Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol103.Issue1.1999:
Journal of Computational and Applied Mathematics 103 (1999) 99–114
A posteriori estimators for nonlinear elliptic partial di erential
equations
Felix Christian, Guimar˜aes Santos ∗
Computational Mechanics Group, Department of Mechanical Engineering, UFPE, Recife PE 50740-530, Brazil
Received 9 October 1997; received in revised form 3 July 1998
Abstract
Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of nite element discrete solutions to linear elliptic partial di erential equations. For
each estimator there is a set of restrictions de ned in such a way that the analysis of its performance is made possible.
Usually, the available estimators may be classi ed into two types, i.e., the implicit estimators (based on the solution of a
local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the
performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error.
Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than
for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in
the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators,
originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We
also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear
c 1999 Elsevier Science B.V. All
problems in order to build new reliable and robust estimators for nonlinear problems.
rights reserved.
Keywords: Finite Element Method; Nonlinear PDEs; A posteriori error estimators
1. Introduction
This work deals with the relationship between the approximation error of nite element solutions
to strongly nonlinear elliptic partial di erential equations in some norms, with the error estimators
computed for some suitably de ned linear elliptic partial di erential equations. It will be proved
in what follows that, provided the problem data are smooth, it is possible to build linear elliptic
∗
Corresponding author. E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 4 4 - 1
100
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
problems such that their nite element error is asymptotically equal to the nite element error for the
nonlinear problem, and, provided the estimator being used is asymptotically exact for smooth linear
problems, then, it will also be asymptotically exact for that auxiliary linear problem and, consequently
for the original nonlinear problem. This work is more precisely developed and more general than
the theory presented in [12]. The estimators considered are in a very large class, including virtually
all implicit estimators, i.e., those estimators computed through the solution of local elliptic problems
(either patchwise or elementwise). This is important in the sense that one may use estimators for
nonlinear problems in the same fashion as it is done for linear problems, with the understanding that
the same advantages and disadvantages of any particular estimator originally developed for linear
problems will occur when used for nonlinear problems.
Works concerning error estimators for nonlinear problems are not equal in number and accomplishments to those concerning linear problems. In this work we do not have the intention of reviewing
the literature in that eld, but it is relevant to cite some important work, in which either similar
or di erent strategies were used, when compared with our approach. The idea of computing estimators through linear problems can be traced back to the abstract works of Krasnosel’skii and
collaborators [9], and, when related to a formal framework of the nite element method, to the
works of Babuska and Rheinboldt [2, 11]. More recently, regarding strongly nonlinear elliptic partial di erential equations with quadratic growth, Tsuchyia has also cited the relation between error
estimation for linear problems and for nonlinear problems [13] (see also [10]). Verfurth has developed a method of estimating the norm of the residual, which is equivalent to the error of the
nonlinear problem [14]. The disadvantage of the strategy related to estimating the residual (explicit
error estimators), is that the estimator can only be proved to be equivalent to the error, therefore, including some multiplying constants which may be either small or large depending on the
problem.
In this work we deal only with regular points, because it allows for a more direct approach,
making it easier to convey the main ideas. Extensive numerical experiments will be presented in
[12], including examples with known solutions.
′
Let F : W01;p1 × Rm → W −1;p2 be given and consider the following problem:
Pr 1. Find (u0 ; 0 ) ∈ (W01;p1 × Rm ) such that
F(u0 ; 0 ) = 0
on
;
where
⊂ R2 is open and bounded. Here
F(u; ) = −3 · [a(3u; u; ; x)] + b(3u; u; ; x) + c(u; ; x) − f(; x);
where u :
→ R; a : R2 × R × Rm × R2 → R2 ; b : R2 × R × Rm × R2 → R; c : R × Rm × R2 → R;
f : Rm × R2 → R are given smooth enough functions.
In this work we are interested in the a posteriori numerical analysis, so we are going to assume
that the following hypothesis holds
Hypothesis 1.1. There exists a nonempty set ⊂ Rm such that, for all 0 ∈ ; Pr.1 has at least
one solution point u0 (0 ) in some given admissible closed convex set K ⊂ W01;p1 .
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
101
The discrete problem is set as
DPr. 1. Find (uh ; h ) ∈ (S h ( h ; p;
) ∩ K) × Rm such that
hF(uh ; h ); vh i = 0;
for all vh ∈ S h ( h ; p;
) ∩ Wd1;p2 :
The main issue now is to establish some restrictions on the di erential equations we are going
to deal with. Actually, there are further issues which will not be covered here, but the reader will
nd them in [3], where a more complete description of the hypothesis will be found. For a more
detailed analysis on the di erentiability structure required in the hypothesis stated below see [3]; for
the existence and convergence results we refer to [3].
′
Let us be speci c and assume that F : W01;p1 × Rm → W −1;p2 , 1 ¡ p1 ¡ ∞, 1 ¡ p2 , is de ned by
F(u; ) = Q(u; ) − R(u; ) − f():
(1)
Here Q(:; ) : W01;p1 → W −1;p2 is an isomorphism and a strongly nonlinear operator for all ∈ Rm ;
′
′
R : W01;p1 × Rm → W −1;p2 is a compact and smooth nonlinear operator, and f() ∈ W −1;p2 .
Also, assume
′
Hypothesis 1.2. F : W01;p1 × Rm → W −1;p2 satis es the following properties:
(i) F is a -Fredholm operator of index i(F) = m from = (W01;p1 ; H01 ; W01;∞ ) into ∗ =
′
(W −1;p2 ; H −1 ; W −1;∞ ).
(ii) The extension
′
Du QH (w; ) : H01 → H −1
is a coercive and bounded linear operator for all w ∈ W 1;∞ and ∈ Rm , with the constants
of boundedness and coercivity being bounded uniformly away from ∞ and 0, respectively, in
bounded sets of (w; ) ∈ W 1;∞ × Rm . Furthermore, its coecients are in L∞ .
(iii) For all u1 ; u2 ∈ W01;∞ and ∈ Rm , there exists C = C(ku1 kW 1;∞ ; ku2 kW 1;∞ ; ||), such that
kDu QH (u1 ; ) − Du QH (u2 ; )kL(H01 ; H −1 ) 6Cku1 − u2 kW 1;∞
and
kDu R(u1 ; ) − Du R(u2 ; )kL(H01 ; H −1 ) 6Cku1 − u2 kW 1;∞ :
Furthermore, all the coecients of Du F(u0 ; 0 ) are as smooth as the gradient of u0 .
(iv) For all (u; ) ∈ W01;∞ × Rm , all the existing derivatives of F with respect to the function
and the parameter at (u; ) are Holder-continuous with respect to . Moreover, the existing
derivatives of F with respect to the parameter are in W −1;p , for all needed values of p, and
are Holder-continuous with respect to the function.
′
(v) For all (u; ) ∈ W01;∞ ×Rm , the linear operator Du R(u; ) : W01;p → W −1; r is a compact operator
for all p, r ′ 6p6r, with r ¿ 2 as large as needed.
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
The main results in this paper depend on some properties of linearized operators. So we de ne:
Hypothesis 1.3. (Continuous inf–sup condition). Let ∞ ¿ r¿2 be any number, 1=r + (1=r ′ ) = 1,
′
and B : H01 × H01 → R be as above. Then, B : W01; t × W01; t → R is bounded and satis es
inf
u∈W01; t
sup
v∈W01; t
′
B(u; v)
¿
¿ 0;
kukW 1; t kvkW 1; t′
sup {B(u; v)} ¿ 0
u∈W01; t
′
for all v ∈ W01; t ;
for all t ∈ [r ′ ; r].
Hypothesis 1.4. (Discrete inf–sup condition). Let ∞ ¿ r¿2 be any number, 1=r + (1=r ′ ) = 1; and
′
B : H01 × H01 → R be as above. Then, B : W01; t × W 1; t → R is bounded and satis es
inf
uh ∈S h
(
sup
vh ∈S h
B(uh ; vh )
kuh kW 1; t kvh kW 1; t′
sup {B(uh ; vh )} ¿ 0
)
¿ ¿ 0;
for all vh ∈ S h ;
uh ∈S h
for all t ∈ [r ′ ; r]. Also, 6= (h).
First, for any q ∈ [r ′ ; r], h ¿ 0, wh ∈ S h and w ∈ W 1;∞ , de ne
h = h (h; q; w) = max
inf
vh ∈S h
n
h
−n=q
o
kw − vh kW 1; q ; inf h {kw − vh kW 1;∞ } :
vh ∈S
The following results are statements concerning existence and convergence of discrete solutions
to DPr.1. For that, let K = Du (u0 ; 0 ) and B(: ; : :) = hK(:); (::)i.
Theorem 1.5. Let F : W01;p1 × Rm −→ W −1;p2 satisfy Hypothesis 1.2. Let the linear operator
K : H01 −→ H −1 be de ned as above and let the bilinear form B : H01 × H01 −→ R be de ned based
on K. Let
∈ Dt , for some t ¿ 2. Let (u0 ; 0 ) be a strong regular solution point to Pr.1, such
that u0 ∈ W01;p1 ∩ W 1+;p , with ¿ n=p.
Then, there is an r ¿ 2; such that, if one can nd q ∈ [r ′ ; r]; satisfying
′
h (h; q; u0 ) → 0;
1; q
one can also nd h0 ¿ 0; and ¿ 0; such that, for all 0 ¡ h6h0 ; there exists a unique uh ∈ W˜ 0 ∩
B (uh ); such that (uh ; 0 ) solves DPr.1. Here the sequence {uh }h→0 ∈ S h is to be suitably chosen.
Proof. This result is just a particular case (for regular solution points) of a more general theorem
presented in [3].
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
103
Remark. The sequence {uh }h→0 in the above theorem is to be chosen such that the objectives of
the analysis are met [3, 12].
