the solutions u and u h of (1.3) and (3.10) satisfy for i = 0 and 1:

I1~ -u~tl-/,G,--< C [ II27 -uh~ II _ ,§ I~,G, + ~ + h2 I/ull4,n], (3.43)

where G O and G 1 are concentric spheres, Gi~ ~G/+ 1 ~ c O h, and ~t= Eu.

Proof. For i = 0 and 1, let co/e C~~ be such that

coi=l o n G / , co i > 0 o n G ' i and (3.44)

coi = 0 on f2 \G'/

where G'/ is a sphere satisfying G / c c G ' i c c G / + ~. Setting e = ~ - u ~ and so it follows that

Ilell--i,Gi ~-~ Ilcolell-/,G,+ ~

= sup I(coi e, r/i)~, +~ I (3.45) ,,~H'(G,+,) IIr//ll~,G,+ ~

Introduce z~ such that

Az /= t h in G/+ 1 z / = 0 on aG~+ 1. (3.46)

F rom (A2) and elliptic regularity theory it follows that

IIz~ll~+ 2,G,+~ < C II~/ill/,~,+,. (3.47)

364 J.W. Barrett and C.M. EUiott

As zi~Hi+2(Gi+l)c~H~(Gi+l) and ~oi~C~(G'i) it follows that

(co i e, qi)G, +1 = a ( z i , ~o i e ) G ` +1

=a(e, O)i zi) + (e, r zi �9 g (Di)Gi + l

+ (e, V. (~z~ g coi))~, +1

and so from (3.45) we obtain

Hell-i,~, < C Ilell _,+ 1),G,+ 1 la(e,o~izi)l

+ sup . (3.48) z, Eu'+2r I)~Bo~tG,+ 1> Ilzilli+2,G,+l

In addition, it follows that

a (e, wi zi) = a (e, c% z i - n h ( ~ zi)) + a (e, n h (w i zi)) (3.49) and that

a(e , 7~h(O) i Zi)) = ~lh( e, 7~h(O)i Zi))

= [a h(u~,.h(o~, ~ , ) ) - a ~(u~,.~(~o,~,))]

+ E/h (nn (~oi ~i)) - lh(nh(o~i ~i))2. (3.50)

Hence the desired result (3.43) follows from combining (3.48), (3.49) and (3.50) with (A1) and the bounds (3.15) and (3.18). []

Theorem 3.3. Let the assumptions (A1), (A2) and (A5) hold and ~=h 2. 7hen the solutions u and u~ o f (1.3) and (3.10) satisfy:

Ilt~--u~lio,r~E <-- - Ch2[llull,,~+ I1~113,r~.], (3.51)

where f2o~ =f21 c ~ f 2 h and ~ = E u .

h Proof. Setting e = f i - u ~ we have from (3.43) for f2o___G o that

Ilell O,no--< C I-Ilell- x,G, + h2 Ilull~,~] < C[llell_ 2,G~ + h2 Ilull4,o].

The desired result (3.51) then follows from (3.41). []

4. Numerical Example

We now report on a numerical example. The problem chosen was

-V2R=0 in ~x2+y2~l u=g on OO,

where g was chosen so that the solution U=x2--y 2. Due to symmetry the problem was solved in a single quadrant. For our trial space we took piecewise linears on uniform right-angled triangles, resulting from a uniform partition of the complete square [0, 1] x [0, 1] into squares with sides of length h = 1/,1 and then into triangles by joining the SW to the NE vertex. The computational

