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Journal of Mathematical Sciences, Vol. xxx, No. y, 2003

APPROXIMATION OF ABSTRACT DIFFERENTIAL EQUATIONS

Davide Guidetti, Bulent Karas¨ en, and Sergei Piskarev ¨ oz

UDC 517.988.8

1. Introduction This review pap er is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and pro jection methods can b e considered from the p oint of view of general approximation schemes (see, e.g., [207, 210, 211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant pap ers and b ooks. We can refer to the bibliography [218], which contains ab out 3000 references. Unfortunately, no b ooks or reviews on general approximation theory app ear for differential equations in abstract spaces during last 20 years. Any information on the sub ject can b e found in the original pap ers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describ e the general approximation scheme, different typ es of convergence of op erators, and the relation b etween the convergence and the approximation of sp ectra. Also, such a convergence analysis can b e used if one considers elliptic problems, i.e., the problems which do not dep end on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter­Kato and Lax­Richtmyer theorems from the general and common p oint of view and related problems. The approximation of ill-p osed problems is considered in Sec. 4, which is based on the theory of approximation of local C -semigroups. Since the backward Cauchy problem is very imp ortant in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature b efore to the b est of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parab olic equations in Cn ([0,T ]; En ),
Cn ([0,T ]; En ), Lpn ([0,T ]; En ), and Bn ([0,T ]; C (h )) spaces.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 113, Functional Analysis, 2002. 1072­3374/03/xxxy­0001 $ 27.00 c 2003 Plenum Publishing Corporation 1


The last section, Sec. 6 deals with semilinear problems. We consider approximations of Cauchy problems and also the problems with p eriodic solutions. The approach describ ed here is based on the theory of rotation of vector fields and the principle of compact approximation of op erators. 2. General Approximation Scheme Let B (E ) denote the Banach algebra of all linear b ounded op erators on a complex Banach space E . The set of all linear closed densely defined op erators on E will b e denoted by C (E ). We denote by (B ) the sp ectrum of the op erator B, by (B ) the resolvent set of B, by N (B ) the null space of B , and by R(B ) the range of B . Recall that B B (E ) is called a Fredholm op erator if R(B ) is closed, dim N (B ) < and codim R(B ) < , the index of B is defined as ind B = dim N (B ) - codim R(B ). The general approximation scheme [83­85, 187, 207, 210] can b e describ ed in the following way. Let En and E b e Banach spaces, and let {pn } b e a sequence of linear b ounded op erators pn : E En ,pn B (E, En ), n N = {1, 2, ··· }, with the following prop erty: pn x
E
n

x

E

as n for any x E.

Definition 2.1. A sequence of elements {xn },xn En ,n N, is said to b e P -convergent to x E iff xn - pn x
E
n

0 as n ; we write this as xn x.

Definition 2.2. A sequence of elements {xn },xn En ,n N, is said to b e P -compact if for any N N there exist N N and x E such that xn x, as n in N . Definition 2.3. A sequence of b ounded linear op erators Bn B (En ),n N, is said to b e PP -convergent to the b ounded linear op erator B B (E ) if for every x E and for every sequence {xn },xn En ,n N, such that xn x one has Bn xn Bx. We write this as Bn B. For general examples of notions of P -convergence, see [82, 187, 203, 211]. Remark 2.1. If we set En = E and pn = I for each n N, where I is the identity op erator on E , then Definition 2.1 leads to the traditional p ointwise convergence of b ounded linear op erators which is denoted by Bn B. Denote by E + the p ositive cone in a Banach lattice E. An op erator B is said to b e positive if for any x+ E + , it follows Bx+ E + ; we write 0 B.

Definition 2.4. A system {pn } is said to b e discrete order preserving if for all sequences {xn },xn En , and any element x E , the following implication holds: xn x implies x+ x+ . n 2


It is known [99] that {pn } preserves the order iff pn x+ - (pn x)+

E

n

0 as n for any x E.

In the case of unb ounded op erators, and, in general, we know infinitesimal generators are unb ounded, we consider the notion of compatibility. Definition 2.5. A sequence of closed linear op erators {An }, An C (En ), n N, is said to b e compatible with a closed linear op erator A C (E ) iff for each x D(A) there is a sequence {xn },xn D(An ) En ,n N, such that xn x and An xn Ax. We write this as (An ,A) are compatible. In practice, Banach spaces En are usually finite dimensional, although, in general, say, for the case of a closed op erator A, we have dim En and An
B (En )

as n .

2.1. Approximation of spectrum of linear operators. The most imp ortant role in approximations of equation Bx = y and approximations of sp ectra of an op erator B is played by the notions of stable and regular convergence. These notions are used in different areas of numerical analysis (see [10, 15, 81, 86­ 89, 207, 210, 212, 222]). Definition 2.6. A sequence of op erators {Bn },Bn B (En ),n N, is said to b e stably convergent to an
- op erator B B (E ) iff Bn B and Bn 1 B (En )

= O(1),n . We will write this as: Bn B stably.

Definition 2.7. A sequence of op erators {Bn },Bn B (En ), is called regularly convergent to the op erator B B (E ) iff Bn B and the following implication holds: xn
E
n

= O(1) & {Bn xn } is P -compact = {xn } is P -compact .

We write this as: Bn B regularly. Theorem 2.1 ([210]). For Bn B (En ) and B B (E ) the fol lowing conditions are equivalent: (i) Bn B regularly, Bn are Fredholm operators of index 0 and N (B ) = {0}; (ii) Bn B stably and R(B ) = E ; (iii) Bn B stably and regularly;
- (iv) if one of conditions (i)­(iii) holds, then there exist Bn 1 B (En ),B -1 - B (E ), and Bn 1 B -1

regularly and stably. This theorem admits an extension to the case of closed op erators B C (E ),Bn C (En ) [213]. Let C b e some op en connected set, and let B B (E ). For an isolated p oint (B ), the corresp onding maximal invariant space (or generalized eigenspace) will b e denoted by W (; B ) = P ()E , 1 where P () = (I - B )-1 d and is small enough so that there are no p oints of (B ) in 2i | -|= the disc { : | - | } different from . The isolated p oint (B ) is a Riesz point of B if 3


I - B is a Fredholm op erator of index zero and P () is of finite rank. Denote by W (, ; Bn ) =
|n -|<,n (Bn )

W (n ,Bn ), where n (Bn ) are taken from a -neighb orhood of . It is clear that

1 (In - Bn )-1 d . The following theorems state the 2i | -|= complete picture of the approximation of the sp ectrum. W (, ; Bn ) = Pn ()En , where Pn () = Theorem 2.2 ([82, 208, 209]). Assume that Ln () = I - Bn and L() = I - B are Fredholm operators of index zero for any and Ln () L() stably for any (B ) = . Then (i) for any 0 (B ) , there exists a sequence {n }, n (Bn ), n N, such that n 0 as n ; (ii) if for some sequence {n },n (Bn ), n N, one has n 0 as n , then 0 (B ); (iii) for any x W (0 ,B ), there exists a sequence {xn }, xn W (0 ,; Bn ), n N, such that xn x as n ; (iv) there exists n0 N such that dim W (0 ,; Bn ) dim W (0 ,B ) for any n n0 . Remark 2.2. The inequality in (iv) can b e strict for all n N as is shown in [207]. Theorem 2.3 ([210]). Assume that Ln () and L() are Fredholm operators of index zero for al l . Suppose that Ln () L() regularly for any and (B ) = . Then statements (i)­(iii) of Theorem 2.2 hold and also (iv) there exists n0 N such that dim W (0 ,; Bn ) = dim W (0 ,B ) for al l n n0 ; (v) any sequence {xn },xn W (0 ,; Bn ), n N, with xn of this sequence belongs to W (0 ,B ). Remark 2.3. Estimates of |n - 0 |, gap W (0 ,; Bn ), W (0 ,B ) ^ and |n - 0 | are given in [210],
E
n

= 1 is P -compact and any limit point

^ where n denotes the arithmetic mean (counting algebraic multiplicities) of the sp ectral values of Bn that contribute to W (0 ,; Bn ). For the notion of gap and its prop erties, see [105]. 2.2. Regions of convergence. Theorems 2.2 and 2.3 have b een generalized to the case of closed op erators in [213] by using the following notions introduced by Kato [105]. Definition 2.8. The region of stability s = s ({An }), An C (Bn ), is defined as the set of all C such that (An ) for almost all n and such that the sequence { (In - An )-1 }n s ({An }) and such that the sequence of op erators {(In - An )-1 }n op erator S () B (E ). 4
N

is b ounded.

The region of convergence c = c ({An }), An C (En ), is defined as the set of all C such that
N

is PP -convergent to some


It is clear that S (·) is a pseudo-resolvent, and S (·) is a resolvent of some op erator iff N (S ()) = {0} for some (cf. [105]). Definition 2.9. A sequence of op erators {Kn }, Kn C (En ), is called regularly compatible with an op erator K C (E ) if (Kn ,K ) are compatible and, for any b ounded sequence xn
E
n

= O(1) such that

xn D(Kn ) and {Kn xn } is P -compact, it follows that {xn } is P -compact, and the P -convergence of {xn } to some x and that of {Kn xn } to some y as n in N N imply that x D(K ) and Kx = y. Definition 2.10. The region of regularity r = r ({An },A), is defined as the set of all C such that (Kn ,K ), where Kn = In - An and K = I - A are regularly compatible. The relationships b etween these regions are given by the following statement. Proposition 2.1 ([213]). Suppose that c = and N (S ()) = {0} at least for one point c so that S () = (I - A)-1 . Then (An ,A) are compatible and c = s (A) = s r = r (A). It is shown in [213] that the conditions (An ,A) are compatible, I - An and I - A are Fredholm operators with index zero for any and (A) = imply (i)­(iv) of Theorem 2.2 when (A) and imply (i)­(iii) of Theorem 2.2 and (iv)­(v) of Theorem 2.3, when r . Definition 2.11. A Riesz p oint 0 (A) is said to b e strongly stable in Kato's sense if dim W (0 ,; Bn ) dim W (0 ,B ) for all n n0 . Theorem 2.4 ([213]). The Riesz point 0 (A) is strongly stable in Kato's sense iff 0 r (A). Investigations of approximation of sp ectra and typ es of convergence, but not those of general approximation scheme are given in [13, 40, 52, 130, 131, 142, 145]. 2.3. Convergence in Anselone's conditions. Throughout this subsection we assume that En = E and pn = I for all n N. Hence the symb ol P will b e omitted in the notation of this subsection. Let us recall that if Bn B compactly (see Definition 2.12), then for any = 0 we have I - Bn I - B regularly [207]. When Bn B compactly and B is a compact op erator, Anselone [10] has proved that (Bn - B )Bn 0, (Bn - B )B 0 as n . (2.1)
s

Considering an approximation of a weakly singular compact integral op erator, Ahues [4] has proved that these convergence prop erties (2.1) are sufficient to state that a Riesz p oint is strongly stable in Kato's sense. 5


Theorem 2.5 ([7]). Assume that B B (E ) is compact and that Bn B. If (Bn - B )Bn 0 as n ; then for any nonzero 0 (B ), assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v) of Theorem 2.3 hold. Theorem 2.6 ([7]). Assume that Bn B and (2.1) holds. Then for any nonzero Riesz point 0 (B ), assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v) of Theorem 2.3 hold. Corollary 2.1 ([5]). Assume that Bn B, I - Bn are Fredholm operators of index zero for {z :
k |z - 0 | }, and (Bn - B )Bn 0 as n for some k N. Then for any nonzero Riesz point

0 (B ), assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v) of Theorem 2.3 hold. Theorem 2.7 ([7]). Assume that B is compact, Bn B , and Bn (Bn - B ) 0. Then 0 In - Bn 0 I - B regularly for any 0 = 0.
k Theorem 2.8 ([7]). Assume that B is compact, Bn B , and Bn (Bn - B ) 0 for some k N. Then

0 In - Bn 0 I - B regularly for any 0 = 0. Let r (B ) b e a sp ectral radius of op erator B B (E ). Theorem 2.9 ([18]). Let E be a Banach lattice. Let 0 (Bn ) and r (Bn ) r (B ) as n . The conclusion on the order of convergence of eigenvectors in Theorem 2.9 also is given in [17]. The application of Theorems 2.5­2.8 to the numerical solution of a mathematical model used in the jet printer industry is considered in [6, 118]. 2.4. Compact convergence of resolvents. We now consider the imp ortant class of op erators which have compact resolvents. We will use this prop erty of generator as an assumption in Sec. 6. In this case, it is natural to consider approximate op erators which "preserve" this prop erty. Definition 2.12. A sequence of op erators {Bn }, Bn B (En ), n N, converges compactly to an op erator B B (E ) if Bn B and the following compactness condition holds: xn
E
n

Bn ,B B (E ) be such that Bn B and

(Bn - B )+ 0 as n . Suppose that r (B ) is a Riesz point of (B ). Then r (Bn ) is a Riesz point of

= O(1) = {Bn xn } is P -compact.