Corollary 1.6. Let the hypotheses of Theorem 1.5 be true. Then, for all h ¿ 0 small enough,
kuh − u0 kW 1; s 6Ckuh − u0 kW 1; s ;
for all s ∈ [r ′ ; r]; such that h (h; s; u0 ) → 0, as h → 0;
kuh − u0 kW 1; s 6C[hn=s−(n=r) kuh − u0 kW 1; r + kuh − u0 kW 1; s ];
for all s ∈ [r; ∞], where {uh }h→0 is a sequence chosen as in Theorem 1.5. Here C = C(u0 ; 0 ) and
r ¿ 2 is as obtained in Theorem 1.5.
Furthermore, taking uh = Ph u0 ; and for all s ∈ [r ′ ; ∞]; such that ¿ n=min{s; p}; the above
inequalities imply that
kuh − u0 kW 1; s 6Ch ku0 kW 1+; p ;
where, 1 = min{0; n=s − (n=r)} and = min{q; 2 }; where q¿1 is the polynomial order of approximation of the shape functions in each nite element, and 2 = + (n=s) − n=min{r; p} if s¿r and
2 = + (n=s) − n=min{s; p} if s ¡ r; and, C 6= C(u0 ; h).
Proof. This result is a particular case (for regular solution points) of a more general theorem
presented in [3].
2. A posteriori estimators
In this section we develop a procedure for relating computable a posteriori error estimators for a
suitably de ned auxiliary linear problem with the exact error (in the norm of W 1; s , s ∈ [r ′ ; r], r¿2)
for the nonlinear problem (between a given solution to Pr.1 and the corresponding discrete solution to
DPr.1). A large class of estimators will be considered, namely, implicit estimators, obtained through
a solution of a suitably de ned local problem and de ned either elementwise or patchwise. In order
to make the procedures clear, we will consider only strong regular solution points. The procedures
regarding simple turning points will be presented in later works.
A rst linear auxiliary problem will be de ned by a bilinear form B1 : H01 × H01 → R and a righthand side f1 . Similarly, a second linear auxiliary problem will be de ned by B0 : H01 × H01 → R
and f0 . For the exact, discrete and error equations of both problems we refer to LP.0, DLP.0 and
Er.0 below, respectively.
In what follows, the expression ! ∩ T 6= ∅ will mean that the interior of the region de ned by !
has an empty intersection with the region de ned by T .
De nition 2.1. Let a mesh h be given. Suppose that a way of building a set of patches ! by
making union of adjacent elements T ∈ h , such that the union of all patches covers
, is given. Let
Vh be that set. De ne
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
(a) The trial space
W! = span{
! j=k(!)
;
j }j=1
and the test space
j=k(!)
V! = span{ !j }j=1
;
de ned over each corresponding patch ! ∈ Vh .
(b) The spaces (de ned elementwise), for each T ∈ h ,
!
j |T ,
j = 1; ::; k(!), for all !, such that ! ∩ T 6= ∅};
ZT = span { !j |T ,
j = 1; ::; k(!), for all !, such that ! ∩ T 6= ∅}:
YT = span {
!
T
It is clearly seen that there exist a decomposition whT = !∩T 6=∅ [wP
h ], for all wh ∈ YT , where
!
!
T
wh ∈ W! with ! ∩ T 6= ∅. Similarly, there exists a decomposition vh = !∩T 6=∅ [vh ], for all vhT ∈ ZT ,
where wh! ∈ W! .
P
Hypothesis 2.2. Let a bilinear form B(: : ; :) : W01; s × W 1; s −→ R be given, for all s ∈ [r ′ ; r], where
r ∈ [2; ∞) is given. Let a mesh h and the set of patches Vh be given as de ned above. Then,
(a) There exists a real number
¿ 0, such that, for each T ∈ h , and for all wh ∈ YT ,
′
#
"
X B (w! ; v! )
!
h
h
;
kwh kW 1; s (T ) 6 sup
kvh! kW 1; s′ (!)
vT ∈ZT
!∩T 6=∅
h
where
6=
(h) does not depend either on T ∈ h , nor on Vh . B! (: : ; :) means the restriction of
B to the patch !. Note that we are using the decomposition of vhT ∈ ZT described just above.
(b) Let R ∈ W −1; r be given, and consider R|! as being a suitably de ned restriction of R to !;
for all ! ∈ Vh . Then, there exist positive constants C1 and C2 ; not depending either on the
h nor on Vh , such that
C1 kRkW −1; s (
) 6
X
T ∈ h
X
!∩T 6=∅
1=s
kR|! kW −1; s (!) 6C2 kRkW −1; s (
) :
Remark. Hypothesis 2.2 means that the given bilinear form B is patchwise elliptic and the patches
do not overlap too much, destroying the stability of the sum of quantities de ned patchwise.
We now de ne the class of implicit estimators.
De nition 2.3. (Implicit estimators). Let a mesh h and a set of patches Vh be given. Let a bilinear
form B(: : ; :) be given, which satis es Hypotheses 1.3, 1.4, and 2.2 for some r ∈ [2; ∞) and spaces
{V! }!∈Vh and {W! }!∈Vh . Let f ∈ W −1; r be given, and de ne w0 ∈ W01; r , wh ∈ S h ( h ) and e ∈ W01; r
to be the solutions of LP.0, DLP.0 and Er.0, respectively. For each T ∈ h , de ne T (x) ∈ YT as
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
105
j=k(!)
}!∩T 6=∅ are constants, which are
T (x) = !∩T 6=∅ [ j=1;:::; k(!) Cj! j! (x)], for x ∈ T , where {{Cj! }j=1
obtained by nding ! ∈ W! , such that
P
P
B! (! ; !j ) = hRh! ; !j i
j = 1; :::; k(!)
for all ! ∈ Vh ;
where Rh! = Rh |! is the restriction of the residual Rh ∈ W −1; r to the patch ! ∈ Vh . Thus, T (x) is
computed by
X
T (x) =
[! (x)];
!∩T 6=∅
for all x ∈ T . The restricted residual R! , the trial and the test spaces should be such that the above
problem has a unique solution.
For some given s¿1, set
T (s) = kT kW 1; s (T ) ;
and
(s) =
(
X
T ∈ h
sT
)1=s
:
The value T is called the elemental estimator for T ∈ h (indicator) and is the (global) estimator.
The auxiliary linear problems are to be de ned as follows.
LP.0. Find w ∈ W01; s , such that
B(w; v) = hf; vi
′
for all v ∈ W01; s :
DLP.0. Find wh ∈ S h , such that
B(wh ; vh ) = hf; vh i
for all vh ∈ S h ;
′
where f ∈ W −1; s . De ning the error by e = w − wh , the error equations for the above problems are
given by
Er.0. Find e ∈ W01; s , such that
B(e; v) = hRh ; vi = B(wh ; v) − hf; vi
′
for all v ∈ W01; s :
The following result shows that the implicit estimators change at most linearly with perturbations
in the coecients of the operators and on the right-hand side.
Theorem 2.4. Let the bilinear forms B0 (: : ; :) : H01 × H01 −→ R and B1 (: : ; :) : H01 × H01 −→ R be
given. Let both bilinear forms satisfy Hypotheses 1.3; 1.4 and 2.2 for some r ∈ [2; ∞) and spaces
{W! }!∈Vh and {V! }!∈Vh . Let f0 ; f1 ∈ W −1; r be given as the right-hand sides for B0 and B1 ;
respectively. Let w0 and w1 ∈ W01; r be solutions of LP:0; w0h ; w1h ∈ S h solutions of DLP.0 and R0h ;
R1h ∈ W −1; r the residuals; all related to B0 and B1 ; respectively. If 0 and 1 are implicit estimators
related to B0 ; f0 and B1 ; f1 ; respectively; then; there exists a constant C; which depends only on
the L∞ (
)-norm of the coecients of both bilinear norms; such that; for each s ∈ [r ′ ; r],
|0 (s) − 1 (s)|6C[kBkL∞ 0 + kRh kW −1; s (
) ]:
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
For the above, kBkL∞ means the L∞ (
)-norm of the di erence between the respective coecients of B0 and B1 ; and Rh is the di erence between the residuals R0h and R1h .
Proof. Let T ∈ h be given. Let 0T ,1T ∈ YT be as in De nition 2.3, related to B0 and B1 , respectively.
Set T = 0T − 1T , and B(: : ; :) = B1 (: : ; :) − B0 (: : ; :)]. Then
X
[hRh! ; ! i! ] =
!∩T 6=∅
X
[B0! (! ; ! ) + B! (!0 ; ! )];
(2)
!∩T 6=∅
for all ! ∈ V! , ! ∩ T 6= ∅. Then, by Hypothesis 2.2 and for each s ∈ [r ′ ; r], we obtain
"
#
X B (! ; v! )
1!
h
h
kT kW 1; s (T ) 6 sup
kvh! kW 1; s′ (!)
vhT ∈ZT !∩T 6=∅
#
"
X hR ; v! i B (! ; v! )
!
h! h !
:
6 sup
+ ! 0 h
!
′
′
T
kv
kv
k
k
1;
s
1;
s
v ∈ZT
(!)
(!)
h W
h W
!∩T 6=∅
h
So, there is C = C(s;
), such that
kT ksW 1; s (T ) 6C
X
[kRh! ksW −1; s (!) + kB! ksL∞ (!) k!0 ksW 1; s (!) ]:
!∩T 6=∅
By adding up over all elements of h and using the summation properties of the residuals stated in
De nition 2.3, we get
X
kT krW 1; s (T ) 6C
T ∈ h
"
kRh ksW −1; s (
)
+
kBksL∞ (
)
X
T ∈ h
k0T ksW 1; s (T )
#
:
Taking the sth-root on both sides of the above expression and using Minkowiskii’s inequality yields
|0 − 1 |6C[kRh kW −1; s (
) + kBkL∞ (
) 0 ];
which immediately gives the desired inequality.
