John W. Barrett and Charles M. Elliott. Department ... J.W. Barrett and C.M. Elliott ... condition (1.7b) is of the Robin type the finite element approximation of (1.8).
Numerische Mathematik
Numer. Math. 49, 343-366 (1986)
9 Springer-Verlag1986
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 L2 norm, then (i) for f 2 = D h convex polyhedral, we show that choosing the penalty parameter e = h x with 2 > K yields optimal H 1 and L2 error bounds if
u~HK+I(Q); (ii) for Of2 being smooth, an unfitted mesh (f2___Dh) 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 L2 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)
cl>c(x)>co>O
a.e. in Q.
(1.1b)
and Consider the elliptic boundary value problem of finding u such that
Au= - V . ( a V u ) + c u = f u=g
on Of2,
in f2,
(l.2a) (1.2b)
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J.W. Barrett and C.M. Elliott
f~L;(O) and g6HZ(f2). The variational form of (1.2) is: find ueHI(O) = {weHI(O): w = g on dO} such that for prescribed data
a(u,v)=l(v)
Vv~H~(O)={weHI(O): w = 0 on dO};
(1.3)
where a (w, v) - (a F w, V v)s~+ (c w, v)e,
t(v) = (f, v)~
(1.4)
(1.5)
and
(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 Oh, 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 du
a~-+ctu=g dv
on dO,
(1.6)
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 Dh, a union of elements, has approximation h K in the L2 norm and if the region of integration is approximated by (P with dist(O, Oh) 0 it follows that a~(',') is continuous and coercive over Hi(f2) and hence (1.8) is a well-posed variational problem. It is a simple matter, see T h e o r e m 2.1 in the next section, to show that the solutions of (1.3) and (1.8) satisfy Ilu-u~llo,~_< Ce Ilufl2,a,
(1.11)
where C is a constant independent of e, and so u~-,u as ~ 0 . As the b o u n d a r y condition (1.7b) is of the Robin type the finite element a p p r o x i m a t i o n of (1.8) can be based on an unfitted mesh if ~?f2 is smooth. The important problem is how to choose the penalty parameter ~ with respect to h and K so that uh~, the finite element approximation of (1.8), converges to u at the optimal rate as h~0. The outline of this paper is as follows: in the next section we define and analyse a finite element approximation to (1.8) in the absence of variational crimes; that is f2h=f2___D h and all integrations are performed 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 approximation for Poisson's equation, a = 1 and c=-0, with homogeneous b o u n d a r y 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)
Ilu-u~llo,a
(1.12b)
and
3 there is no choice of 2 which yields an optimal H 1 estimate.
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Assuming more smoothness on u, ueHr+z(f2), we show in Subsect. these bounds can be improved so that (1.12) holds with 6 = 0 , # 1 = # * = # ~ where p* = min [2, K - 1, K --89 and p* = min [2,/~* + 1, p* +~2, pll *- 89
2.1 that and Po (1.15 a) (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 # ~ = m i n [ 2 , 1+ 89189 Hence the best choice of 2 is ~ leading to /~0-3, _5 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 f2h=-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. However, 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 L2 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 L2 error bound is only O(h ~) we obtain an optimal order interior L2 error bound over a domain f2o c c f21 c c f2h. 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] 2
and for p~[2, ~ 3
Iw-rrhwl,,,p,~ 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
V h < h o.
(2.10)
For all integers s > 0 there exists an extension operator E: H~(f2)---,H~((2) such that Ew=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~(uh, Z) = I~(Z)
Vz~S h9
(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 Subsect. 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 L2 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-
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J.W. Barrett a n d C.M. Elliott
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 L2 error bounds for the Approximation (2.12). Lemma 2.1. 7he solutions u and u~ of (1.3) and (2.12) satisfy:
u-ea---u~
-u~lla + ~ -
1)
0, OD
(2.13)
-ea6v
< z~S inf Ilu--Zlla~+~ -~ h Proof. From (1.2) we have that
a(u,v)=l(v)+ 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 ; lu -un~li.a < [Ch"*/lullg+ 2.~
(2.28)
where #1 and #* are as defined in (2.21), /~*) = m i n [2, p~*)+89
(2.29)
/t~o*) = min [#~*~ + l,/~*) + 89 - 2), #~*)].
(2.30)
and Proof. F r o m (2.13) and (2.20) we obtain u - ~ a ~0u v v-u~h o,oa< ~Ch "'+~lrutrK,a [ Chu~+89 I[U[[g+2,a"
(2.31)
As [lu--uhHo,~a
_-< u
~^u
h
--gaov--U ~
~u
0,(312
+e a ~ v
O,0D
,
(2.32)
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 L2 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 _4 no choice of 2 leads even to an optimal H 1 error bound. Therefore in practice the penalty formulation 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 approximation to (1.8) and show that the bounds (2.27) and (2.28) remain valid if 2 is chosen to be 2. Moreover, although the global L2 error bound is only O(h ~) we obtain an optimal order interior L2 error bound over a domain O o c ~ f21 c c f2h. Before discussing the case of f2 being a convex polyhedral we end this subsection by mentioning the extrapolation m e t h o d of King [8]. Let uh~solve
a,(u~, Z) = l,(z)
VZe Sh
(2.33)
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J.W. Barrett and C.M. Elliott
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 u~j)
-
-
2Ju(j hi2
(2.34 a)
1) __U(j-- 1)
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)ll0m < 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 -V2u=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(h89 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
Finite Element Approximationof the Dirichlet Problem
355
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~OSh such that
VZ ~dSh
(2.40)
sh-- {Z~Sh: Z =Phg on dr2}.
(2.41)
(g --Phg, Z)e~ = 0 and set
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 L2 norms and so we would expect the convergence rate of u~n to u for fixed K to tend to the optimal rate as ,~ increases. This is not 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 f 2 - 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~
V w e H r ' (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,,O, error estimates. The negative norm estimate is required later in obtaining an interior L2 error bound. Lemma 3.2. Let (A2) hold. The solutions u and u~ of (1.3) and (3.10) satisfy:
jl~7-u~l]~, 0 I[~-u~ll-,,o, +
< C {l l ~ - u h l l 0,O,_~ + ITS-- U~It 0,00"
sup
zeH~+ 2(12)nH~(f2)
IIz l1;-+12ml-ll~ -rch zll Lo, flu -u~ll 1,o,
+~-allzh~llo,~,ll~7
-, h ~nz) ~ -a,(u,, h h ~Zh~) I -u~h I{o.on" + la,(u,, + l~(~hS) --lh(r~, ~)l + l~",(~, ~rn2)- ~(rc, ~)I]},
(3.20)
where f~=Eu and ~=Ez. Proof Evidently the inequality
II~--uhlla,