Definition 2.13. The region of compact convergence of resolvents, cc = cc (An ,A), where An C (En ) and A C (E ) is defined as the set of all c (A) such that (In - An )-1 (I - A)-1 compactly. 6


Theorem 2.10. Assume that cc = . Then for any s the fol lowing implication holds: xn
E
n

= O(1) & (In - An )xn

E

n

= O(1) = {xn } is P -compact.

(2.2)

Conversely, if for some c (A) implication (2.2) holds, then cc = . Proof. Let (µIn - An )-1 (µI - A)-1 compactly for some µ cc . Then for xn (I - An )xn
E
n

E

n

= O(1) and

= O(1), from the Hilb ert identity (In - An )-1 - (µIn - An )-1 = (µ - )(In - An )-1 (µIn - An )-1 , (2.3)

we obtain xn = (µIn - An )-1 (In - An )xn - ( - µ)(µIn - An )-1 xn , and it follows that {xn } is P -compact. Conversely, let implication (2.2) hold for some 0 c (A). We show that 0 cc . Taking a b ounded sequence {yn }, n N, we obtain (0 In - An )-1 y to the sequence xn = (0 In - An )-1
nE
n

= O(1) for n N. Let us apply implication (2.2)

yn . It is easy to see that {xn } is P -compact. Hence 0 cc .

Corollary 2.2. Assume that cc = . Then cc = c (A). Proof. It is clear that cc c (A). To prove that cc c (A), let us consider the Hilb ert identity (2.3). Now let µ cc . Then µ cc c (A). Hence, for every c (A) and for any b ounded sequence {xn }, n N, the sequence {(In - An )-1 xn } is P -compact. Comparing Definitions 2.7, 2.8, and 2.13 with implication (2.2), we see that cc r . Theorem 2.11. Assume that cc = . Then r = C. Proof. Take any p oint 1 C. We have to show that (1 In - An ,1 I - A) are regularly compatible. Assume that xn
E
n

= O(1) and that {(1 In - An )xn } is P -compact. To show that {xn } is P -compact, we

take µ cc . Using (2.3) with = 1 , we obtain xn = (µIn - An )-1 (1 In - An )xn +(1 - µ)(µIn - An )-1 xn and, therefore, {xn } is P -compact. Assume now that xn x and (1 In - An )-1 xn y, as n in N N. Then x = (µI - A)-1 y - (1 - µ)(µI - A)-1 x, and it follows that x D(A)and (1 I - A)x = y . 3. Discretization of Semigroups Let us consider the following well-p osed Cauchy problem in the Banach space E with an op erator A C (E ) u (t) = Au(t), t [0, ), u(0) = u0 , 7 (3.1)


where the op erator A generates a C0 -semigroup exp(·A). It is well-known that this C0 -semigroup gives the solution of (3.1) by the formula u(t) = exp(tA)u0 for t 0. The theory of well-p osed problems and numerical analysis of these problems have b een develop ed extensively; see, e.g., [75, 88, 105, 161, 163, 200, 216]. Let us consider on the general discretization scheme for the semidiscrete approximation of the problem (3.1) in the Banach spaces En : un (t) = An un (t), t [0, ), un (0) = u0 n , (3.2)

with the op erators An C (En ) such that they generate C0 -semigroups which are compatible with the op erator A and u0 u0 . n 3.1. The simplest discretization schemes. We have the following version of Trotter­Kato's Theorem on the general approximation scheme. Theorem 3.1 ([203] (Theorem ABC)). The fol lowing conditions (A) and (B) are equivalent to condition (C). (A) Compatibility. There exists (A) n (An ) such that the resolvents converge: (In - An )-1 (I - A)-1 ; (B) Stability. There are some constants M 1 and , independent of n and that M exp(t) for t 0 and any n N; (C) Convergence. For any finite T > 0, one has max n whenever u0 u0 . n The analytic C0 -semigroup case is slightly different from the general case but has the same prop erty (A). Theorem 3.2 ([161]). Let operators A and An generate analytic C0 -semigroups. The fol lowing conditions (A) and (B1 ) are equivalent to condition (C1 ). (A) Compatibility. There exists (A) n (An ) such that the resolvents converge: (In - An )-1 (I - A)-1 ; (B1 ) Stability. There are some constants M2 1 and 2 such that (I - An )-1 8 M2 , Re > 2 ,n N; | - 2 |
t[0,T ]

exp(tAn )

exp(tAn )u0 - pn exp(tA)u0 0 as n


(C1 ) Convergence. For any finite µ > 0 and some 0 < <
(,µ)

, we have 2

max

exp(An )u0 - pn exp(A)u0 0 n

as n whenever u0 u0 . Here, (, µ) = {z ( ) : |z | µ} and ( ) = {z C : | arg z | }. n Definition 3.1. A linear op erator A : D(A) E E is said to have the positive off-diagonal (POD) property if Au, 0 whenever 0 Definition 3.2. An element e E 0 R such that -e x u D(A) and 0
+

E with u, = 0.

is said to b e an order-one in E if for every x E there exists
+

e. For e int E x
e

we can define the order-one norm by x e}.
E

= inf { 0 : -e

An ordered Banach space E is called an order-one space if there exists e int E + such that · Now we can state a version of the Trotter­Kato theorem for p ositive semigroups.

=·

e

.

Theorem 3.3 ([169]). Let the operators An and A from (3.1) and (3.2) be compatible, let E, En be
+ order-one spaces, and let en D(An ) int En . Assume that the operators An have the POD property and

An en

0 for sufficiently large n. Then exp(tAn ) exp(tA) uniformly in t [0,T ].

We can assume without loss of generality that conditions (A) and (B) hold for the corresp onding semigroup case if any discretization processes in time are considered. If we denote by Tn (·) a family 1 (Tn (n ) - In ) B (En ) and Tn (t) = Tn (n )kn , where of discrete semigroups as in [105], i.e., An = n t , as n 0,n , then one obtains the following assertion. kn = n Theorem 3.4 ([203] (Theorem ABC-discr.)). The fol lowing conditions (A) and (B ) are equivalent to condition (C ). (A) Compatibility. There exists (A) n (An ) such that the resolvents converge: (In - An )-1 (I - A)-1 ; (B ) Stability. There are some constants M1 1 and 1 such that Tn (t) M1 exp(1 t) for t R+ = [0, ),n N; (C ) Convergence. For any finite T > 0 one has maxt whenever u0 u0 . n 9
[0,T ]

Tn (t)u0 - pn exp(tA)u0 0 as n , n


Theorem 3.5 ([203]). Assume that conditions (A) and (B) of Theorem 3.1 hold. Then the implicit difference scheme U n (t + n ) - U n (t) = An U n (t + ), U n (0) = u0 , n n is stable, i.e. (In - n An )-k
n

(3.3)

M1 e1 t ,t = kn n R+ , and gives an approximation of the solution

of problem (3.1), i.e., U n (t) (In - n An )-kn u0 exp(tA)u0 P -converges uniformly with respect to n n t = kn n [0,T ] as u0 u0 ,n ,kn ,n 0. n Here, in Theorem 3.5, An = An (In - n An )-1 , and, therefore, (In - n An )-kn = (In + n An )kn . Theorem 3.6 ([203]). Assume that conditions (A) and (B) of the Theorem 3.1 hold and condition n A2 = O(1) n is fulfil led. Then the difference scheme Un (t + n ) - Un (t) = An Un (t), Un (0) = u0 , n n is stable, i.e., (In + n An )k as n ,kn ,n 0. Theorem 3.7 ([161]). Assume that conditions (A) and (B1 ) of Theorem 3.2 hold and condition n An 1/(M +2),n N (3.6)
n

(3.4)

(3.5)

Met ,t = kn n R+ , and gives an approximation of the solution of

problem (3.1), i.e., Un (t) (In + n An )kn u0 u(t) P -converges uniformly with respect to t = kn n [0,T ] n

is fulfil led. Then the difference scheme (3.5) is stable and gives an approximation of the solution of problem (3.1), i.e., Un (t) (In + n An )kn u0 u(t) discretely P -converge uniformly with respect to n t = kn n [0,T ] as u0 u0 , n , kn , n 0. n Let us introduce the following conditions: (B1 ) Stability. There are constants M and such that exp(tAn ) M e t , An exp(tAn ) M t e , t R+ . t

(B1 ) Stability. There are constants M , , and > 0 such that (In - n An )-k M e
kn

, kn An (In - n An )-k M e

kn

, 0 < n < ,n,k N.

Proposition 3.1 ([183]). Conditions (B1 ), (B1 ), and (B1 ) are equivalent. Theorem 3.8. Conditions (A) and (B1 ) are equivalent to condition (C1 ). 10


Theorem 3.9 ([164]). Let the assumptions of Theorem 3.7 and (3.4) be satisfied. Then tAn (In + n An )kn tA exp(tA) uniformly in t = kn n [0,T ]. (3.7)

Conversely, if (In + n An )kn exp(tA) uniformly in t = kn n [0,T ] and (3.7) is satisfied, then condition (C1 ) holds. Theorem 3.10 ([164]). Let condition (B1 ) hold. Then exp(tAn ) - (In - n An )-k If, moreover, the stability condition (3.6) holds, then exp(tAn ) - (In + n An )k
n n

c

n t e. t

c

n t e, t n t e An xn , t

(exp(tAn ) - (In + n An )kn )xn An (exp(tAn ) - (In + n An )kn )xn

cn et An xn , c t = kn n .

In the case of analytic C0 -semigroups for the forward scheme, as we saw, the stability condition n An < 1/(M +2) cannot b e improved even in Hilb ert spaces for self-adjoint op erators. In the case of almost p eriodic C0 -semigroups and the forward scheme for differential equations of first order in time (3.1), one obtains necessary and sufficient stability condition
n

An < 1 [163]. It was discovered that the stability condition

of the forward scheme like (3.5) for the p ositive C0 -semigroups also can b e written in the form n An < 1; see [168]. Stability of difference schemes under some sp ectral conditions were obtained in [26]. The stability of difference schemes for differential equations in Hilb ert spaces in the energy norm are investigated in [179, 180], where schemes with weights were also considered. Semidiscrete approximations are studied also in [180]. 3.2. Rational approximation. Let us denote by Pp (z ) an element of the set of all real p olynomials of Pp (z ) degree no greater than p and by p,q the set of all rational functions rp,q (z ) = and Pq (0) = 1. Then Pq (z ) a Pad´ (p, q )-approximation for e-z is defined as an element Rp,q (z ) p,q such that e |e-z - Rp,q (z )| = O(|z |p
+q +1

) as |z | 0.