Now, let us be speci c and introduce our two auxiliary linear problems, the rst for theoretical
purposes only and the second for the actual computation of the error estimator. As before, and for
the rest of this paper, (u0 ; 0 ) and (uh ; 0 ) will be the solution to Pr.1 and the solution to DPr.1,
′
respectively. Next, let F : W01;p1 → W −1;p2 be given and satisfy the hypothesis of Theorem 1.5. For
any xed r ∈ [2; ∞), as close to 2 as needed, we take s ∈ [r ′ ; r] and set K0 , K0h , Kh : W 1; s → W −1; s ;
′
and B0 , B1 : W01; s × W01; s → R as
K0 = Du F(u0 ; 0 );
K0h =
Z
(3)
1
[Du F(uh + t(u0 − uh ); 0 )]dt;
(4)
0
Kh = Du F(uh ; 0 );
B0 (u; v) = hK0 u; vi
B1 (u; v) = hKh u; vi
(5)
′
for all u ∈ W01 s and v ∈ W01; s ;
(6)
′
for all u ∈ W01 s and v ∈ W01; s :
(7)
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
107
Furthermore, set
eh = u0 − uh ;
(8)
K0 = K0h − K0 ;
(9)
K1 = K0 − Kh ;
(10)
b0 = K 0 u0 :
(11)
The next lemma states some properties for the above operators
Lemma 2.5. Let the hypothesis of Theorem 1.5 and Corollary 1.6 be satis ed. Then, for each
s ∈ [r ′ ; r]:
(i) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; there exists a constant
C 6= C(h); such that
kK0 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(ii) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; there exists a constant
C 6= C(h); such that
kK1 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(iii) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; B0 and B1 satisfy Hypotheses
1.3 and 1.4.
Proof. Items (i) and (ii) follow directly from item (iii) of Hypothesis 1.2 and Corollary 1.6.
Item(iii) is a consequence of the following facts: (a) K0 satis es the inf–sup condition with r = 2,
since (u0 ; 0 ) is a strong regular point; (b) K0 is a compact perturbation of Du Q(u0 ), which satis es
Hypotheses 1.3 and 1.4, for some r ¿ 2 [3]; (c) from item (ii) of the current lemma, it follows that
Kh converges uniformly to K0 for all s ∈ [r ′ ; r]. Then the result follows [3].
The two auxiliary problems will be de ned by the bilinear form B1 (computable) and B0 (abstract),
together with the right-hand sides
f1 = −F(uh ; 0 );
(12)
f0 = b 0 ;
(13)
h
respectively. Let w0h ∈ W01; s and w0h
∈ S h solve LP.0 and DLP.0 with B ≡ B1 and f ≡ f1 , that is,
B1 (w0h ; v) = h−F(uh ; 0 ); vi
h
B1 (w0h
; vh ) = h−F(uh ; 0 ); vh i
′
for all v ∈ W01; s ;
(14)
for all vh ∈ S h :
(15)
Since by de nition, u0 solves LP.0 with B ≡ B0 and f ≡ f0 , let u0h ∈ S h solve the corresponding
discrete problem (DLP.0), i.e.,
B0 (u0h ; vh ) = hb0 ; vh i
for all vh ∈ S h :
(16)
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
Next, let us de ne the error expressions
e0h = u0 − u0h ;
h
;
ewh = w0h − w0h
which are solutions to Er.0 for the abstract and computable auxiliary problems, respectively.
The next lemma establishes some further results regarding the relationship between both linear
problems.
Lemma 2.6. Let the hypothesis of Theorem 1.5 be satis ed. Then; the following statements are
true:
(i) K0 (u0 − uh ) = −F(uh ; 0 ) + K0 eh :
(ii) K0 u0 = b0 = −F(uh ; 0 ) + K0 eh + K0 uh :
(iii) B0 (u0h − uh ; vh ) = B(eh ; vh ) = hK0 eh ; vh i; for all vh ∈ S h :
(iv) There exists a constant C 6= C(h); such that
ku0h − uh kW 1; s 6CkK0 kL(W01; s ;W −1; s ) keh kW 1; s
for all s ∈ [r ′ ; r]
h
(v) w0h
= 0:
(vi) Set
R0h = b0 − K0 u0h ;
and
h
R1h = −F(uh ; 0 ) − Kh w0h
= −F(uh ; 0 ):
Then,
Rh = R0h − R1h = K0 eh + K0 (uh − u0h )
and; hence; there exists a constant C 6= C(h); such that
kRh kW −1; s 6CkK0 kL(W01; s ;W −1; s ) keh kW 1; s :
Proof. (i) This result comes from the observation that K0h eh = −F(uh ; 0 ). Then,
K0 eh = K0h eh + (K0 − K0h )eh = −F(uh ; 0 ) + K0 eh :
(ii) This relation comes directly from the result in (i).
(iii) Since hF(uh ; 0 ); vh i = 0, and B0 (u0 − u0h ; vh ) = 0 for all vh ∈ S h , and from (i) we obtain
B0 (u0h − uh ; vh ) = B0 (u0h + (u0 − u0h ) − uh ; vh ) = B0 (u0 − uh ; vh )
= h−F(uh ) + K0 eh ; vh i = hK0 eh ; vh i:
(iv) From Lemma 2.5, B0 satis es Hypothesis 1.4, for some r ¿ 2, and, then, with the help of
(iii) we get
hK0 eh ; vh i
B0 (uh − u0h ; vh )
= sup
;
kuh − u0h kW 1; s 6 sup
kvh kW 1; s′
kvh kW 1; s′
vh ∈S h
vh ∈S h
where 6= (h) and s ∈ [r ′ ; r]. Thus, the result follows immediately.
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
109
(v) Again, by Lemma 2.5, B1 satis es Hypothesis 1.4 for the same r ¿ 2 as in item (iii) above,
h
and, then, since hF(uh ; 0 ); vh i = 0, for all vh ∈ S h , we nd that w0h
= 0.
(vi) This statement follows directly from (ii) and (iv).
Remark. As we said in the introduction, the strategy of using auxiliary linear problems for estimating
the nite element error for nonlinear problems is not new. Therefore, property (v) in the lemma
above were already known, no matter what linear coercive operator is used. The other remaining
properties are new, and they show important issues. For instance, item (iv) shows that the nite
element solution to the nonlinear problem is a higher-order perturbation of a nite element solution to
a smooth linear problem. This opens a clear space for further investigations on the relation between
superconvergence properties of nonlinear problems and those for linear problems.
Now, de ne 0 (s) and 1 (s) as being the same a posteriori estimator (with respect to the norm of
h
W01; s , s ∈ [r ′ ; r]) applied for estimating the errors e0h = u0 − u0h and ewh = w0h − w0h
= w0h , respectively.
The next theorem will state that both estimators will give the same result, asymptotically speaking
(that is, when h → 0).
Theorem 2.7. Consider F = Q − R − f : W01;p1 × Rm −→ W −1;p2 satisfying all the hypotheses of
′
′
Theorem 1.5. Let the bilinear forms B0 (: : ; :) : W01; s × W01; s −→ R and B1 (: : ; :) : W01; s × W01; s −→ R
be de ned as above, for all s ∈ [r ′ ; r]; satisfying the hypotheses of Theorem 2.4. Consider u0 and
w0h as the solutions to LP.0, related to the bilinear forms B0 and B1 ; and the right-hand sides
h
f0 and f1 ; respectively. Let u0h and w0h
be the respective approximate solutions to DLP.0; and
R0h and R1h be the corresponding residuals; as described in problem Er.0. Let 0 (s) and 1 (s)
(s ∈ [r ′ ; r]) be the result of the same implicit a-posteriori estimator applied to the abstract and the
computable problems, respectively. If the given estimator satis es Hypothesis 2.2; Then,
′
|0 (s) − 1 (s)|6Ckeh kW 1;∞ (keh kW 1;s + 0 ):
Here, as de ned in (8), eh = u0 − uh is the error for the nonlinear problem Pr.1, and C 6= C(h).
Proof. From Theorem 2.4 and Lemma 2.6 we get
|0 (s) − 1 (s)|6C(kBkL∞ 0 + kK0 kL(W01; s ;W −1; s ) keh kW 1; s ):
From Lemma 2.5 we obtain that
kBkL∞ 6Ckeh kW 1;∞ ;
and
kK0 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ :
The three relations above yield the desired result.
The lemma below shows that all errors (nonlinear problem and auxiliary problems) are higherorder perturbations of each other, and that the estimator is asymptotically the same when applied to
both auxiliary problems.
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
Lemma 2.8. Let the hypothesis of Theorem 2.7 be satis ed. Furthermore; let u0 ∈ W 1+;p ; such
that ¿ n=min{r; p}. Then; for all s ∈ [r ′ ; r];
(i) keh − ewh kW 1; s 6Ch” keh kW 1; s ;
(ii) kewh − e0h kW 1; s 6Ch” kewh kW 1; s ;
(iii) |0 (s) − 1 (s)|6Ch” (ke0h kW 1; s + 0 (s));
where ” ¿ 0.