Finite Element Approximation of the Dirichlet Problem

Table 1

365

h= l/J e=h; lu--U~ll .~h Ilu--uh~blo,o~ Ilu--u~ll0.~o maxlu(xj)-uh,(xj)l J 2 nodc~

xje~

4 1 1.75819 0.47962 0.07414 0.33396 8 1 1.04683 0.28778 0.04238 0.19930

16 1 0.58026 0.16018 0.02338 0.11074 32 1 0.30699 0.08494 0.01238 0.05870

4 2 0.85301 0.15715 0.03332 0.10556 8 2 0.38118 0.04211 0.00864 0.02804

16 2 0.18326 0.01066 0.00218 0.00712 32 2 0.09072 0.00268 0.00055 0.00179

4 3 0.71392 0.05188 0.02570 0.02114 8 3 0.36119 0.01125 0.00638 0.00316

16 3 0.18173 0.00312 0.00161 0.00145 32 3 0.09407 0.00092 0.00040 0.00117

4 4 0.72100 0.04537 0.02593 0.01185 8 4 0.38135 0.01554 0.00664 0.00934

16 4 0.24053 0.01441 0.00278 0.01307

domain O h was obta ined by replacing ~?t2 by its chord in each triangle it intersects as described in Sect. 3. Choosing ~ = x 2 _y2 as the extension for g, we

h from (3.10) to u for different values of h and ). with obta ined approximat ions u, e=h 'k The results are presented in Table 1. For the interior estimate the domain O o was chosen to be the square [ - �89189 x [_~,~].1 1

F rom Table 1 we see that for 2 = 2 the H a, global L 2 and interior L 2 errors are converging at the optimal rate. The H a and interior L 2 rates confirm the analysis of Sect. 3, whilst the global L 2 rate is better than that predicted. Overall we see the vast superiority of choosing )~= 2 as opposed to 2 = 1, which is the choice often quoted in the literature. Although, it appears that the H 1 error is converging at the opt imal rate for 2 = 1 , better than that predicted. Moreover, the choice 2 = 3 yields optimal rates in the H a and the interior L 2 norms. Once again this is not predicted by the analysis. However, we see that for 2 = 4 these opt imal rates are lost. In fact in this case decreasing h can lead to an increase in the error.

To conclude, we see that this example certainly confirms the opt imal H a and interior L 2 rate predicted for the choice 2 = 2 and the superiority of this choice over choosing 2 = 1. However, it would appear that the rates predicted for other choices of 2 could be improved.

R e f e r e n c e s

1. Babuska, I.: The finite element method with penalty. Math. Comput. 27, 221-228 (1973) 2. Babuska, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element

method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A.K. Aziz, ed.), pp. 3-363. New York: Academic Press 1972


3. Barrett, J.W., Elliott, C.M.: A finite element method on a fixed mesh for the Stefan problem in a saturated porous medium. In: Numerical Methods for Fluid Dynamics (K.W. Morton and M.J. Baines, eds.), pp. 389-409. London: Academic Press 1982

4. Barrett, J.W., Elliott, C.M.: Total flux estimates for a finite element approximation of elliptic equations. (Submitted for publication.)

5. Barrett, J.W., Eltiott, C.M.: Fixed mesh finite element approximations to a free boundary problem for an elliptic equation with an oblique derivative boundary condition. Comput. Math. Appl. 11, 335-345 (1985)

6. Barrett, J.W., Elliott,. C.M.: Finite element approximation of elliptic equations with a Neumann or Robin condition on a curved boundary. (Submitted for publication)

7. Clement, Ph.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 9, 77-84 (1975)

8. King, J.T.: New error bounds for the penalty method and extrapolation. Numer. Math. 23, 153-165 (1974)

9. King, J.T., Serbin, S.M.: Computational experiments and techniques for the penalty method with extrapolation. Math. Comput. 32, 111-126 (1978)

10. Kufner, A., John, O., Fucik, S.: Function Spaces. Leyden: Nordhoff 1977 11. Necas, J.: Les M6thodes Directes en Th6orie des Equations Elliptiques. Paris: Masson 1967 12. Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comput. 28,

937-958 (1974) 13. Utku, M., Carey, G.F.: Boundary penalty techniques. Comput. Methods Appl. Mech. Eng. 30,

103-118 (1982) 14. Zhong-Ci Shi: On the convergence rate of the boundary penalty method. Int. J. Numer.

Methods Eng. 20, 2027-2032 (1984)

Received September 5, 1985/January 16, 1986

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Finite element approximation of the Dirichlet problem ... · The finite element approximation of (1.3) requires the construction of finite dimensional spaces approximating HI(O )