It is well known that a Pad´ approximation for e-z exists, is unique and is represented by the formula e Rp,q (z ) = Pp,q (z )/Qp,q (z ), where
p

Pp,q (z ) =
j =0

(p + q - j )!p!(-z )j , Qp,q (z ) = (p + q )!j !(p - j )!

q

(p + q - j )!q !z j (p + q )!j !(q - j )!.
j =0

11


In [174, 175], details of the location of p oles and the order of convergence of rational approximations in different regions are given. Definition 3.3. A rational approximation rp,q (·) p,q for e-z is said to b e (a) A-acceptable if |rp,q (z )| < 1 for Re(z ) > 0; (b) A( )-acceptable if |rp,q (z )| < 1 for z ( ) = {z : - < arg(z ) < , z = 0}. It is well known that Rq,q (z ),Rq
-1,q

(z ), and Rq

-2,q

(z ) are A-acceptable. But for q 3 and p = q - 3,

the Pad´ functions are not A-acceptable. e Theorem 3.11 ([175]). For any q 2 and p 0, the Pad´ approximation of e-z has no poles in the e sector S
p,q -1

= z : | arg(z )| < cos

q-p-2 p+q

;

in particular, for p q p +4 al l poles lie in the left half-plane. Since r (·) p,q is an approximation of e-z , it is natural to construct the op erator-function r (n An )k which can b e considered as an approximation of exp(tAn ) for t = kn . For simplicity, we assume in this section that exp(tAn ) M, t R+ . Theorem 3.12 ([44]). Let condition (B) be satisfied. There is a constant C depending on r such that if r is A-acceptable, then r (n An )k CM k for n > 0,k N. Remark 3.1. The term k in Theorem 3.12 cannot b e removed in general; moreover, there are examples [55, 97], which show that the inequality r (n An )k c k, k N, holds. We say that r (·)
p,q

is accurate of order 1 d p + q if |e-z - r (z )| = O(|z |d+1 ) as |z | 0.

Theorem 3.13 ([44]). Let condition (B) be satisfied. Then there is a constant C depending on r such that, if r is A-acceptable and accurate of order d, then r (n An )k u0 - exp(tAn )u0 CM n n
d n

Ad+1 u0 for n > 0,k N,u0 D(Ad+1 ). n n n n

Theorem 3.14 ([44]). Let condition (B1 ) be satisfied. Then there is a constant C depending on r , such that if r is A-acceptable and accurate of order d, then r (n An )k u0 - exp(tAn )u0 CM n n 12
d n

Ad u0 for n > 0,k N,u0 D(Ad ). nn n n


Theorem 3.15 ([162, 185]). Let condition (B1 ) be satisfied. Then there is a constant C depending on r , such that if r is A-acceptable and accurate of order d with |r ()| < 1 or condition (3.6) is satisfied, then r (n An )k u0 - exp(tAn )u0 CM n n t
n d-



A u0 for n > 0, 0 d, t = kn ,k N. nn

In [54, 152, 154], the analogs of Theorems 3.13­3.15 were proved for multistep methods. Let us recall that constant M2 in condition (B1 ), which defines , 0 < < , by M2 sin < 1 [110] 2 is such that (In - An )-1 M for any (/2+ ). | - | (3.8)

Theorem 3.16 ([55, 150]). Let condition (B1 ) be satisfied. Then there is a constant C depending on r , such that if r is A( )-acceptable, accurate of order d, and (/2 - , /2] for from condition (3.8), then r (n An )k CM for n > 0,k N, and r (n An )k - exp(tAn ) - k exp(-
-b n - - akn (-An )-b ) CM (kn d + kn 1/b ), t = kn n ,

where = r () and a, b are some positive constants. It is p ossible to show [151] that
k j =1

r (

n,j

An ) is a stable approximation for exp(
n,j

k j =1 n,j

A) with

a variable stepsize, but under condition 0 < c n,i /

C < , i, j N.

3.3. Richardson's extrapolation method. Let us consider schemes (3.3) and (3.5) which have the
order of convergence O(n ) and denote Unn (kn ) = Un (t)u0 and U n (kn ) = U n (t)u0 ,t = kn n . The following n n n

approach to the limit is valid. Theorem 3.17 ([167]). Assume that condition (B) is satisfied. Then for V n (t) = 2U n (kn ) - U one has
2 V n (t) - un (t) n Met t2 A3 u0 , t = kn n . nn If, in addition, scheme (3.5) is stable, then for Vn (t) = 2Unn (kn ) -Unn (2kn ),t = kn n , 2 Vn (t) - un (t) n Met t2 A3 u0 , t = kn n . nn /2 n n /2 n

(2kn ),

Let us consider the Crank­Nicolson scheme ~ ~ ~ ~ Un (kn + n )+ Un (kn ) ~ Un (kn + n ) - Un (kn ) = An , Un (0) = In , k N0 , n 2 (3.9) 13


Theorem 3.18 ([167]). Assume that condition (B) is satisfied and that scheme (3.9) is stable. Then 4 ~ /2 1 ~ n (t) = Unn (2kn ) - Unn (kn ) satisfies 3 3
4 n (t) - un (t) cn et t2 A6 u0 , t = kn n . nn In general, we set Vnn (t) = Rp,q (n An )kn u0 , t = kn n . n

Theorem 3.19 ([167]). Assume that condition (B) is satisfied, p = q and the scheme which corresponds 1 22q /2 Vnn (t)+ 2q Vnn (t), to Vnn is stable. Then for n (t) = - 2q 2 -1 2 -1 n (t) - un (t) c
2q +2 t e n

t3/2 2q A n n

+3 0 un

+ t3

2q -3 n

A4q n

+2 0 un

, t = kn n .

Theorem 3.20 ([167]). Assume that condition (B1 ) is satisfied, p = q and n An const. Then for 1 22q /2 Vnn (t)+ 2q Vnn (t), 0 2q and n (t) = - 2q 2 -1 2 -1 n (t) - un (t) c
2q +2 n et 2q +2-



t

A u0 , t = kn n . nn

3.4. Lax-type equivalence theorems with orders. The Lax equivalence theorem on the convergence of the solution of the approximation problem to the solution of the given well-p osed Cauchy problem states that the stability of the method is necessary and sufficient for the convergence provided it is compatible. Recently, Lax's theorem with orders, which make it p ossible to consider "unstable" approximations, was obtained. Definition 3.4. C0 -semigroups exp(tAn ) and exp(tA) are said to b e compatible of order O((n )) on a linear manifold U E with resp ect to the semigroup exp(·A) if exp(tA)U D(A) and there is a constant C such that (An pn - pn A)exp(tA)x Cn (n )et |x|U for any x U , where |· | denotes the seminorm on U . Definition 3.5. C0 -semigroups exp(tAn ) is said to b e stable of order O(Mn en t ) if there are constants Mn and n such that exp(tAn ) Mn en t for any t R+ . The following is a slight modification of [47­50] and [66­68], which was proved in [164]. 14 (3.11) (3.10)


Theorem 3.21. Let a C0 -semigroup exp(·An ) be compatible of order O((n )) on a linear manifold U E with respect to a semigroup exp(·A), exp(tA)U U , and let | exp(tA)x|U M |x|U . The fol lowing assertions are equivalent: Cn (exp(tAn )pn - pn exp(tA)) x 2Mn en t K t(n ),x; E, U ; 2 Mx , x E, n t (ii) exp(tAn )pn - pn exp(tA) x Mn e Cn t(n )|x|U , t = kn n [0,T ],x U ; 2 (iii) exp(tAn ) Mn en t , (An pn - pn A)exp(tA)x Cn n (n )et |x|U for any x U ,t R+ , (i) where Mx is a constant depending only on x and K (t, x; E, U ) = inf
y U

x-y

E

+t|y |U

is Peetre functional.

Definition 3.6. A family of discrete semigroups {Un (kn n )} is said to b e compatible of order O((n )) on a linear manifold U E with resp ect to the semigroup exp(·A) if U = E and (Un (n )pn - pn exp(n A)) exp(tA)x Cn (n )|x|U for any x U . (3.12)

Theorem 3.22. Let exp(tAn )Un Un , let condition (B) hold, and let | exp(tAn )x|Un Cet |xn |Un for any xn Un and t > 0. Then the fol lowing conditions are equivalent: Cn kn n (n ),xn ,En , Un ,n,kn N; (a) (Un (kn n ) - exp(kn n An ))xn Mn K 2 Mx , xn En , n (b) (Un (kn n ) - exp(kn n An ))xn Mn Cn t te (n )|xn |Un , xn Un ; 2 Cn Mn n et (n )|xn |Un , where (c) Un (kn n ) B(En ) Mn , (Un (n ) - exp(n A)) exp(tAn )xn 2 kn n = t [0,T ]. Definition 3.7. A family of discrete semigroups {U (kn n )} is said to b e stable of order O(1/(n Un (kn n )
B (En ) -1

)) if

C/(n

-1

) for n, kn N, 0 < n ,n kn [0,T ].

(3.13)

Theorem 3.23. Let a discrete semigroup {U (kn n )} be compatible of order O((n )) on a linear manifold U E with respect to the semigroup exp(·A). The fol lowing assertions are equivalent: (i) Un (kn n )
B (En )

C/(n

-1

); C K (kn n (n ),x; E, U ),n,kn N; (n-1 ) x E, C (n-1 ) kn n (n )|x|U , kn n [0,T ],x U , M, x

(ii) (Un (kn n )pn - pn exp(kn n A))x (iii) (Un (kn n )pn - pn exp(kn n A))x where Mx is a constant depending only on x.

15


Theorem 3.24. Let | exp(tA)x|U C |x|U for any x U and t [0,T ]. Then the fol lowing conditions are equivalent: (i) The family of operators {U (kn n )} is compatible of order O((n )) on a linear manifold U E with respect to the semigroup exp(·A) and stable of order O(1/(n-1 )); C K (kn n (n ),x; E, U ),n,kn N; (ii) (Un (kn n )pn - pn exp(kn n A))x (n-1 ) x E, C Mx , (iii) (Un (kn n )pn - pn exp(tA))x -1 ) (n kn n (n )|x|U , t = kn n [0,T ], x U . On an extension of Lax­Richtmyer theory see [157, 181]. For a particular case where E = Lp (Rd ) and the op erator A P (D) = consider the Cauchy problem u(x, t) = P (D)u(x, t), u(x, 0) = u0 (x),x R+ , t with P (D) such that (3.14) is well-p osed in the sense u(·,t) ^ Let us denote P ( ) =
||r Lp (R) ||r

p D on E , one can

(3.14)
Lp (R)

c u0 (·)

,t R+ . ^ exp(tP )
M
p

p (i ) . It is well known that (3.14) is well-p osed iff



C, t R+ , where Mp is the space of Fourier multipliers. The semidiscrete approximation of (3.14) is given by un (x, t) = Ph (D)un (x, t), un (x, 0) = u0 (x),x R+ , n t where Ph (Dh ) = h-r
||r

(3.15) b ei


p (h)

I

^ b un (x + h, t) and Ph ( ) = h-r


||r

p (h)

,h

. The op eraQ(h ),r =

I

^ ^ tor Ph (Dh ) is said to b e compatible with the op erator P (D)of order µ if Ph ( )- P ( ) = hµ | |r ^ deg Ph ( ),Q is an infinitely differentiable function, and |Q()| Q0 > 0 for 0 < ||
0

+µ

.

Theorem 3.25 ([43]). Let P (D) and Ph (Dh ) be compatible of order µ, and (3.14) and let (3.15) be wel l posed. Then for every T > 0, there exists C > 0 such that (etP chµ u0
W
2,r +µ h

(Dh )

- exp(tP (D))u0

L2 (Rd )



(Rd )

, and for 0 < s < r + µ, (etP (etP
h h

(Dh )

- exp(tP (D))u0

L2 (Rd ) L (Rd )

ch ch

sµ µ+r sµ µ+r

u0 u0

s B2

, ,

(Dh )

- exp(tP (D)))u0 n

B2

d/2+s

where Bp = Bp,



is the Besov space.