Proof. (i) This inequality follows from
keh − ewh kW 1; s 6 sup
v∈W 1; s′
= sup
v∈W 1; s′
= sup
v∈W 1; s′
B0 (eh − ewh ; v)
kvkW 1; s′
B0 (eh ; v) − B1 (ewh ; v) + B(ewh ; v)
kvkW 1; s′
hK0 eh ; vi + B(ewh ; v)
kvkW 1; s′
6 C(kK0 kL(W −1; s ;W −1; s ) keh kW 1; s + kBkL∞ kewh kW 1; s );
from
kewh kW 1; s =
kw0h kW 1; s 6
sup
v∈W 1; s′
(
B1 (w0h ; v)
kvkW 1; s′
)
6kF(uh ; 0 )kW −1; s 6Ckeh kW 1; s
and, from Corollary 1.6,
kBkL∞ 6Ckeh kW 1;∞ 6Ch−n=min{r;p}
kK0 kL(W −1; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(ii) Similarly, the second inequality is obtained by
ke0h − ewh kW 1; s 6 sup
v∈W 1; s′
= sup
v∈W 1; s′
6 sup
v∈W 1; s′
B0 (e0h − ewh ; v)
kvkW 1; s′
B0 (e0h ; v) − B1 (ewh ; v) + B(ewh ; v)
kvkW 1; s′
hK0 eh ; vi + B(ewh ; v)
kvkW 1; s′
6 C(kRh kW −1; s + kBkL∞ kewh kW 1; s )
6 C(kK0 kL(W −1; s ;W −1; s ) keh kW 1; s + kBkL∞ kewh kW 1; s ):
The rest follows from the previous item.
(iii) From Lemma 2.6 we have that
keh − e0h kW 1; s = kuh − u0h kW 1; s 6CkK0 kL(W −1; s ;W −1; s ) keh kW 1; s
6 Ch−n=min{r;p} keh kW 1; s :
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
111
So, for h ¿ 0 small enough, we obtain
keh kW 1; s 6
ke0h kW 1; s
:
1 − Ch−n=min{r;p}
From Theorem 2.7 and the inequality above we obtain the third and last inequality. Finally we set
” = − (n=min{r; p}).
Lemma 2.9. Let the hypothesis of Theorem 2.7 be satis ed. Furthermore, let u0 ∈ W 1+;p ; such that
¿ n=min{r; p}. Then; if 0 (s); s ∈ [r ′ ; r]; is asymptotically exact; so will be 1 (s); i.e:; if there
exists a constant C 6= C(h) and ”0 ¿ 0; such that; for all small enough h ¿ 0;
|0 (s) − ke0h kW 1; s |6Ch”0 ke0h kW 1; s ;
then, the same is true for 1 (s); that is; there exists a constant C 6= C(h); and ” ¿ 0; such that
|1 (s) − kewh kW 1; s |6Ch” kewh kW 1; s :
Proof. From Lemma 2.8 there exists ”1 ¿ 0, such that
|1 (s) − kewh kW 1; s | 6 |kewh kW 1; s − ke0h kW 1; s | + |ke0h kW 1; s − 0 (s)| + |0 (s) − 1 (s)|:
6 Ch”1 kewh kW 1; s + Ch”1 ke0h kW 1; s + Ch”1 (ke0h kW 1; s + 0 (s)):
Since 0 (s) is asymptotically exact
0 6(1 + Ch”0 )ke0h kW 1; s :
From Lemma 2.8 we get
ke0h kW 1; s 6(1 + Ch”1 )kewh kW 1; s :
The three inequalities above yield the desired result, by taking ” = min{”0 ; ”1 }.
Lemma 2.10. Let the hypotheses of Lemma 2.9 be satis ed for given implicit estimators 0 (s) and
1 (s), s ∈ [r ′ ; r]. Then, for all h ¿ 0 small enough, the estimator 1 (s) is asymptotically equal to
keh kW 1; s ; for all s ∈ [r ′ ; r]. That is, there exists ” ¿ 0 and a constant C 6= C(h); such that
|1 (s) − keh kW 1; s |6Ch” keh kW 1; s ;
for all s ∈ [r ′ ; r].
Proof. From Lemma 2.8
|1 (s) − keh kW 1; s |6|1 (s) − kewh kW 1; s | + |kewh kW 1; s − keh kW 1; s |:
From Lemma 2.8, we get that there exists ”0 ¿ 0, such that
|kewh kW 1; s − keh kW 1; s |6Ch”0 keh kW 1; s ;
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
and from Lemmas 2.9 and 2.8
|1 (s) − kewh kW 1; s |6Ch”1 kewh kW 1; s 6Ch”1 (1 + Ch”0 )keh kW 1; s :
The above inequalities provide the statement of the lemma, by taking ” = min{”0 ; ”1 }.
Remark. The de nition of asymptotic exactness given in the statement of Lemma 2.10 may be
weakened by supposing that there exists a function Q(h), with Q(h) → 0 as h → 0, and such that
|1 (s) − keh kW 1; s |6Q(h)keh kW 1; s ;
for all s ∈ [r ′ ; r].
As an example, in order to illustrate the procedures described in this section, we analyse the
′
following partial di erential equation, F : W01;p1 × R → W −1;p1 , 1=p1 + (1=p1′ ) = 1:
F(u; ) = −3 · [(1 + |3u|2 )
p1 −2
2
3u] + u − f:
Here we will assume that p1 ¿ 1 and that the domain
⊂ Rn is as smooth as we wish. The above
p−2
de nition implies that Q(u; ) = −3 · [(1 + |3u|2 ) 2 3u], R(u; ) = −u and f() = f. Furthermore,
d =@
(no Neumann boundary condition). It is observed that when ¿ 0, then Q−R is a uniformly
coercive monotone operator and, then, Pr.1 has a unique solution for each such . Also, provided u0
is smooth enough it is not a dicult task to show that Hypothesis 1.2 is satis ed. The smoothness
of u0 depends on the smoothness of @
and of f, which are assumed to be as smooth as needed.
Now, we observe that, for all ∈ H01 ,
K0 = Du F(u0 ; 0 ) = −3 · [A(x) · 3 ] + 0 ;
where A is the matrix
A(x) = (1 + |3u0 |2 )(p1 −4)=2 [(p1 − 2)3u0 3u0 + (1 + |3u0 |2 )I ]:
It is easily seen that, for all ∈ Rn ,
(A · ) · ¿
||2 [1 + (p1 − 1)|3u0 |2 ];
||2 (1 + |3u0 |2 )
1 ¡ p1 62;
p1 ¿ 2:
Then, for all 0 ¿ 0, Du F(u0 ; 0 ) : H01 → H −1 is a uniformly coercive elliptic linear operator
with smooth coecients. If (uh ; 0 ) is the nite element solution to DPr:1, which exists following
Theorem 1.5, then it converges, following Corollary 1.6, with rate min{q; −n=r}, in the W 1;∞ -norm,
where q¿1 is the polynomial order of approximation of the shape functions in each element.
In a similar fashion as we did for K0 , we obtain that, for all ∈ H01
Kh = Du F(uh ; 0 ) = −3 : [Ah (x) : 3 ] + 0 ;
where Ah is the matrix
Ah (x) = (1 + |3uh |2 )
p1 −4
2
[(p1 − 2)3uh 3uh + (1 + |3uh |2 )I ]:
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
113
It is easily seen that, for all ∈ Rn ,
(Ah · ) · ¿
||2 [1 + (p1 − 1)|3uh |2 ];
||2 (1 + |3uh |2 )
1 ¡ p1 62;
p1 ¿ 2;
and thus, Kh : H01 → H −1 is a uniformly coercive linear elliptic operator.
Now, the strategy to obtain 1 (s), s ∈ [r ′ ; r] is to estimate ewh , considering the following error
equation:
B1 (ewh ; v) = h−F(uh ; 0 ); vi
′
for all v ∈ W 1; s :
Recall that B1 (: ; : :) = hKh (:); (: :)i. There are several options for computing the implicit estimator
1 . For a review of some of them see [7, 16]. The best choices will be among those which may
be asymptotically exact, provided some smoothness requirements are satis ed. Particularly, those
requirements are met by our abstract and smooth linear problem, de ned by the bilinear form B0
and the right-hand side f0 . Hence, by Lemma 2.10, 1 (s), computed by such a method, will be
asymptotically exact with respect to the error eh = u0 − uh in the W01; s -norm.
We would like to make two basic concluding remarks. First, the use of auxiliary linear problems
for estimating the error for nonlinear problems has been in use for some time. What we have proved
is that it is possible to justify the use of such estimators designed for linear problems in nonlinear
problems (asymptotically exact implicit estimators preserve that property for the nonlinear problems,
provided some standard assumptions are satis ed). Second, the majority of the estimators for linear
problems are considered in the norm of some suitable Hilbert space, but a large number of nonlinear
problems are not posed on such spaces. Nevertheless, it is our conjecture that the estimators should
behave well in other norms, provided some assumptions (e.g., Hypotheses 1.3 and 1.4) about the
operator and solution set hold, together with some restrictions on the mesh and on the nite element
spaces.
References
[1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2] I. Babuska, W.C. Rheinboldt, Computational error estimates and adaptive processes for some nonlinear structural
problems, Comput. Meth. Appl. Mech. Eng. 34 (1982) 895–937.
[3] F.C.G. Santos, The linearization method for the numerical analysis of nite element solutions to quasilinear elliptic
partial di erential equations, Research Report 001-1998, Computational Mechanics Group, UFPE, 1998.
[4] F.C.G. Santos, A posteriori estimation for quasilinear elliptic partial di erential equations, Research Report 002-1998,
Computational Mechanics Group, UFPE, 1998.
[5] I. Babuska, Stroubolis, C.S. Upadhyay, A model study of the quality of A-posteriori estimators for linear elliptic
problems: error estimation in the interior of patchwise uniform grids of triangles, Technical Note BN-1147, Institute
for Physical Science and Technology, University of Maryland, 1993.
[6] N.A. Bobylev, Yu.M. Burman, Morse lemma for functionals of variational calculus, Funktsional’nyi Analiz i Ego
Prilozheniya, 25(3) (1991)1-11, (English Translation).
[7] G. Caloz, J. Rappaz, Numerical analysis for nonlinear and bifurcation problems, To appear in: L.J. Lions, (Eds.),
Handbook of Numerical Analysis, P.G. Ciarlet, North-Holland, Amsterdam.