It is remarked in [32] that for a quite general case, Bp,q = (Lp (R),D(A)),q .

16


k If we consider a full discretization scheme for (3.14) in the form Lh Un

+1

k = Bh Un ,k = 0, 1, 2,... ,

where Lh v =


a (h)v (x + h) and Bh v =
-1

b (h)v (x + h), then a discrete semigroup can b e cona (h)e
, h

structed as Un (kn )u0 = F n ^ with order µ if Un ( ) = en

^ ^ ^ ^ k un ^ Un ( )^0 , Un ( ) = Bn ( )/Ln ( ), Bn ( ) = + | |r ) as , h 0.

(the time step

n

is connected with h by n /hr = const). Such a finite-difference op erator Un (kn ) approximates (3.14)
^ P ( )

+ O(hr

+µ

+µ

Theorem 3.26 ([43]). Let (3.14) be wel l-posed and let Un (kn ) be stable in E = L2 (Rd ) and approximate (3.14) with order µ > 0. Then for any T > 0, there is a constant c > 0 such that (Un (t) - exp(tP (D)))u0 n and for 0 < s < r + µ (Un (t) - exp(tP (D)))u0 (Un (t) - exp(tP (D)))u0 (Un (t) - exp(tP (D)))u0
L2 (Rd ) L (Rd ) L (Rd ) L2 (Rd )

chµ u0 n

W

2,r +µ

(Rd )

,

ch

sµ µ+r

u0

s B2

, , , t = kn [0,T ].

chµ u0 ch
sµ µ+r

B2,

d/2+µ+r 1 d/2+s

u0

B2

Conversely, the order of convergence implies the smoothness of u0 ; see [32, 43]. n The time discretization of parab olic problems with memory by the backward Euler method was considered in [27]. The stability and error estimates take place in the Banach space framework, and the results are used for obtaining error estimates in the L2 and maximum norms for piecewise-linear finite-element discretizations in two space dimensions.

4. Backward Cauchy Problem In a Banach space E , let us consider the backward Cauchy problem: v (t) = Av (t), t [0,T ], v (T ) = v T , where the element v (0) is unknown. At least in two imp ortant cases it is not a well-p osed problem; namely, if A is unb ounded and generates an analytic C0 -semigroup or if the C0 -semigroup exp(·A) is compact. Indeed, in these situations, the problem exp(TA)x = v
T

(4.1)

is ill p osed [51, 100, 196] in the sense that the

op erator exp(-TA) is not b ounded on E and, moreover, D(exp(-TA)) = E in general. This means that in general the Cauchy problem (4.1) has a solution only for some (but not every) initial data v T and the solution v (0), if it is exists, does not dep end continuously on the initial data. After changes of variables, 17


setting v () = u(T - ), one can rewrite the problem (4.1) in the form u (t) = -Au(t), t [0,T ], u(0) = u0 , where u0 = v T is given and u(T ) is the element to b e found. In this section, we consider the approximation of (4.2) with op erator A, generating an analytic C0 -semigroup. Definition 4.1. A b ounded linear op erator R such that () 0 as 0 and
,T

(4.2)

on the space E is called a regularizator for the Cauchy u - exp(-TA)u0 0 as 0.

problem (4.2) if for any > 0 and any u0 E for which a solution of (4.2) exists, there exists = () > 0 su p
u -u
0

R


(),T

In [140], it is proved that for the existence of a linear regularizator of the problem (4.2) that commutes with op erator A, it is necessary and sufficient that -A generate C -semigroups S (t), 0 t T , such that C strongly converges to the identity op erator I as 0.

There are many regularizators, which can b e considered for problem (4.2). For example, in [165], it was shown that if -A2 generates a cosine op erator function, then the method of quasi-reversibility, which n is given by the Cauchy problems un, (t) = -An un, (t) - A2 un, (t), un, (0) = u0 , n n is a regularization method for (4.2), and un, (T ) - pn u(T ) C 1/ log(1/) - log log(1/) - o(log
-1

u0 - pn u0 / + , where = () = n

(1/)) . In this case S (t) exp(-tA)exp(-T A2 )is a C -semigroup

with C = exp(-T A2 ) and C I as 0. Moreover, the generator of this C -semigroup is -A. It has b een shown in [60] that the stochastic differential equation du (t) = -Au (t)dt - Au (t)dw(t), u(0) = u0 , where w(·) is the standard one-dimensional Wiener process, yields a stochastic regularization of (4.2). Explicitly, the op erator-function U (t)u0 = 1 2i e
-t- w (t)-w (0) - 1 2 2 |t| 2

(4.3)

( - A)-1 u0 d, t > 0,

which represents a solution of (4.3) for any u0 Ac (A), p ossesses the following prop erties:
0

lim U (T )u0 - exp(-TA)u0 = 0, exp c2 |t| + c3 |t|-µ )2 + b(, |t|) for any > 0.

(4.4) (4.5)

U (t)

c1 |t|

18


Here, the function b(, t) is b ounded in the parameters and t and Ac (A) is the set of entire vectors of the op erator A. By virtue of the inequality U (T )u - exp(-TA)u0 U (T ) u - u0 + U (T )u0 - exp(-TA)u0 , (4.6)

this means that there is a dep endence on = () such that U (T ) b ecomes a regularizator. The op erator function t exp (T - t)A U (T ), 0 t T , is a C -semigroup with C = exp(TA)U (T ). One can see that C I as 0, and that the generator of this C -semigroup is -A. 4.1. C -semigroups and ill-posed problems. Let C b e a b ounded linear op erator on the Banach space E, i.e., C B (E ), and let T > 0 b e some finite numb er. Definition 4.2 ([191]). A family of b ounded op erators {S (t) : 0 t < T } is called a local C -semigroup on E if (i) S (t + s)C = S (t)S (s) for t, s, t + s [0,T ); (ii) S (0) = C ; (iii) S (·) is strongly continuous on [0,T ). Clearly, S (·) is a commutative family. A local C -semigroup is said to b e nondegenerate if the condition S (t)x = 0 for all t (0,T ) implies x = 0. It is seen from Definition 4.2 that a local C -semigroup is nondegenerate [63] if and only if C is injective, i.e., N (C ) = {0}. Concerning construction with N (C ) = {0} see [112], [113]. It is very interesting question how to apply the case of noninjective C, i.e., degenerate C -semigroups, for ill-p osed problems. Unfortunately this approach still is not realized. Starting from now on, we will consider only the case where C B (E ) is an injective op erator. Definition 4.3. The generator of {S (t) : 0 t < T } is defined as the limit -Gx := C
-1 h0+

lim

1 (S (h)x - Cx),x D(G), h 1 (S (h)x - Cx) R(C )}. h
-1

with the natural domain D(G) := {x E : lim

h0+

Proposition 4.1 ([182]). The operator G is closed, R(C ) D(G) and C

GC = G.

We denote the C -semigroup S (·) with the generator -G by S (·G). Next, let (0,T ). We set


L ()x :=
0

e-t S (tG)xdt, x E, > 0.

(4.7)

This is the so-called local Laplace transform of S (·G). 19


Proposition 4.2. Let S (·G) be a local C -semigroup and let L (·) be the local Laplace transform of S (·G). Then, for any x E , one has L ()x D(G) and ( + G)L ()x = Cx - e- S (G)x for al l [0,T ) and > 0. (4.8)

In the case of local C -semigroups, the sp ectrum (-G) can b e located on the half-line [0, ). Therefore, in this case the Laplace transform of the local C -semigroup does not exist in general, and we follow the ideas of [30, 182, 191]. The function L () with prop erty (4.8) is called an asymptotic resolvent. Theorem 4.1 ([182]). Let A be a closed linear operator on E and let C B (E ) be injective. (i) If the operator A is the generator of a local C -semigroup {S (t); 0 t < T } on E , then there exists an asymptotic C -resolvent L () of -A such that dm m! L ()x M m+1 x , x E, m d with 0 m/ , > a, m N {0}, and the operator A satisfies C D(C
-1 -1

(4.9) AC = A.

(ii) If -A has an asymptotic resolvent which satisfies (4.9), and CD(A) is dense in D(A), AC ) D(A), i.e., Cx D(A) and AC x R(C ) imply x D(A), then the part A0 of A in E0 := D(A) generates a local C -semigroup on E0 with C equal to C0 := C |E0 . In particular, under the assumption that CD(A) = E , the operator -A generates a local C -semigroup on E if and only if C
-1

AC = A and there exists an asymptotic C -resolvent satisfying (4.9). In this case,

A has a dense domain. Remark 4.1. An asymptotic C -resolvent L () of op erator -A is compact for some C (and then for any large enough) if and only if S (·A) is compact and uniformly continuous in t. Indeed, if S (·A) is compact then by (4.7) and [219], it follows that L () is compact. Conversely, taking derivative of L () in and using the fact that S (·A) is uniformly continuous in t we have that S (·A) is compact as the uniform limit of compact op erators. This fact could b e used in the approximation of semilinear equations in case of the C -semigroups approach (see Sec. 6). Let us consider the abstract Cauchy problem, which is given by (4.2). Definition 4.4. A function u(·) is called a solution of (AC P ; T, y ) if u(·) is continuously differentiable in t [0,T ), u(t) D(A) for all 0 t < T , and u(·) satisfies (4.2). We denote by (AC P ; T, C D(A)) the problem (AC P ; T, y ) with y CD(A). Definition 4.5. The Cauchy problem (AC P ; T, C D(A)) is said to b e generalized wel l-posed if for every y CD(A), there is a unique solution u(·; y ) of (AC P ; T, y ) such that u(t; y ) M (t) C
-1

y

for

0 t < T and y CD(A), where the function M (t) is b ounded on every compact subinterval of [0,T ). 20


It should b e stressed here that the generalized well-p osedness in the sense of Definition 4.5 is more general than that in the case of the problem in (3.1). Moreover, we can state that this generalized well-p osedness is a solvability condition of (4.2) for which a regularizator exists. Theorem 4.2 ([182]). Let C be a bounded linear injection on E , and let A be a closed linear operator. Then the fol lowing assertions are equivalent: (I) The operator -A is the generator of a local C -semigroup; (I I) C
-1

AC = A, and the problem v (t) = -Av (t)+ Cx, t [0,T ),v(0) = 0, has a unique solution

for every x E. If either (A) = or A has a dense domain, (I) and (I I) are also equivalent to (III) C C
-1 -1

AC = A, and the problem (AC P ; T, C D(A)) is generalized wel l-posed. Moreover, u(t; y ) =

S (tA)y, t [0,T ), is a unique solution for every initial value y CD(A). Since local C -semigroups are regularizators of the ill-p osed problem (4.2) it is very imp ortant to

present the approximation theory of local C -semigroups. 4.2. Semidiscrete approximation theorem. Within the general discretization scheme, let us consider the semidiscrete approximation of the problem (4.2) in the Banach spaces En : un (t) = -An un (t), t [0,T ), un (0) = u0 , n where the op erators -An are generators of local Cn -semigroups which are compatible with the op erator -A and u0 u0 . We understand compatibility in the sense of the general approximation scheme as the n ~ ~ PP -convergence of Cn C and the PP -convergence of resolvents (In - An )-1 (I - A)-1 for some ~ ~ (A) (An ). Recall that in our general case (4.1), such a does exist, since conditions (A) and (B) from Theorem 3.1 are naturally assumed to b e satisfied. Theorem 4.3 ([214] (Theorem ABC-C)). Under the assumption CD(A2 ) = E , the fol lowing conditions (Ac ) together with (Bc ) are equivalent to condition (Cc ). (Ac ) Compatibility. Cn C and operators An and A are compatible; (Bc ) Stability. For any 0 < < T there is some constant M independent of n such that S (tAn ) M for 0 t and n N; (Cc ) Convergence. For any 0 < < T , we have
t[0, ]

(4.10)

max

S (tAn )x0 - pn S (tA)x0 = 0, as n , n 21


whenever x0 x0 . n Remark 4.2. In the case of exp onentially b ounded C -semigroups [64, 65] we can trivially change condition (Ac ) to the condition ~ ~ ~ (A') Cn C and ( - An )-1 Cn ( - A)-1 C for some C; see [224] for details. Since, the construction can b e done just as with condition (A'), in this case, we do ~ ~ ~ not need to assume that ( - An )-1 ( - A)-1 for some . Remark 4.3. We have set the condition CD(A2 ) = E for simplicity. For the general case one obtains the convergence on the set CD(A2 ). In the case of integrated semigroups, such situations have b een well investigated; see, e.g., [33, 35]. Actually, the pap er [33] is devoted to the following effect observed in the study of convergence of semigroups. Supp ose we are given a sequence of uniformly b ounded semigroups {exp(tAn ),t 0},n 1 ( exp(tAn ) M, t R+ ) acting on the Banach space E. Assume furthermore that the limit lim
n

(I - An )-1 x = S ()x exists for any x E. If R(S ()) = E (4.11)

(R(S ()) is common for all > 0), the semigroups in question strongly converge, by the Trotter­Kato theorem. One can also show that if condition (4.11) is relaxed, the limit
n

lim exp(tAn )x

(4.12)

exists for all x R(S ()) (see e.g. [115], [70, p. 34], or [33, 35]). As observed by T.G. Kurtz [115] for any x E , there exists the limit
t n 0

lim

exp(sAn )xds.

(4.13)

In general, however, one cannot exp ect that (4.12) holds for x R(S ()). This effect is of course related to Arendt's theorem, or rather to the generation theorem for "absolutely continuous integrated semigroups" presented in [37]. Let us consider the following semidiscretization of problem (4.3) in Banach spaces En : dun, (t) = -An un, (t)dt - An un, (t)dw(t), un, (0) = u0 , n where u0 u0 , the op erators An generate analytic semigroups, and {(, F , P) ,w(t)} is the standard n one-dimensional Wiener process (Brownian motion). As usual, the symb ol E[·] denotes the mathematical exp ectation. We emphasize that the situation where (An ), (A) C \ 22 3 4 is considered. (4.14)


Theorem 4.4 ([214]). Let the conditions (A) and (B1 ) of Theorem 3.2 be satisfied, and let n > 0 be a sequence which converges to 0 as n . Then there exists a sequence n such that un,n (t) u(t) for every t [0,T ] as n . Here un,n (·) is a solution of (4.14) and u(·) is a solution of (4.2) with u0 Ac (A). The convergence is understood in the fol lowing sense: sup
u0 -pn u n
0

n

un,n (t) - pn u(t) 0,

P-almost surely as n 0.

4.3. Approximation by discrete C -semigroups. Following Sec. 3, we denote by {Tn (·)} a family of discrete semigroups, on En , resp ectively, i.e., Tn (t) = Tn (n )kn , where kn = [t/n ]. We define the generator 1 of Tn (·) by the formula -An = (Tn (n ) - In ) and consider the process n 0, kn , n . We n assume that Cn B (En ) is an injective op erator such that Tn Cn = Cn Tn . The discrete Cn -semigroup Un (·) is defined as Un (t) = Tn (t)Cn . In this subsection we also assume that the dimension of each of the spaces En is finite, but dim (En ) as n . Theorem 4.5 ([214] (Theorem ABC-C-discr)). Under condition (A) of Theorem 3.1 and the assumption CD(A2 ) = E , the fol lowing conditions (Acd ) and (Bcd ) together are equivalent to condition (Ccd ). (Acd ) Compatibility. Cn C, the operators An ,A are compatible, and An B (En ), n N; (Bcd ) Stability. For any 0 < < T , there is some constant M , independent of n, such that Un (t) M for al l 0 t < T and n N is satisfied uniformly for any choice of {n } and {kn } as long as n 0, and kn = [t/n ]; (Ccd ) Convergence. For any 0 < < T , maxt[0, whenever x0 x0 . n Theorem 4.6 ([214]). Let the conditions (Ac ) and (Bc ) be satisfied. Assume that condition (A) of Theq - with q < 1. orem 3.1 and the assumption CD(A2 ) = E are satisfied and also that n A2 Cn 1 n M T Then Un (t) M (1 - q )-1 for 0 t < T and any n N uniformly for any choice of {n } and {kn } with n 0, as long as kn = 0 < < T , maxt[0,
] ]

Un (t)x0 - pn S (tA)x0 0 as n 0,n , n

Un (t)x0 - pn S (tA)x0 n

t . Moreover, for any n 0 as n 0, n , whenever x0 x0 . n

Remark 4.4. In fact, the scheme U n (t) (I + n An )-kn Cn with t = kn n can b e constructed even under condition (3.4). Indeed, n An = n An (In - An )-1 - n A2 (In - An )-1 , and by the choice of n we can make the second term less than , and then by choosing n appropriately for a fixed , we obtain n An 2 , so that the scheme U n (·) is well defined. 23


Remark 4.5. In contrast to the well-p osed case, for ill-p osed problems it looks that the implicit and explicit methods of discretization in time are not so different in the sense of stability advantages (compare with Theorems 3.5 and 3.6). Moreover, under condition (3.4), it follows from the identity
2 (In - n An )kn Cn = (In - n A2 )kn (In + n An )-kn Cn n

and inequality
2 (In ± n A2 )k n
n

Cetn

A2 n

,t = kn n ,

that the stability prop erties of the implicit and explicit methods are the same. There are a lot of stochastic finite-difference schemes which could b e written for problem (4.14). For example, some of the simplest are Un, (t + n ) - Un, (t) = -n An Un, (t) - w(t)An Un, (t), ¯ ¯ ¯ ¯ Un, (t + n ) - Un, (t) = -n An Un, (t + n ) - w(t)An Un, (t), ¯ where w(t) = w(t) - w(t - n ) , t = kn n , and Un, (0) = Un, (0) = In . Theorem 4.7 ([214]). Let the conditions (A) and (B1 ) of Theorem 3.2 be satisfied. Assume that the stability conditions (3.4) and n A2 ec An = O(1) are fulfil led for some constant c > 0. Then for n = n n the scheme (4.15) has the stable behavior in the fol lowing sense: n := sup sup E
nN

(4.15) (4.16)

t Un,n (t)u0 - exp -tAn + n (w(t) - w(0))An - 2 A2 u0 n n 2nn

: u0 1 n

0,

and converges in the fol lowing sense: E Un,n (t)u0 - pn u(t) n For the scheme U
n,n

C

n

A exp(-TA)u0 + un,n (t) - pn un (t) + Cn u0 , 0 < t T. n

(·), similar notions are employed.

We can also study the convergence of more sophisticated numerical methods. For example, in [45], in order to approximate (4.14), the following Runge­Kutta scheme was considered: Y1 = Un, (t)+ n An Un, (t),

Un, (t + n ) - Un, (t) = -n An Un, (t)+ w(t)An Un, (t) n (w(t))2 + - 1 (n An Y1 - An Un, (t)). 2 n 24

(4.17)


Thus, the solution can b e written in the form Un, (t + n ) = (In - n An - where Zn = 2 n 2 kn k An ) k 2
n

=1

- - (In + Zn 1 w(t)An +(Zn 1 /2)2 w(t)2 A2 )Un, (0), n

(4.18)

In - n An -

2 n 2 An . 2

Theorem 4.8 ([45]). Let conditions (A) and (B) of Theorem 3.2 be satisfied. Assume that the stability conditions (3.4) and n A2 ec An = O(1) are fulfil led for some constant c > 0. Then for n = n , n scheme (4.18) has the stable behavior in the fol lowing sense: n := sup E Un,n (t)u0 - e( n
t -tAn +n (w (t)-w (0))An - 2 2 A2 nn

) u0 n

: u0 1 0, n

and converges in the fol lowing sense: E C
n

Un,n (t)u0 - pn u(t) n 0 < t T.

A exp(-TA)u0 + un,n (t) - pn un (t) + Cn u0 , n

In the case of the well-p osed problem du (t) = Au (t)dt + Au (t)dw(t), u(0) = u0 , (4.19)

where the op erator A generates an analytic C0 -semigroup, the semidiscrete and full-discretization schemes do not need additional stability assumptions and the order of convergence will b e defined just by the compatibility prop erty of the scheme. More precisely, the term et under the integral leads to the absolute convergence of the integral indep endently of the b ehavior of on any compact set. For example, we have the following assertion. Theorem 4.9 ([45]). Let the conditions (A) and (B ) of Theorem 3.2 be satisfied. Assume that the stability condition (3.4) is fulfil led for some constant C > 0. Then for any n [0, ], the scheme like (4.18) has the stable behavior in the fol lowing sense: sup E Un,n (t)u0 - e(tAn n
t +n (w (t)-w (0))An - 2 2 A2 nn

) u0 n

: u0 1 n , n

and converges in the fol lowing sense: E Un,n (t)u0 - pn exp(tA)u0 n Cn A exp(tA)u0 + un,n (t) - pn un (t) + Cn u0 , n 0 < t T. 25


5. Coercive Inequalities In a Banach space E , let us consider the following inhomogeneous Cauchy problem: u (t) = Au(t)+ f (t), t [0,T ], u(0) = u0 , where the op erator A generates C0 -semigroup and f (·) is some function from [0,T ] into E . Problem (5.1) can b e considered in various functional spaces. The most p opular situations are the following settings: the well-p osedness in C ([0,T ]; E ),C second article in this volume). We say that problem (5.1) is well p osed, say in C ([0,T ]; E ), if, for any f (·) C ([0,T ]; E ) and any u0 D(A), (i) problem (5.1) is uniquely solvable, i.e., there exists u(·) which satisfies the equation and b oundary condition (5.1), u(·) is continuously differentiable on [0,T ], u(t) D(A) for any t [0,T ] and Au(·) is continuous on [0,T ]; (ii) the op erator (f (·),u0 ) u(·) as an op erator from C ([0,T ]; E ) â D(A) to C ([0,T ]; E ) is continuous. In the case u0 0, the coercive well-p osedness in C ([0,T ]; E ) means that c f (·)
C ([0,T ];E ) ,0

(5.1)

([0,T ]; E ), and Lp ([0,T ]; E ) spaces (see [9, 24, 139], and also the

Au(·)

C ([0,T ];E )



. In general, the coercive well-p osedness in the space ([0,T ]; E ) for problem (5.1) means

that it is well-p osed in the space ([0,T ]; E ) and u (·)
([0,T ];E )

+ Au(·)

([0,T ];E )

C ( f (·)

([0,T ];E )

+ u0

F

),

where F is some subspace of E. For results of the coercive well-p osedness see [9, 24, 139]. The semidiscrete approximation of (5.1) are the following Cauchy problems in Banach spaces En : un (t) = An un (t)+ fn (t), t [0,T ], un (0) = u0 , n with op erators An which generate C0 -semigroups, An and A are compatible, u0 u0 and fn f in n appropriate sense. Following Sec. 3, it is natural to assume that conditions (A) and (B1 ) are satisfied. Here we are going to describ e the discretization of (5.2) in time. The simplest difference scheme (Rothe scheme) is Un - U n
k k -1 n 0 n

(5.2)

= An U n + k , k 1, ..., n u0 n ,

k

T n

, (5.3)

U= 26


where, for example, in the case of fn (·) C ([0,T ]; En ), one can set k = fn (kn ),k {1, ..., K },K = n T , and in the case fn (·) L1 ([0,T ]; En ), one can set n k n 1 = n
tk

fn (s)ds, tk = kn , k {1, ..., K }.
tk
-1

5.1. Coercive inequality in Cn ([0,T ]; En ) spaces. Denote by Cn ([0,T ]; En ) the space of elements n = {k }K such that k En ,k {0, ..., K }, with the norm n k =0 n semigroup. Theorem 5.1 ([24]). Let condition (B1 ) be satisfied. Cn ([0,T ]; En ), i.e., U
n Cn ([0,T ];En ) n Cn ([0,T ];En )

= max0k

K

k n

E

n

.