[8] M. Crouzeix, J. Rappaz, On Numerical Approximation in Bifurcation Theory, RMA 13, Masson, Paris, 1989.
[9] G.M. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Ya. B. Rutitskii, V.Ya. Stetsenko, Approximate Solutions of
Operator Equations, Wolters-Noordho , Groninger, 1972.
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[10] J. Poussin, J. Rappaz, Consistency, stability, A-priori and A-posteriori errors for Petrov-Galerkin methods applied to
nonlinear problems, Nr. 04.92, Echole
Polytechnique Federale de Lausanne, 1992.
[11] W.C. Rheinboldt, Numerical Analysis of Parameterized Nonlinear Equations, Wiley, New York, 1986.
[12] F.C.G. Santos, Numerical analysis of parameterized partial di erential equations, Ph.D. Thesis, University of
Maryland, 1995.
[13] T. Tsuchiya, A-priori and A-posteriori error estimates of nite element solutions of parameterized nonlinear equations,
Ph.D. Thesis, University of Maryland, 1990.
[14] R. Verfurth, A review of A-posteriori error estimation and adaptive mesh re nement, Tech. Rep., Institute Fur
Angewandte Mathematik, Universitat Zurich, 1993.
A posteriori estimators for nonlinear elliptic partial di erential
equations
Felix Christian, Guimar˜aes Santos ∗
Computational Mechanics Group, Department of Mechanical Engineering, UFPE, Recife PE 50740-530, Brazil
Received 9 October 1997; received in revised form 3 July 1998
Abstract
Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of nite element discrete solutions to linear elliptic partial di erential equations. For
each estimator there is a set of restrictions de ned in such a way that the analysis of its performance is made possible.
Usually, the available estimators may be classi ed into two types, i.e., the implicit estimators (based on the solution of a
local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the
performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error.
Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than
for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in
the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators,
originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We
also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear
c 1999 Elsevier Science B.V. All
problems in order to build new reliable and robust estimators for nonlinear problems.
rights reserved.
Keywords: Finite Element Method; Nonlinear PDEs; A posteriori error estimators
1. Introduction
This work deals with the relationship between the approximation error of nite element solutions
to strongly nonlinear elliptic partial di erential equations in some norms, with the error estimators
computed for some suitably de ned linear elliptic partial di erential equations. It will be proved
in what follows that, provided the problem data are smooth, it is possible to build linear elliptic
∗
Corresponding author. E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 4 4 - 1
100
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
problems such that their nite element error is asymptotically equal to the nite element error for the
nonlinear problem, and, provided the estimator being used is asymptotically exact for smooth linear
problems, then, it will also be asymptotically exact for that auxiliary linear problem and, consequently
for the original nonlinear problem. This work is more precisely developed and more general than
the theory presented in [12]. The estimators considered are in a very large class, including virtually
all implicit estimators, i.e., those estimators computed through the solution of local elliptic problems
(either patchwise or elementwise). This is important in the sense that one may use estimators for
nonlinear problems in the same fashion as it is done for linear problems, with the understanding that
the same advantages and disadvantages of any particular estimator originally developed for linear
problems will occur when used for nonlinear problems.
Works concerning error estimators for nonlinear problems are not equal in number and accomplishments to those concerning linear problems. In this work we do not have the intention of reviewing
the literature in that eld, but it is relevant to cite some important work, in which either similar
or di erent strategies were used, when compared with our approach. The idea of computing estimators through linear problems can be traced back to the abstract works of Krasnosel’skii and
collaborators [9], and, when related to a formal framework of the nite element method, to the
works of Babuska and Rheinboldt [2, 11]. More recently, regarding strongly nonlinear elliptic partial di erential equations with quadratic growth, Tsuchyia has also cited the relation between error
estimation for linear problems and for nonlinear problems [13] (see also [10]). Verfurth has developed a method of estimating the norm of the residual, which is equivalent to the error of the
nonlinear problem [14]. The disadvantage of the strategy related to estimating the residual (explicit
error estimators), is that the estimator can only be proved to be equivalent to the error, therefore, including some multiplying constants which may be either small or large depending on the
problem.
In this work we deal only with regular points, because it allows for a more direct approach,
making it easier to convey the main ideas. Extensive numerical experiments will be presented in
[12], including examples with known solutions.
′
Let F : W01;p1 × Rm → W −1;p2 be given and consider the following problem:
Pr 1. Find (u0 ; 0 ) ∈ (W01;p1 × Rm ) such that
F(u0 ; 0 ) = 0
on
;
where
⊂ R2 is open and bounded. Here
F(u; ) = −3 · [a(3u; u; ; x)] + b(3u; u; ; x) + c(u; ; x) − f(; x);
where u :
→ R; a : R2 × R × Rm × R2 → R2 ; b : R2 × R × Rm × R2 → R; c : R × Rm × R2 → R;
f : Rm × R2 → R are given smooth enough functions.
In this work we are interested in the a posteriori numerical analysis, so we are going to assume
that the following hypothesis holds
Hypothesis 1.1. There exists a nonempty set ⊂ Rm such that, for all 0 ∈ ; Pr.1 has at least
one solution point u0 (0 ) in some given admissible closed convex set K ⊂ W01;p1 .
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
101
The discrete problem is set as
DPr. 1. Find (uh ; h ) ∈ (S h ( h ; p;
) ∩ K) × Rm such that
hF(uh ; h ); vh i = 0;
for all vh ∈ S h ( h ; p;
) ∩ Wd1;p2 :
The main issue now is to establish some restrictions on the di erential equations we are going
to deal with. Actually, there are further issues which will not be covered here, but the reader will
nd them in [3], where a more complete description of the hypothesis will be found. For a more
detailed analysis on the di erentiability structure required in the hypothesis stated below see [3]; for
the existence and convergence results we refer to [3].
′
Let us be speci c and assume that F : W01;p1 × Rm → W −1;p2 , 1 ¡ p1 ¡ ∞, 1 ¡ p2 , is de ned by
F(u; ) = Q(u; ) − R(u; ) − f():
(1)
Here Q(:; ) : W01;p1 → W −1;p2 is an isomorphism and a strongly nonlinear operator for all ∈ Rm ;
′
′
R : W01;p1 × Rm → W −1;p2 is a compact and smooth nonlinear operator, and f() ∈ W −1;p2 .
Also, assume
′
Hypothesis 1.2. F : W01;p1 × Rm → W −1;p2 satis es the following properties:
(i) F is a -Fredholm operator of index i(F) = m from = (W01;p1 ; H01 ; W01;∞ ) into ∗ =
′
(W −1;p2 ; H −1 ; W −1;∞ ).
(ii) The extension
′
Du QH (w; ) : H01 → H −1
is a coercive and bounded linear operator for all w ∈ W 1;∞ and ∈ Rm , with the constants
of boundedness and coercivity being bounded uniformly away from ∞ and 0, respectively, in
bounded sets of (w; ) ∈ W 1;∞ × Rm . Furthermore, its coecients are in L∞ .
(iii) For all u1 ; u2 ∈ W01;∞ and ∈ Rm , there exists C = C(ku1 kW 1;∞ ; ku2 kW 1;∞ ; ||), such that
kDu QH (u1 ; ) − Du QH (u2 ; )kL(H01 ; H −1 ) 6Cku1 − u2 kW 1;∞
and
kDu R(u1 ; ) − Du R(u2 ; )kL(H01 ; H −1 ) 6Cku1 − u2 kW 1;∞ :
Furthermore, all the coecients of Du F(u0 ; 0 ) are as smooth as the gradient of u0 .
(iv) For all (u; ) ∈ W01;∞ × Rm , all the existing derivatives of F with respect to the function
and the parameter at (u; ) are Holder-continuous with respect to . Moreover, the existing
derivatives of F with respect to the parameter are in W −1;p , for all needed values of p, and
are Holder-continuous with respect to the function.
′
(v) For all (u; ) ∈ W01;∞ ×Rm , the linear operator Du R(u; ) : W01;p → W −1; r is a compact operator
for all p, r ′ 6p6r, with r ¿ 2 as large as needed.
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
The main results in this paper depend on some properties of linearized operators. So we de ne:
Hypothesis 1.3. (Continuous inf–sup condition). Let ∞ ¿ r¿2 be any number, 1=r + (1=r ′ ) = 1,
′
and B : H01 × H01 → R be as above. Then, B : W01; t × W01; t → R is bounded and satis es
inf
u∈W01; t
sup
v∈W01; t
′
B(u; v)
¿
¿ 0;
kukW 1; t kvkW 1; t′
sup {B(u; v)} ¿ 0
u∈W01; t
′
for all v ∈ W01; t ;
for all t ∈ [r ′ ; r].
Hypothesis 1.4. (Discrete inf–sup condition). Let ∞ ¿ r¿2 be any number, 1=r + (1=r ′ ) = 1; and
′
B : H01 × H01 → R be as above. Then, B : W01; t × W 1; t → R is bounded and satis es
inf
uh ∈S h
(
sup
vh ∈S h
B(uh ; vh )
kuh kW 1; t kvh kW 1; t′
sup {B(uh ; vh )} ¿ 0
)
¿ ¿ 0;
for all vh ∈ S h ;
uh ∈S h
for all t ∈ [r ′ ; r]. Also, 6= (h).
First, for any q ∈ [r ′ ; r], h ¿ 0, wh ∈ S h and w ∈ W 1;∞ , de ne
h = h (h; q; w) = max
inf
vh ∈S h
n
h
−n=q
o
kw − vh kW 1; q ; inf h {kw − vh kW 1;∞ } :
vh ∈S
The following results are statements concerning existence and convergence of discrete solutions
to DPr.1. For that, let K = Du (u0 ; 0 ) and B(: ; : :) = hK(:); (::)i.