We recall that coercive well-p osedness in C ([0,T ]; E ) implies [24] that A generates an analytic C0 -

Then problem (5.3) is stable in the space

C



n Cn ([0,T ];En )

+ u0 n

.

Theorem 5.2 ([24]). Let condition (B1 ) be satisfied. Then problem (5.3) is almost coercive stable in the space Cn ([0,T ]; En ), i.e., An U
n Cn ([0,T ];En )

M

An u0 n

E

n

+min log(1/n ), 1+ log An



n Cn ([0,T ];En )

.

It should b e noted that if (5.1) is coercive well p osed in the space C ([0,T ]; E ), then [69] the op erator A should b e b ounded or the space E should contain a subspace isomorphic to c0 . This means that problem (5.3) is not coercive well p osed in Cn ([0,T ]; En ) space in general. For the explicit scheme
k k Un - Un -1 k = An Un -1

+ k , k {1, ..., K }, n

(5.4)

0 Un

=

u0 n

,

Theorem 5.2 can b e reconstructed, but under a stability condition. Theorem 5.3 ([24]). Let condition (B1 ) is satisfied, and let n log 1 An for sufficiently smal l n > 0. Then problem (5.4) is almost coercive stable in the space Cn ([0,T ]; En ), i.e., An Un M where un =
0 Cn ([0,T ];En )

+ Un

Cn ([0,T ];E

n,1-

1 log 1 n

An u0 n


E

n,1-

1 log 1 n

+min log(1/n ), 1+ log An
1 1- n



nC

n

([0,T ];En )

,

E

n,

An exp(tAn )un

E

dt

1-

.

27


Remark 5.1. The space En, with equivalent norm coincides with the real interp olation space (En ,D(An ))1-1/p,p ; see [139].
, , 5.2. Coercive inequality in Cn0 ([0,T ]; En ) spaces. Denote by Cn0([0,T ]; En ), 0 < < 1, the space

of the elements n with the norm
, n Cn0 ([0,T ];En )

= max

0k K

k n

E

n

+

1k
max

k+l - k n n

E

n

(n k) (ln )- .

Theorem 5.4 ([183]). Let condition (B1 ) hold.
, Cn0 ([0,T ]; En ) with 0 < < 1, i.e.,

Then the scheme (5.3) is coercive wel l-posed in

An U

, n C 0 ([0,T ];En ) n



M (1 - )

An u0 n

E

n

+

, n Cn0 ([0,T ];En )

.

Roughly sp eaking, assumption (B1 ) is necessary and sufficient for the coercive well-p osedness in
, Cn0 ([0,T ]; En ) space.

5.3. Coercive inequality in Lpn ([0,T ]; En ) spaces. Denote by Lpn ([0,T ]; En ), 1 p < , the space of elements n with the norm
K



n

Lpn

([0,T ];En )

=
j =0

k n

p E

1/p
n



n

.

Theorem 5.5 ([183]). Let condition (B1 ) hold. Let the difference scheme (5.3) be coercive wel l posed in
0 Lpn ([0,T ]; En ) for some 1 < p0 < . Then it is coercive wel l posed in Lpn ([0,T ]; En ) for any 1 < p <

and An U
n Lpn ([0,T ];En )

+ max

0k K

U

k nE

n,1-1/p



Mp2 p-1
,0



n Lpn ([0,T ];En )

+U

0 n 1-1/p

.

It should b e noted that in contrast to the case of C in Lp , we need some additional assumptions.

-space, the analyticity of the semigroup exp(·A)

is not enough for the coercive well-p osedness in Lp space [127], therefore, to state coercive well-p osedness

Theorem 5.6 ([183]). Let 1 < p, q < , 0 < < 1, and let condition (B1 ) hold. Then the difference scheme (5.3) is coercive wel l posed in Lpn ([0,T ]; En,,q ), i.e., An U
n Lpn ([0,T ];E
n,,q

)

+ max

0k K

k Un

E

n,1-1/p



Mp2 (p - 1)(1 - )



n Lpn ([0,T ];E

n,,q

)

0 + Un

1-1/p

,

where En,,q is the interpolation space (En ,D(An )),q with the norm


un 28

E

n,,q

=
0

An (In - An )-1

q E

n

d

1/q

.


For the general Banach space E , we have the following results. Assume that A is the generator of the analytic semigroup exp(tA),t R+ , of linear b ounded op erators with an exp onentially decreasing norm as t . This means that stability condition (B1 ) holds with 0. Theorem 5.7 ([23]). Let condition (B1 ) hold. Then the solution of difference scheme (5.3) is almost coercive stable, i.e., An U
n Lpn ([0,T ];En )

M

An U

0 nE

n

+min{log

1 , 1+ | log An n

B (En )

|} n

Lpn ([0,T ];En )

holds for any p 1, where M does not depend on n ,u0 , and n . n Of course, for schemes like
k k Un - Un n -1

= An = u0 n .

k k Un + Un 2

-1

+ k , n {1, ..., K }, n

(5.5)

0 Un

the coercive well-p osedness can b e considered. Theorem 5.8 ([23]). Let condition (B1 ) hold. Then the solution of difference scheme (5.5) is almost coercive stable, i.e., the estimate An
j j Un + Un 2 -1 Lpn ([0,T ];En )

M

An u0 n

E

n

+min log

1 , 1+ | log An n

En E

n

|

n

Lpn ([0,T ];En )

holds for any p 1, where M does not depend on n ,u0 , and n . n Theorem 5.9 ([23]). Let condition (B1 ) hold and condition (3.6) be satisfied. Then the solution of difference scheme (5.5) is almost coercive stable, i.e., the estimate An Un
Lpn ([0,T ];En )

M

An u0 n

E

n

+min log

1 , 1+ | log An n

En E

n

|

n

Lpn ([0,T ];En )

holds for any p 1, where M does not depend on n ,u0 , and n . n The necessary and sufficient conditions for the coercive well-p osedness of problem (5.1) in Lp ([0,T ]; E ) were obtained in [101, 220, 221]. More precisely, a Banach space E has the UMD prop erty iff the Hilb ert transform Hf (t) = extends to a b ounded op erator on and quotient spaces of Lq Lp 1 p.v.
-

1 f (s)ds t-s

(R; E ) for some (all) p (1, ). It is well known, that all subspaces

(,µ) with 1 < q < have this prop erty.

The Poisson semigroup on L1 (R) is not coercive well p osed on the Lp (R,E ) space if E is not an UMD space (see [127]). Hence the assumptions on E to b e an UMD space is necessary in some sense. 29


But it was an op en problem whether every generator of an analytic semigroup on Lq (,µ), 1 < q < , provided the coercive well-p osedness in Lp (R; E ). Recently, Kalton and Lancien [103] gave a strong negative answer to this question. If every b ounded analytic semigroup on a Banach space E is such that problem (5.1) is coercive well p osed, then E is isomorphic to a Hilb ert space. If A generates a b ounded analytic semigroup {exp(zA) : | arg(z )| }, on a Banach space E , then the following three sets are b ounded in the op erator norm: (i) {( - A)-1 : iR, = 0}; (ii) {exp(tA),tA exp(tA) : t > 0}; (iii) {exp(zA) : | arg z | }. In Hilb ert spaces, this already implies the coercive well-p osedness in Lp (R+ ; E ), but only in Hilb ert spaces E . The additional assumption that we need in more general Banach spaces E is the R-b oundedness. A set T B (E ) is said to b e R-b ounded if there is a constant C < such that for all Z1 ,... ,Zk T and x1 ,... ,xk E, k N,
1 0 k

rj (u)Zj (xj ) du C
j =0 0

1

k

rj (u)xj du,
j =0

(5.6)

where {rj } is a sequence of indep endent symmetric {-1, 1}-valued random variables, e.g., the Rademacher functions rj (t) = sign(sin(2j t)) on [0, 1]. The smallest C such that (5.6) is fulfilled, is called the Rb oundedness constant of T and is denoted by R(T ). Theorem 5.10 ([221]). Let A generate a bounded analytic semigroup exp(tA) on a UMD-space E . Then problem (5.1) is coercive wel l posed in the space Lp (R+ ; E ) if and only if one of the sets (i), (ii) or (iii) above is R-bounded. The interpretation of the discrete coercive inequality and a discrete semigroup defines the convolution op erator of the form An
k j =0 k Tn -j

Qn n n with some b ounded op erator Qn B (En ), which usually has a

smoothness prop erty as it is clear from the proofs of Theorems 5.7 and 5.8. Here, Tn (n )k is a discrete semigroup, say, as in Sec. 3.1. The b oundedness of the convolution op erator in Lpn (Z+ ; En ) space implies the discrete coercive well-p osedness in Lpn (Z+ ; En ). Also, in this section, we assume that Banach spaces En satisfy the collective UMD-prop erty, i.e., we assume that the Hilb ert transforms Hn fn (t) = 30 1 p.v.
-

1 fn (s)ds t-s


extend to a b ounded op erators on Lp (R; En ) for some (all) p (1, ) such that all of them are b ounded by a constant which does not dep end on n. This assumption holds for example if all En can b e emb edded in a fixed space Lp () with 1 < p < . Definition 5.1. A discrete semigroup Tn (·) with a generator An generates the coercive well-p osedness on Lpn (Z+ ; En ) space if the corresp onding convolution op erator n An on the Lpn (Z+ ; En ) space. Theorem 5.11 ([23]). Assume that for convolution operator
k k j =0 k Tn -j

Qn j n

n

is continuous

n An
j =0

k Tn

-j

Qn j n

n

,

the fol lowing conditions hold: 10 the set {An ( - Tn )-1 Qn n : || = 1, = 1, = -1} is R-bounded; 20 the set {( - 1)( +1)An ( - Tn )-2 Qn n : || = 1, = 1, = -1} is R-bounded. Then the discrete semigroup Tn (·) generates the coercive wel l-posedness on the Lpn (Z+ ; En ) space. Theorem 5.12 ([23]). Let En be UMD Banach spaces. Also, assume that the set {( - An )-1 : iR, = 0} is R-bounded with the R-boundedness constant independent of n. Then the solution of difference scheme (5.3) is coercive stable, i.e., An U
n Lpn (Z+ ;En )

M n

Lpn (Z+ ;En )

(5.7)

holds for any p 1, where M is independent of n ,u0 , and n . n Remark 5.2. It should b e noted that Theorem 3.2 can b e reformulated in terms of R-b oundedness with the change of condition (B1 ) by the following condition: there is a 0 < < /2 such that the set {( - An )-1 : ( + /2)} is R-b ounded with the R-b oundedness constant indep endent of n. Condition (C1 ) can b e written, due to [221, Theorem 4.2], in the following form: exp(tAn ) exp(tA) converges for any t R and there is 0 < < /2 such that the set {exp(zAn ) : z ( )} is R-b ounded with the R-b oundedness constant indep endent of n. Therefore, one of our assumption in Theorems 5.12 and 5.13 is in some sense condition (B1 ) changed by the R-b oundedness condition. Theorem 5.13 ([23]). Let En be UMD Banach spaces. Also, assume that the set {( - A)-1 : iR, = 0} is R-bounded with the R-boundedness constant independent of n. Then the solution of difference 31


scheme (5.5) is coercive stable, i.e.,
k k U + Un An n 2 -1 Lpn ([0,T ];En )

M n

Lpn ([0,T ];En )

(5.8)

holds for any p 1, where M does not depend on n ,u0 , and n . n Remark 5.3. Analyzing the proofs of Theorems 5.12 and 5.13, it is easy to see that one can set An = An in statements (5.7) and (5.8). Moreover, statement (5.8) can b e written in the form An Un
Lpn ([0,T ];En )

M n

Lpn ([0,T ];En )

.

n The proof of this fact is based on the relation An = An (In - An )-1 . 2 It is p ossible to consider a more general Pad´ difference scheme [24] for p = q - 1 or p = q - 2. In e this case, the difference scheme is written in the form
k k Un - Un n -1

= (An Un )k

-1

0 + p,q,k , Un = u0 , 1 k K. n n

(5.9)

k -1 Rp,q (n An ) - I p Un and k - p,q,k En Mn +q . To formulate the coercive n n n statements of Secs. 5.1­5.3, we just need to change the op erator An by An . We know from Theorem 3.16

where (An Un )k =

that under condition (B1 ) with p = q , the Pad´ approximation is stable, but, in general, it is not coercive e stable. To obtain the coercive inequality, we need condition (3.6). Spaces where the problem considered can also b e very different [24].
¯ 5.4. Coercive inequality in Bn ([0,T ]; Ch (h )) Ch ([0,T ]; Ch (h )). From the p oint of view of the

numerical analysis, it is very interesting to consider problem (5.1) in the space ([0,T ]; E ) such that E is smoother than C () (elements of such a space can easily b e well approximated) and ([0,T ]; E ) is like C ([0,T ]; E ) or the space of b ounded functions. An interesting fact is that such a situation is actually p ossible at least for a strongly elliptic op erator of order 2 with coefficients of class C (). Since op erator (pn v )i = v (ih) is very concrete in such space E , i.e., it takes values in the grid p oints, we omit pn in the notation of this section. Theorem 5.14 ([39]). Let be an open bounded subset of Rd lying to one side of its topological boundary , which is a submanifold of Rd of dimension d - 1 and class C A = A(x, Dx ) =
||2 2+

, for some (0, 2) \{1}. Let

a (x)Dx

be a strongly el liptic operator of order two (thus, Re
||=2 d

a (x) | |2 for some > 0 and for any , 2 such that for

(x, ) â R ) with coefficients of class C (). Then there exist µ 0 and 0 32


any C with || µ and | Arg | 0 , the problem v -Av = y, 0 v = 0, has a unique solution v belonging to C ||1+
2

2+

(), for any y C () and for a certain M > 0, +v
C
2+

v

C ()

+ || v

C ()

()

M

y

C ()

+ ||

2

0 y

C ( )

,

(5.10)

where 0 is the trace operator on . It is clear from (5.10) that the op erator A does not generate a C0 -semigroup in E = C () space in general, but, following, say, [139], one can construct a semigroup exp(tA),t 0, which is analytic. Let I = Z, and let E b e a Banach space with norm or (U )j instead of U (j ) for any j I , we set B (I ; E ) := {U : I E : sup Uj < +}, U
j I B (I ;E )

· . For a grid function U : I E , writing Uj

:= sup Uj .
j I

It is easily seen that B (I ; E ) is a Banach space with the norm

·

B (I ;E )

. If the set I is some interval,

say, I = (a, ), we denote by B (I ; E ) the set of all b ounded functions from I into E. For the grid function U : I E and h > 0, we define the op erator h by formula (h U )j := h-1 (Uj
m For any m Z we set (h U )j := h-m m i=0 +1

- Uj ).

m (-1)m-i Uj +i . If U B (I ; E ), we set i
r h U B ( I ;E )

U Finally, let (0, 1). We define [U ]C and if m N0 , U
h

m Ch (I ;E )

:= max

: 0rm .

(I ;E )

:= sup

(k - j )h

-

Uk - Uj

: j, k I ,j < k ,

m Ch + (I ;E )

:= max

U

m Ch (I ;E )

m , [h U ]C

h

(I ,E )

.

0 0 In the same context, we denote by B (I ; E ) the space Ch (I ; E ). If E = C, we write simply B (I ) or Ch (I ).

~ ~ Let f B (N; E ). We denote by f the extension of f to N0 such that f0 = 0. For a nonnegative real numb er , we define f
Ch,0 (N;E )

~ := f

Ch (N0 ;E )

.

(5.11)

L . For j I := {1, ..., n - 1}, we are given complex n numb ers aj ,bj ,bj , and cj satisfying the following conditions (): Now let L > 0,n N,n 3, and let h = 33


(1) there exists > 0 such that Re(aj ) for every j I ; (2) max |aj |, |bj |, |bj |, |cj | Q for every j I with Q > ; (3) there exists : [0,L] [0, +) such that (0) = 0 and is continuous at 0 such that for j, k I with j k, |ak - aj | (k - j )h . For C, we study the following problem:
2 Uj - aj (h U )j -1

- bj (h U )j - bj (h U )j

-1

- cj Uj = f

j

for j = 1, ..., n - 1, (5.12)

U0 = Un = 0. For this purp ose, we set I := {0, 1, ..., n - 1,n} and for U B (I ; E ), define U j if j I , ~j = U 0 if j {0,n}. We introduce the op erator Ah in B (I ; E ) defined by
2~ (Ah U )j := aj (h U )j -1

~ ~ + bj (h U )j + bj (h U )j

-1

~ + cj Uj for j I .

(5.13)

Further, we assume that ( 1) there exists > 0 such that Re(aj ) for every j I ; ( 2) max a
Ch (I )

,b

Ch (I )

,b

Ch (I )

,b

Ch (I )

Q with Q > .

Proposition 5.1 ([95]). Assume that assumptions ( ) are satisfied for some (0, 2) \{1}. Fix 0 . Then there exists µ0 > 0 such that { C : || µ0 , | Arg()| 0 } (Ah ), 0, - arccos Q where Ah is the operator defined in (5.13). Moreover, for every r [0, 2] there exists c > 0 depending only on L, , Q, and r such that for every f B (I ; E ) and any F B (I ; E ) with F |I = f , one has ( - Ah )-1 f
+ Ch,0r (I ;E )

c||

r 2

-1

F

Ch (I ;E )

+ || 2 max{ F0 , Fn } .



Let us consider the following mixed Cauchy­Dirichlet parab olic problem: u (t, x) = Au(t, x)+ f (t, x), t [0,T ],x [0,L], t u(t, x ) = 0, u(0,x) = 0, t [0,T ],x {0,L}, x [0,L], (5.14)

where A is a second-order differential op erator and L > 0. We say that problem (5.14) has a strict solution if there exists a continuous function u(t, x) having the first derivative with resp ect to t and derivatives of 34


order less than or equal to 2 with resp ect to x which are continuous up to b oundary of [0,T ] â [0,L], i.e. uC
1

[0,T ]; C () C [0,T ]; C 2 () and the equations in (5.14) are satisfied.

Theorem 5.15 ([91]). Consider problem (5.14) under the fol lowing assumptions: (I) T and L are positive real numbers; 1 ; (I I) (0, 1) \ 2 (III) 2u u Au(x) = a(x) 2 (t, x)+ b(x) (t, x)+ c(x)u(t, x), x x with a, b, c C 2 ([0,L]); (IV) a is real-valued and min a = > 0; (V) f C ([0,T ] â [0,L]), t f (t, ·) B [0,T ]; C 2 ([0,L]) ; t f (t, 0) and t f (t, L) belong to C ([0,T ]); f (0, 0) = f (0,L) = 0. Then there exists a unique strict solution u(·) of problem (5.14). u B [0,T ]; C 2+2 ([0,L]) and B [0,T ]; C 2 ([0,L] . t Such a solution belongs to

Now let I b e a set which can dep end on a p ositive parameter h and the Banach space Xh = B (I ). Next, we introduce a linear op erator Ah in Xh dep ending on h. In each case, (Ah ) contains { C \{0} : || R and | Arg()| 0 }, where R > 0 and 0 ( , ), and there exists M > 0 such that 2 ( - Ah )-1
L(Xh )

M ||-1

for in the sp ecified subset of C. Here, R, 0 , and M are indep endent of h. Then we consider another ~ ~ ~ ~ ~ set I such that I I ; we set Xh := B (I ). We define an extension op erator Eh from Xh to Xh : in all our concrete cases, this is the extension with zero. Next, for (0, 1), we introduce the norms ·
2+2,h

·

2,h

and

~ in Xh . The first of these norms is connected with

·

X

and the op erator Ah by the following

prop erty: there exist two p ositive constants c1 and c2 indep endent of h such that for every U Xh , c1 Eh U
2,h

U

(Xh ,D(Ah ))

c2 Eh U

2,h

.

~ Then, for every, h we consider the restriction op erator Rh L(Xh ,Xh ) such that Rh Eh = IXh . Let us ~ also introduce a seminorm ph in Xh : in concrete cases, we have ph (U ) = U |I\I ~ ~ if || R and | Arg()| 0 , for every G Xh , then || Eh ( - Ah )-1 Rh G
2,h ~ B (I\I )

. We assume that

+ Eh ( - Ah )-1 Rh G

2+2,h

M

G

2,h

+ || ph (G) . 35


~ Another inequality we imp ose is the following. If || R, | Arg()| 0 and G Xh , then Ah ( - Ah )-1 Rh G
Xh

M ||-

G

2,h

+ || ph (G) .

Such an inequality can b e easily deduced in each of our examples. In the formulation b elow, we remove the parameter h. In the case of the backward Euler scheme (5.3), we have ~ Theorem 5.16 ([95]). Let X and X be Banach spaces with norms · · ·
X

and

·

~ X

, respectively, and let

~ ~ A B (X ), E B (X, X ), and R B (X, X ) be such that RE = IX . Assume, moreover, that (0, 1) ~ are norms in X , while p is a seminorm in the same space. Final ly, assume that , , M > 0 such that the fol lowing conditions are satisfied: there exist R > 0, 0 2 (a) { C : || R, | Arg()| 0 } (A) and, for in this set, and
2

and

2+2

( - A)-1 (b) for every F X , EF
2

B (X )

M (1 + ||)-1 ;

M F

(X,D(A))

;

~ (c) for every V X, C with || R and | Arg()| 0 , (1 + ||)-1 E ( - A)-1 RV +(1 + ||) A( - A)-1 RV (d) p(V ) V (e) RV
X 2 X 2

+ E ( - A)-1 RV V
2

2+2

M

+(1+ ||) p(V ) ;

~ for every V X and p(E F ) = 0 for every F X ;
2

~ for every V X. T Let T > 0, K N,K 2, and = . Assume that R < 1. K ~ Let G B ({0, 1, ..., K }; X ) be such that G0 = 0; consider problem (5.3) with k = RGk for k = V 1, ..., K and U 0 = 0. Then, for U B ({0, 1, ..., K }; X ) which solves (5.3), one has EU
k 2+2

c

0k K

max

Gk

2

+

0k1
max

((k2 - k1 ) )- p(Gk2 - Gk1 ) ,

(5.15)

for k = 0, 1, ..., K, where c is a positive constant depending only on , R, 0 , M , and T and is independent of n and G. We now consider the Crank­Nicolson scheme: we replace (5.3) by (5.5). Theorem 5.16 has the following analog. Theorem 5.17 ([94]). Assume that the assumptions of Theorem 5.16 are satisfied and, moreover, (f ) A 36
B (X )

S with some S > 0;


(g) if || 2S , then E ( - A)-1 RV (h) E RV
2 2

M

V

2

+

-

~ p(V ) for every V X ;

M

V

2

+

-

~ p(V ) for every V X ;

(i) 2R < 1. ~ Let G B ({0, 1, ..., K }; X ) be such that G0 = 0; consider problem (5.5) with k = RGk for k = 1, ..., K and U 0 = 0. If U B {0, 1, ..., K }; X EU
k 2+2

solves (5.5) for k = 0, 1, ..., K , then +
0k1
c

0k K

max

Gk

2

max

((k2 - k1 ) )- p(Gk2 - Gk1 ) ,

(5.16)

where c is a positive constant depending only on , R, 0 ,M ,S , and T and is independent of n and G. An application of Theorems 5.16 and 5.17 to the discretization of problem (5.14) is the following. L T and h := . We assume that K 2 and n 3. For j = 0, 1, ..., n, we set Let K, n N. We set := K n 1 aj := a(jh), bj = b(jh), cj := c(jh), Nn-1 := {1, ..., n - 1}, Nn := {0, 1, ..., n - 1,n}, and 2 X := B (Nn If V X as b efore, for i Nn
-1 -1

~ ), X := B (Nn ).