Theorem 1.5. Let F : W01;p1 × Rm −→ W −1;p2 satisfy Hypothesis 1.2. Let the linear operator
K : H01 −→ H −1 be de ned as above and let the bilinear form B : H01 × H01 −→ R be de ned based
on K. Let
∈ Dt , for some t ¿ 2. Let (u0 ; 0 ) be a strong regular solution point to Pr.1, such
that u0 ∈ W01;p1 ∩ W 1+;p , with ¿ n=p.
Then, there is an r ¿ 2; such that, if one can nd q ∈ [r ′ ; r]; satisfying
′
h (h; q; u0 ) → 0;
1; q
one can also nd h0 ¿ 0; and ¿ 0; such that, for all 0 ¡ h6h0 ; there exists a unique uh ∈ W˜ 0 ∩
B (uh ); such that (uh ; 0 ) solves DPr.1. Here the sequence {uh }h→0 ∈ S h is to be suitably chosen.
Proof. This result is just a particular case (for regular solution points) of a more general theorem
presented in [3].
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
103
Remark. The sequence {uh }h→0 in the above theorem is to be chosen such that the objectives of
the analysis are met [3, 12].
Corollary 1.6. Let the hypotheses of Theorem 1.5 be true. Then, for all h ¿ 0 small enough,
kuh − u0 kW 1; s 6Ckuh − u0 kW 1; s ;
for all s ∈ [r ′ ; r]; such that h (h; s; u0 ) → 0, as h → 0;
kuh − u0 kW 1; s 6C[hn=s−(n=r) kuh − u0 kW 1; r + kuh − u0 kW 1; s ];
for all s ∈ [r; ∞], where {uh }h→0 is a sequence chosen as in Theorem 1.5. Here C = C(u0 ; 0 ) and
r ¿ 2 is as obtained in Theorem 1.5.
Furthermore, taking uh = Ph u0 ; and for all s ∈ [r ′ ; ∞]; such that ¿ n=min{s; p}; the above
inequalities imply that
kuh − u0 kW 1; s 6Ch ku0 kW 1+; p ;
where, 1 = min{0; n=s − (n=r)} and = min{q; 2 }; where q¿1 is the polynomial order of approximation of the shape functions in each nite element, and 2 = + (n=s) − n=min{r; p} if s¿r and
2 = + (n=s) − n=min{s; p} if s ¡ r; and, C 6= C(u0 ; h).
Proof. This result is a particular case (for regular solution points) of a more general theorem
presented in [3].
2. A posteriori estimators
In this section we develop a procedure for relating computable a posteriori error estimators for a
suitably de ned auxiliary linear problem with the exact error (in the norm of W 1; s , s ∈ [r ′ ; r], r¿2)
for the nonlinear problem (between a given solution to Pr.1 and the corresponding discrete solution to
DPr.1). A large class of estimators will be considered, namely, implicit estimators, obtained through
a solution of a suitably de ned local problem and de ned either elementwise or patchwise. In order
to make the procedures clear, we will consider only strong regular solution points. The procedures
regarding simple turning points will be presented in later works.
A rst linear auxiliary problem will be de ned by a bilinear form B1 : H01 × H01 → R and a righthand side f1 . Similarly, a second linear auxiliary problem will be de ned by B0 : H01 × H01 → R
and f0 . For the exact, discrete and error equations of both problems we refer to LP.0, DLP.0 and
Er.0 below, respectively.
In what follows, the expression ! ∩ T 6= ∅ will mean that the interior of the region de ned by !
has an empty intersection with the region de ned by T .
De nition 2.1. Let a mesh h be given. Suppose that a way of building a set of patches ! by
making union of adjacent elements T ∈ h , such that the union of all patches covers
, is given. Let
Vh be that set. De ne
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
(a) The trial space
W! = span{
! j=k(!)
;
j }j=1
and the test space
j=k(!)
V! = span{ !j }j=1
;
de ned over each corresponding patch ! ∈ Vh .
(b) The spaces (de ned elementwise), for each T ∈ h ,
!
j |T ,
j = 1; ::; k(!), for all !, such that ! ∩ T 6= ∅};
ZT = span { !j |T ,
j = 1; ::; k(!), for all !, such that ! ∩ T 6= ∅}:
YT = span {
!
T
It is clearly seen that there exist a decomposition whT = !∩T 6=∅ [wP
h ], for all wh ∈ YT , where
!
!
T
wh ∈ W! with ! ∩ T 6= ∅. Similarly, there exists a decomposition vh = !∩T 6=∅ [vh ], for all vhT ∈ ZT ,
where wh! ∈ W! .
P
Hypothesis 2.2. Let a bilinear form B(: : ; :) : W01; s × W 1; s −→ R be given, for all s ∈ [r ′ ; r], where
r ∈ [2; ∞) is given. Let a mesh h and the set of patches Vh be given as de ned above. Then,
(a) There exists a real number
¿ 0, such that, for each T ∈ h , and for all wh ∈ YT ,
′
#
"
X B (w! ; v! )
!
h
h
;
kwh kW 1; s (T ) 6 sup
kvh! kW 1; s′ (!)
vT ∈ZT
!∩T 6=∅
h
where
6=
(h) does not depend either on T ∈ h , nor on Vh . B! (: : ; :) means the restriction of
B to the patch !. Note that we are using the decomposition of vhT ∈ ZT described just above.
(b) Let R ∈ W −1; r be given, and consider R|! as being a suitably de ned restriction of R to !;
for all ! ∈ Vh . Then, there exist positive constants C1 and C2 ; not depending either on the
h nor on Vh , such that
C1 kRkW −1; s (
) 6
X
T ∈ h
X
!∩T 6=∅
1=s
kR|! kW −1; s (!) 6C2 kRkW −1; s (
) :
Remark. Hypothesis 2.2 means that the given bilinear form B is patchwise elliptic and the patches
do not overlap too much, destroying the stability of the sum of quantities de ned patchwise.
We now de ne the class of implicit estimators.
De nition 2.3. (Implicit estimators). Let a mesh h and a set of patches Vh be given. Let a bilinear
form B(: : ; :) be given, which satis es Hypotheses 1.3, 1.4, and 2.2 for some r ∈ [2; ∞) and spaces
{V! }!∈Vh and {W! }!∈Vh . Let f ∈ W −1; r be given, and de ne w0 ∈ W01; r , wh ∈ S h ( h ) and e ∈ W01; r
to be the solutions of LP.0, DLP.0 and Er.0, respectively. For each T ∈ h , de ne T (x) ∈ YT as
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
105
j=k(!)
}!∩T 6=∅ are constants, which are
T (x) = !∩T 6=∅ [ j=1;:::; k(!) Cj! j! (x)], for x ∈ T , where {{Cj! }j=1
obtained by nding ! ∈ W! , such that
P
P
B! (! ; !j ) = hRh! ; !j i
j = 1; :::; k(!)
for all ! ∈ Vh ;
where Rh! = Rh |! is the restriction of the residual Rh ∈ W −1; r to the patch ! ∈ Vh . Thus, T (x) is
computed by
X
T (x) =
[! (x)];
!∩T 6=∅
for all x ∈ T . The restricted residual R! , the trial and the test spaces should be such that the above
problem has a unique solution.
For some given s¿1, set
T (s) = kT kW 1; s (T ) ;
and
(s) =
(
X
T ∈ h
sT
)1=s
:
The value T is called the elemental estimator for T ∈ h (indicator) and is the (global) estimator.
The auxiliary linear problems are to be de ned as follows.
LP.0. Find w ∈ W01; s , such that
B(w; v) = hf; vi
′
for all v ∈ W01; s :
DLP.0. Find wh ∈ S h , such that
B(wh ; vh ) = hf; vh i
for all vh ∈ S h ;
′
where f ∈ W −1; s . De ning the error by e = w − wh , the error equations for the above problems are
given by
Er.0. Find e ∈ W01; s , such that
B(e; v) = hRh ; vi = B(wh ; v) − hf; vi
′
for all v ∈ W01; s :
The following result shows that the implicit estimators change at most linearly with perturbations
in the coecients of the operators and on the right-hand side.
Theorem 2.4. Let the bilinear forms B0 (: : ; :) : H01 × H01 −→ R and B1 (: : ; :) : H01 × H01 −→ R be
given. Let both bilinear forms satisfy Hypotheses 1.3; 1.4 and 2.2 for some r ∈ [2; ∞) and spaces
{W! }!∈Vh and {V! }!∈Vh . Let f0 ; f1 ∈ W −1; r be given as the right-hand sides for B0 and B1 ;
respectively. Let w0 and w1 ∈ W01; r be solutions of LP:0; w0h ; w1h ∈ S h solutions of DLP.0 and R0h ;
R1h ∈ W −1; r the residuals; all related to B0 and B1 ; respectively. If 0 and 1 are implicit estimators
related to B0 ; f0 and B1 ; f1 ; respectively; then; there exists a constant C; which depends only on
the L∞ (
)-norm of the coecients of both bilinear norms; such that; for each s ∈ [r ′ ; r],
|0 (s) − 1 (s)|6C[kBkL∞ 0 + kRh kW −1; s (
) ]:
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
For the above, kBkL∞ means the L∞ (
)-norm of the di erence between the respective coecients of B0 and B1 ; and Rh is the di erence between the residuals R0h and R1h .
Proof. Let T ∈ h be given. Let 0T ,1T ∈ YT be as in De nition 2.3, related to B0 and B1 , respectively.
Set T = 0T − 1T , and B(: : ; :) = B1 (: : ; :) − B0 (: : ; :)]. Then
X
[hRh! ; ! i! ] =
!∩T 6=∅
X
[B0! (! ; ! ) + B! (!0 ; ! )];
(2)
!∩T 6=∅
for all ! ∈ V! , ! ∩ T 6= ∅. Then, by Hypothesis 2.2 and for each s ∈ [r ′ ; r], we obtain
"
#
X B (! ; v! )
1!
h
h
kT kW 1; s (T ) 6 sup
kvh! kW 1; s′ (!)
vhT ∈ZT !∩T 6=∅
#
"
X hR ; v! i B (! ; v! )
!
h! h !
:
6 sup
+ ! 0 h
!
′
′
T
kv
kv
k
k
1;
s
1;
s
v ∈ZT
(!)
(!)
h W
h W
!∩T 6=∅
h
So, there is C = C(s;
), such that
kT ksW 1; s (T ) 6C
X
[kRh! ksW −1; s (!) + kB! ksL∞ (!) k!0 ksW 1; s (!) ]:
!∩T 6=∅
By adding up over all elements of h and using the summation properties of the residuals stated in
De nition 2.3, we get
X
kT krW 1; s (T ) 6C
T ∈ h
"
kRh ksW −1; s (
)
+
kBksL∞ (
)
X
T ∈ h
k0T ksW 1; s (T )
#
:
Taking the sth-root on both sides of the above expression and using Minkowiskii’s inequality yields
|0 − 1 |6C[kRh kW −1; s (
) + kBkL∞ (
) 0 ];
which immediately gives the desired inequality.
Now, let us be speci c and introduce our two auxiliary linear problems, the rst for theoretical
purposes only and the second for the actual computation of the error estimator. As before, and for
the rest of this paper, (u0 ; 0 ) and (uh ; 0 ) will be the solution to Pr.1 and the solution to DPr.1,
′
respectively. Next, let F : W01;p1 → W −1;p2 be given and satisfy the hypothesis of Theorem 1.5. For
any xed r ∈ [2; ∞), as close to 2 as needed, we take s ∈ [r ′ ; r] and set K0 , K0h , Kh : W 1; s → W −1; s ;
′
and B0 , B1 : W01; s × W01; s → R as
K0 = Du F(u0 ; 0 );
K0h =
Z
(3)
1
[Du F(uh + t(u0 − uh ); 0 )]dt;
(4)
0
Kh = Du F(uh ; 0 );
B0 (u; v) = hK0 u; vi
B1 (u; v) = hKh u; vi
(5)
′
for all u ∈ W01 s and v ∈ W01; s ;
(6)
′
for all u ∈ W01 s and v ∈ W01; s :
(7)
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
107
Furthermore, set
eh = u0 − uh ;
(8)
K0 = K0h − K0 ;
(9)
K1 = K0 − Kh ;
(10)
b0 = K 0 u0 :
(11)
The next lemma states some properties for the above operators
Lemma 2.5. Let the hypothesis of Theorem 1.5 and Corollary 1.6 be satis ed. Then, for each
s ∈ [r ′ ; r]:
(i) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; there exists a constant
C 6= C(h); such that
kK0 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(ii) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; there exists a constant
C 6= C(h); such that
kK1 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(iii) There exists a small enough h0 ¿ 0; such that for all 0 ¡ h ¡ h0 ; B0 and B1 satisfy Hypotheses
1.3 and 1.4.
Proof. Items (i) and (ii) follow directly from item (iii) of Hypothesis 1.2 and Corollary 1.6.
Item(iii) is a consequence of the following facts: (a) K0 satis es the inf–sup condition with r = 2,
since (u0 ; 0 ) is a strong regular point; (b) K0 is a compact perturbation of Du Q(u0 ), which satis es
Hypotheses 1.3 and 1.4, for some r ¿ 2 [3]; (c) from item (ii) of the current lemma, it follows that
Kh converges uniformly to K0 for all s ∈ [r ′ ; r]. Then the result follows [3].
The two auxiliary problems will be de ned by the bilinear form B1 (computable) and B0 (abstract),
together with the right-hand sides
f1 = −F(uh ; 0 );
(12)
f0 = b 0 ;
(13)
h
respectively. Let w0h ∈ W01; s and w0h
∈ S h solve LP.0 and DLP.0 with B ≡ B1 and f ≡ f1 , that is,
B1 (w0h ; v) = h−F(uh ; 0 ); vi
h
B1 (w0h
; vh ) = h−F(uh ; 0 ); vh i
′
for all v ∈ W01; s ;
(14)
for all vh ∈ S h :
(15)
Since by de nition, u0 solves LP.0 with B ≡ B0 and f ≡ f0 , let u0h ∈ S h solve the corresponding
discrete problem (DLP.0), i.e.,
B0 (u0h ; vh ) = hb0 ; vh i
for all vh ∈ S h :
(16)
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
Next, let us de ne the error expressions
e0h = u0 − u0h ;
h
;
ewh = w0h − w0h
which are solutions to Er.0 for the abstract and computable auxiliary problems, respectively.
The next lemma establishes some further results regarding the relationship between both linear
problems.
Lemma 2.6. Let the hypothesis of Theorem 1.5 be satis ed. Then; the following statements are
true:
(i) K0 (u0 − uh ) = −F(uh ; 0 ) + K0 eh :
(ii) K0 u0 = b0 = −F(uh ; 0 ) + K0 eh + K0 uh :
(iii) B0 (u0h − uh ; vh ) = B(eh ; vh ) = hK0 eh ; vh i; for all vh ∈ S h :
(iv) There exists a constant C 6= C(h); such that
ku0h − uh kW 1; s 6CkK0 kL(W01; s ;W −1; s ) keh kW 1; s
for all s ∈ [r ′ ; r]
h
(v) w0h
= 0:
(vi) Set
R0h = b0 − K0 u0h ;
and
h
R1h = −F(uh ; 0 ) − Kh w0h
= −F(uh ; 0 ):
Then,
Rh = R0h − R1h = K0 eh + K0 (uh − u0h )
and; hence; there exists a constant C 6= C(h); such that
kRh kW −1; s 6CkK0 kL(W01; s ;W −1; s ) keh kW 1; s :
Proof. (i) This result comes from the observation that K0h eh = −F(uh ; 0 ). Then,
K0 eh = K0h eh + (K0 − K0h )eh = −F(uh ; 0 ) + K0 eh :
(ii) This relation comes directly from the result in (i).
(iii) Since hF(uh ; 0 ); vh i = 0, and B0 (u0 − u0h ; vh ) = 0 for all vh ∈ S h , and from (i) we obtain
B0 (u0h − uh ; vh ) = B0 (u0h + (u0 − u0h ) − uh ; vh ) = B0 (u0 − uh ; vh )
= h−F(uh ) + K0 eh ; vh i = hK0 eh ; vh i:
(iv) From Lemma 2.5, B0 satis es Hypothesis 1.4, for some r ¿ 2, and, then, with the help of
(iii) we get
hK0 eh ; vh i
B0 (uh − u0h ; vh )
= sup
;
kuh − u0h kW 1; s 6 sup
kvh kW 1; s′
kvh kW 1; s′
vh ∈S h
vh ∈S h
where 6= (h) and s ∈ [r ′ ; r]. Thus, the result follows immediately.
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
109
(v) Again, by Lemma 2.5, B1 satis es Hypothesis 1.4 for the same r ¿ 2 as in item (iii) above,
h
and, then, since hF(uh ; 0 ); vh i = 0, for all vh ∈ S h , we nd that w0h
= 0.
(vi) This statement follows directly from (ii) and (iv).
Remark. As we said in the introduction, the strategy of using auxiliary linear problems for estimating
the nite element error for nonlinear problems is not new. Therefore, property (v) in the lemma
above were already known, no matter what linear coercive operator is used. The other remaining
properties are new, and they show important issues. For instance, item (iv) shows that the nite
element solution to the nonlinear problem is a higher-order perturbation of a nite element solution to
a smooth linear problem. This opens a clear space for further investigations on the relation between
superconvergence properties of nonlinear problems and those for linear problems.
Now, de ne 0 (s) and 1 (s) as being the same a posteriori estimator (with respect to the norm of
h
W01; s , s ∈ [r ′ ; r]) applied for estimating the errors e0h = u0 − u0h and ewh = w0h − w0h
= w0h , respectively.
The next theorem will state that both estimators will give the same result, asymptotically speaking
(that is, when h → 0).
Theorem 2.7. Consider F = Q − R − f : W01;p1 × Rm −→ W −1;p2 satisfying all the hypotheses of
′
′
Theorem 1.5. Let the bilinear forms B0 (: : ; :) : W01; s × W01; s −→ R and B1 (: : ; :) : W01; s × W01; s −→ R
be de ned as above, for all s ∈ [r ′ ; r]; satisfying the hypotheses of Theorem 2.4. Consider u0 and
w0h as the solutions to LP.0, related to the bilinear forms B0 and B1 ; and the right-hand sides
h
f0 and f1 ; respectively. Let u0h and w0h
be the respective approximate solutions to DLP.0; and
R0h and R1h be the corresponding residuals; as described in problem Er.0. Let 0 (s) and 1 (s)
(s ∈ [r ′ ; r]) be the result of the same implicit a-posteriori estimator applied to the abstract and the
computable problems, respectively. If the given estimator satis es Hypothesis 2.2; Then,
′
|0 (s) − 1 (s)|6Ckeh kW 1;∞ (keh kW 1;s + 0 ):
Here, as de ned in (8), eh = u0 − uh is the error for the nonlinear problem Pr.1, and C 6= C(h).
Proof. From Theorem 2.4 and Lemma 2.6 we get
|0 (s) − 1 (s)|6C(kBkL∞ 0 + kK0 kL(W01; s ;W −1; s ) keh kW 1; s ):
From Lemma 2.5 we obtain that
kBkL∞ 6Ckeh kW 1;∞ ;
and
kK0 kL(W01; s ;W −1; s ) 6Ckeh kW 1;∞ :
The three relations above yield the desired result.
The lemma below shows that all errors (nonlinear problem and auxiliary problems) are higherorder perturbations of each other, and that the estimator is asymptotically the same when applied to
both auxiliary problems.
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
Lemma 2.8. Let the hypothesis of Theorem 2.7 be satis ed. Furthermore; let u0 ∈ W 1+;p ; such
that ¿ n=min{r; p}. Then; for all s ∈ [r ′ ; r];
(i) keh − ewh kW 1; s 6Ch” keh kW 1; s ;
(ii) kewh − e0h kW 1; s 6Ch” kewh kW 1; s ;
(iii) |0 (s) − 1 (s)|6Ch” (ke0h kW 1; s + 0 (s));
where ” ¿ 0.
Proof. (i) This inequality follows from
keh − ewh kW 1; s 6 sup
v∈W 1; s′
= sup
v∈W 1; s′
= sup
v∈W 1; s′
B0 (eh − ewh ; v)
kvkW 1; s′
B0 (eh ; v) − B1 (ewh ; v) + B(ewh ; v)
kvkW 1; s′
hK0 eh ; vi + B(ewh ; v)
kvkW 1; s′
6 C(kK0 kL(W −1; s ;W −1; s ) keh kW 1; s + kBkL∞ kewh kW 1; s );
from
kewh kW 1; s =
kw0h kW 1; s 6
sup
v∈W 1; s′
(
B1 (w0h ; v)
kvkW 1; s′
)
6kF(uh ; 0 )kW −1; s 6Ckeh kW 1; s
and, from Corollary 1.6,
kBkL∞ 6Ckeh kW 1;∞ 6Ch−n=min{r;p}
kK0 kL(W −1; s ;W −1; s ) 6Ckeh kW 1;∞ 6Ch−n=min{r;p} :
(ii) Similarly, the second inequality is obtained by
ke0h − ewh kW 1; s 6 sup
v∈W 1; s′
= sup
v∈W 1; s′
6 sup
v∈W 1; s′
B0 (e0h − ewh ; v)
kvkW 1; s′
B0 (e0h ; v) − B1 (ewh ; v) + B(ewh ; v)
kvkW 1; s′
hK0 eh ; vi + B(ewh ; v)
kvkW 1; s′
6 C(kRh kW −1; s + kBkL∞ kewh kW 1; s )
6 C(kK0 kL(W −1; s ;W −1; s ) keh kW 1; s + kBkL∞ kewh kW 1; s ):
The rest follows from the previous item.
(iii) From Lemma 2.6 we have that
keh − e0h kW 1; s = kuh − u0h kW 1; s 6CkK0 kL(W −1; s ;W −1; s ) keh kW 1; s
6 Ch−n=min{r;p} keh kW 1; s :
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
111
So, for h ¿ 0 small enough, we obtain
keh kW 1; s 6
ke0h kW 1; s
:
1 − Ch−n=min{r;p}
From Theorem 2.7 and the inequality above we obtain the third and last inequality. Finally we set
” = − (n=min{r; p}).
Lemma 2.9. Let the hypothesis of Theorem 2.7 be satis ed. Furthermore, let u0 ∈ W 1+;p ; such that
¿ n=min{r; p}. Then; if 0 (s); s ∈ [r ′ ; r]; is asymptotically exact; so will be 1 (s); i.e:; if there
exists a constant C 6= C(h) and ”0 ¿ 0; such that; for all small enough h ¿ 0;
|0 (s) − ke0h kW 1; s |6Ch”0 ke0h kW 1; s ;
then, the same is true for 1 (s); that is; there exists a constant C 6= C(h); and ” ¿ 0; such that
|1 (s) − kewh kW 1; s |6Ch” kewh kW 1; s :
Proof. From Lemma 2.8 there exists ”1 ¿ 0, such that
|1 (s) − kewh kW 1; s | 6 |kewh kW 1; s − ke0h kW 1; s | + |ke0h kW 1; s − 0 (s)| + |0 (s) − 1 (s)|:
6 Ch”1 kewh kW 1; s + Ch”1 ke0h kW 1; s + Ch”1 (ke0h kW 1; s + 0 (s)):
Since 0 (s) is asymptotically exact
0 6(1 + Ch”0 )ke0h kW 1; s :
From Lemma 2.8 we get
ke0h kW 1; s 6(1 + Ch”1 )kewh kW 1; s :
The three inequalities above yield the desired result, by taking ” = min{”0 ; ”1 }.
Lemma 2.10. Let the hypotheses of Lemma 2.9 be satis ed for given implicit estimators 0 (s) and
1 (s), s ∈ [r ′ ; r]. Then, for all h ¿ 0 small enough, the estimator 1 (s) is asymptotically equal to
keh kW 1; s ; for all s ∈ [r ′ ; r]. That is, there exists ” ¿ 0 and a constant C 6= C(h); such that
|1 (s) − keh kW 1; s |6Ch” keh kW 1; s ;
for all s ∈ [r ′ ; r].
Proof. From Lemma 2.8
|1 (s) − keh kW 1; s |6|1 (s) − kewh kW 1; s | + |kewh kW 1; s − keh kW 1; s |:
From Lemma 2.8, we get that there exists ”0 ¿ 0, such that
|kewh kW 1; s − keh kW 1; s |6Ch”0 keh kW 1; s ;
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F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
and from Lemmas 2.9 and 2.8
|1 (s) − kewh kW 1; s |6Ch”1 kewh kW 1; s 6Ch”1 (1 + Ch”0 )keh kW 1; s :
The above inequalities provide the statement of the lemma, by taking ” = min{”0 ; ”1 }.
Remark. The de nition of asymptotic exactness given in the statement of Lemma 2.10 may be
weakened by supposing that there exists a function Q(h), with Q(h) → 0 as h → 0, and such that
|1 (s) − keh kW 1; s |6Q(h)keh kW 1; s ;
for all s ∈ [r ′ ; r].
As an example, in order to illustrate the procedures described in this section, we analyse the
′
following partial di erential equation, F : W01;p1 × R → W −1;p1 , 1=p1 + (1=p1′ ) = 1:
F(u; ) = −3 · [(1 + |3u|2 )
p1 −2
2
3u] + u − f:
Here we will assume that p1 ¿ 1 and that the domain
⊂ Rn is as smooth as we wish. The above
p−2
de nition implies that Q(u; ) = −3 · [(1 + |3u|2 ) 2 3u], R(u; ) = −u and f() = f. Furthermore,
d =@
(no Neumann boundary condition). It is observed that when ¿ 0, then Q−R is a uniformly
coercive monotone operator and, then, Pr.1 has a unique solution for each such . Also, provided u0
is smooth enough it is not a dicult task to show that Hypothesis 1.2 is satis ed. The smoothness
of u0 depends on the smoothness of @
and of f, which are assumed to be as smooth as needed.
Now, we observe that, for all ∈ H01 ,
K0 = Du F(u0 ; 0 ) = −3 · [A(x) · 3 ] + 0 ;
where A is the matrix
A(x) = (1 + |3u0 |2 )(p1 −4)=2 [(p1 − 2)3u0 3u0 + (1 + |3u0 |2 )I ]:
It is easily seen that, for all ∈ Rn ,
(A · ) · ¿
||2 [1 + (p1 − 1)|3u0 |2 ];
||2 (1 + |3u0 |2 )
1 ¡ p1 62;
p1 ¿ 2:
Then, for all 0 ¿ 0, Du F(u0 ; 0 ) : H01 → H −1 is a uniformly coercive elliptic linear operator
with smooth coecients. If (uh ; 0 ) is the nite element solution to DPr:1, which exists following
Theorem 1.5, then it converges, following Corollary 1.6, with rate min{q; −n=r}, in the W 1;∞ -norm,
where q¿1 is the polynomial order of approximation of the shape functions in each element.
In a similar fashion as we did for K0 , we obtain that, for all ∈ H01
Kh = Du F(uh ; 0 ) = −3 : [Ah (x) : 3 ] + 0 ;
where Ah is the matrix
Ah (x) = (1 + |3uh |2 )
p1 −4
2
[(p1 − 2)3uh 3uh + (1 + |3uh |2 )I ]:
F.C.G. Santos / Journal of Computational and Applied Mathematics 103 (1999) 99–114
113
It is easily seen that, for all ∈ Rn ,
(Ah · ) · ¿
||2 [1 + (p1 − 1)|3uh |2 ];
||2 (1 + |3uh |2 )
1 ¡ p1 62;
p1 ¿ 2;
and thus, Kh : H01 → H −1 is a uniformly coercive linear elliptic operator.
Now, the strategy to obtain 1 (s), s ∈ [r ′ ; r] is to estimate ewh , considering the following error
equation:
B1 (ewh ; v) = h−F(uh ; 0 ); vi
′
for all v ∈ W 1; s :
Recall that B1 (: ; : :) = hKh (:); (: :)i. There are several options for computing the implicit estimator
1 . For a review of some of them see [7, 16]. The best choices will be among those which may
be asymptotically exact, provided some smoothness requirements are satis ed. Particularly, those
requirements are met by our abstract and smooth linear problem, de ned by the bilinear form B0
and the right-hand side f0 . Hence, by Lemma 2.10, 1 (s), computed by such a method, will be
asymptotically exact with respect to the error eh = u0 − uh in the W01; s -norm.
We would like to make two basic concluding remarks. First, the use of auxiliary linear problems
for estimating the error for nonlinear problems has been in use for some time. What we have proved
is that it is possible to justify the use of such estimators designed for linear problems in nonlinear
problems (asymptotically exact implicit estimators preserve that property for the nonlinear problems,
provided some standard assumptions are satis ed). Second, the majority of the estimators for linear
problems are considered in the norm of some suitable Hilbert space, but a large number of nonlinear
problems are not posed on such spaces. Nevertheless, it is our conjecture that the estimators should
behave well in other norms, provided some assumptions (e.g., Hypotheses 1.3 and 1.4) about the
operator and solution set hold, together with some restrictions on the mesh and on the nite element
spaces.
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