(5.17)

, we set ~ ~ ~ ~ ~ Vi+1 - 2Vi + Vi+1 Vi+1 - Vi-1 ~ + bi + ci Vi , h2 2h V i 0 if 1 i n - 1, if i {0,n}. (5.18)

(Ah V )i := ai where

~ Vi = (E V )i = Next, we define

~ R B (X, X ), RV := V |Nn-1 ~ ~ for every V X . Then, again for V X and V V
2

(5.19)

0,

1 , we set 2 - i1 )h)-2 |Vi2 - Vi1 |},
0in-2

:= max{ V

ma ~ X , 0i
(5.20)

2+2

:= max{ V max

2 |(h V )i |, max |(h V )i |,

0i1
2 2 ((i2 - i1 )h)-2 |(h V )i2 - (h V )i1 |},

(5.21)

with (h V )i := Vi+1 - Vi Vi+2 - 2Vi+1 + Vi 2 for 0 i n - 1, (h V )i := for 0 i n - 2, h h p(V ) := max{|V0 |, |Vn |}. (5.22) 37


One has the following result. Theorem 5.18 ([94]). With the notation (5.17), (5.18), (5.19) and (5.20), the assumptions (a)­(e) of Theorem 5.16 are satisfied, with R, 0 ,M independent of h. If we impose the further condition n h2 , the same also holds for assumptions (f )­(h) of Theorem 5.17 (even with S independent of n). As a consequence, we have the following theorem. Theorem 5.19 ([95]). Consider the problem (5.14) under the assumptions of Theorem 5.15. With the notation (5.17), (5.18), (5.19), and (5.20), set Gk := f (kn ,j h) for k {1, ..., K }, j = 0, ..., n. Set j k := RGk and denote by G0 the zero in B (Nn ). Then if n is sufficiently smal l, the problem ~k ~k Uj - Uj ~0 Uj
n -1

(5.23)

= ai

~ ~ ~ ~ ~ Uik - 2Uik + Uik 1 U k - Uik 1 +1 - - ~ + bi i+1 + ci Uik + k , j h2 2h

(5.24)

= 0,

for j 1,... ,n - 1, k {1, ..., K } has a unique solution such that ~ U
k
2+ Ch 2 (Nn )

c

f

B ([0,T ];C

2

([0,L]))

+max{ f (·, 0)

C ([0,T ])

, f (·,L)

C ([0,T ])

}

(5.25)

with c independent of h and n . An analogous result holds for the Crank­Nicholson scheme (5.5). Then we set Gk := f j under the further condition (5.23). Remark 5.4. It follows from Theorem 5.18 that, for scheme (5.5) with u0 = 0 and under condition (5.23) n Ah U
2 Ch (Nn-1 )

k-

1 n ,j h 2

c

f

B ([0,T ];C

2

([0,L]))

+max{ fn (·, 0)

C ([0,T ])

, f (·,L)

C ([0,T ])

},

(5.26)

with c indep endent of h and n . In the quoted pap ers Theorems 5.16 and 5.17 are also applied to the discretization of the heat equation in a square. A counterexample in [94] and [93] shows that condition (5.23) cannot b e removed in general. Finally, estimates of the order of convergence are given in [93]. The coercive inequalities and their discrete analogs in the spaces C and Lp for elliptic problems of the form, e.g., u (t) = Au(t)+ f (t), have b een considered in [20, 21]. 38 u(0) = 0, u(T ) = uT ,


6. Approximations of Semilinear Equations In a Banach space E , let us consider the semilinear Cauchy problem u (t) = Au(t)+ f (t, u(t)), u(0) = u0 , (6.1)

with the op erator A, generating an analytic C0 -semigroup of typ e (A) < 0, where the function f is smooth enough. The existence and uniqueness of solution of problem (6.1) have b een discussed, e.g., in [9, 31, 96, 98, 139]. 6.1. Approximations of Cauchy problem. By a semidiscrete approximation of problem (6.1), we mean the following Cauchy problems in the Banach spaces En : un (t) = An un (t)+ fn (t, un (t)), un (0) = u0 , n (6.2)
n

where the op erators An generate analytic semigroups in En , An and A are compatible, the functions f approximate f and u0 u0 . n

¯ Let b e an op en set in a Banach space F , and let B : F b e a compact op erator having no fixed p oints on the b oundary of . Then for the vector field F (x) = x -B x, the rotation (I -B ; ) is defined; it is an integer-valued characteristic of this field. Let x b e a unique isolated fixed p oint of the op erator B in the ball S
r
0

of radius r0 centered at x . Then (I -B ; Sr ) = (I -B ; Sr0 ) for 0 < r r0 ,

and this common value of the rotations is called the index of the fixed p oint x and is denoted by ind x . Theorem 6.1 ([166]). Assume that conditions (A) and (B1 ) hold and compact resolvents (I - A)-1 , (In - An )-1 converge: (In - An )-1 (I - A)-1 compactly for some (A) and u0 u0 . n Assume that (i) the functions fn and f are bounded and sufficiently smooth, so that there exists a unique mild solution u (·) of the problem (6.1) on [0,T ] (in this situation ind u (·) = 1); (ii) fn (t, xn ) f (t, x) uniformly with respect to t [0,T ] as xn x; (iii) the space E is separable. Then for almost al l n, problems (6.2) have mild solutions u (t), t [0,T ], in a neighborhood of n pn u (·). Each sequence {u (t)} is P -compact and u (t) u (t) uniformly with respect to t [0,T ]. n n Let us consider the time discretization with resp ect to the explicit difference scheme: Un (t + n ) - Un (t) = An Un (t)+ fn (t, Un (t)), Un (0) = u0 , t = kn ,k = {0, ··· ,K }. n n (6.3)

39


Theorem 6.2 ([166]). Assume that the conditions of Theorem 6.1 and condition (3.6) are satisfied. Then the functions Un (t) from (6.3) give an approximate mild solution u (·) of problem (6.1) and, moreover, Un (t) u (t) uniformly with respect to t [0,T ]. Let us define the op erator (un )(t) = un (t) -
t 0

exp((t - s)An )f (s, un (s))ds.

Remark 6.1. If we assume that the conditions of Theorem 6.1 hold and the functions f (·) and fn (·) have Fr´chet derivatives in some balls containing the solutions u and u and, moreover, ase n sume that fnu (t, pn u (t)) are uniformly continuous with resp ect to the first and second arguments and f
n
un

(t, un (t)) fu (t, u (t)) uniformly with resp ect to t [0,T ] for un u , then [166] for almost all n

the problems (6.2) have mild solutions u (t), t [0,T ], in the neighb orhood of pn u (·). Each sequence n {u (·)} is P -compact and u (t) u (t) uniformly with resp ect to t [0,T ] and, moreover, for sufficiently n n large n n0 and some T T we have c1 n (u ,u0 ) u - pn u n n
F
n

c2 n (u ,u0 ), n

where the constants c1 and c2 are indep endent of n, Fn = C ([0,T ]; En ), and
n

(u ,u0 ) = max n

t[0,T ]

(pn u )(t) - exp(tAn )u0 n
k -1 l=1

E

n

.

Let Un (t) = (In + n An )k and

n

(un )(t) = un (t) -

Un ((k - l)n )fn (ln ,un (ln ))n .

Remark 6.2. If we assume that the conditions of Theorem 6.2 hold and the functions f (·) and fn (·) have Fr´chet derivatives in some balls containing the solutions u (·) and u (·) and, moreover, ase n sume that f f
n
un

n

un

(t, pn u (t)) are uniformly continuous with resp ect to the first and second arguments and

(t, un (t)) fu (t, u (t)) uniformly with resp ect to t [0,T ] as un u and condition (3.6) holds,

then [166] the functions Un (t) from (6.3) give an approximate mild solution of the problem (6.1) and
Un (t) u (t) uniformly with resp ect to t [0,T ] and, moreover, for sufficiently large n n0 and some

T T , we have
c1 n (u ,u0 ) Un - pn u n F
n n

c2 n (u ,u0 ), n
0kn T

where the constants c1 and c2 are indep endent of n, Fnn = {un (kn ) : n

max

un (kn )

E

n

< } and

(u ,u0 ) = max n

t[0,T ]

n

(pn u )(t) -Un (t)u0 n

E

n

.

Schemes which have higher order of convergence than (6.3) are considered in [146, 166]. The Runge­ Kutta methods for semilinear equations were considered in [79], [135­137, 146, 149]. 40


6.2. Approximation of periodic problem. In a Banach space E , let us consider the semilinear T p eriodic problem v (t) = Av(t)+ f t, v(t) , v(t) = v (T + t),t R+ , (6.4)

with the op erator A, generating an analytic compact C0 -semigroup, where the function f is smooth enough and f (t, x) = f (t + T, x) for any x E and t R+ . Let u(·; u0 ) b e a solution of the Cauchy problem (6.1) with the initial data u(0; u0 ) = u0 . This function u(·; u0 ) is also a mild solution, i.e., it satisfies the integral equation u(t) = exp(tA)u0 +
0 t

exp (t - s)A f s, u(s) ds, t R+ .

(6.5)

Then the shift op erator K(u0 ) = u(T ; u0 ) can b e defined, and it maps E into E. If u(·; x ) is a p eriodic solution of (6.1), then x is a zero of the compact vector field defined by I -K, i.e., K(x ) = x . Remark 6.3. We assume here that the op erator I - exp(TA) it is just enough to assume that I - exp(tA) exp t(A - )
-1 -1

exists and is b ounded. Meanwhile,

B (E ) holds for t t0 with some t0 > 0. This
-1

assumption is not restrictive, since, without loss of generality, we can change A by A - I and obtain Me-t for > 0,t 0. It follows [29] that I - exp(tA) B (E ) for any t > 0.

Remark 6.4. We say that function f is smooth enough in the sense that it is at least continuous in b oth arguments,
t[0,T ], x c

su p

f (t, x) C2 and such that there exists the global mild solution of the problem
1

u (t) = Au(t)+ f t, u(t) , u(0) = u0 , t R+ . Definition 6.1. The solution u(·) of the Cauchy problem (6.1) is said to b e stable in the Lyapunov sense if for any > 0 there is > 0 such that the inequality u(0) - u(0) implies max ~ where u(·) is a mild solution of (6.1) with the initial value u(0). ~ ~
0t<

u(t) - u(t) , ~

Definition 6.2. The solution u(·) of the Cauchy problem (6.1) is said to b e uniformly asymptotical ly stable at the p oint u(0) if it is stable in the Lyapunov sense, and for any mild solution u(·) of (6.1) with ~ u(0) - u(0) , it follows that lim ~ is a function u
(0), t

u(t) - u(t) = 0 uniformly in u(·) B (u(0); ), i.e., there ~ ~ u
(0),

(·) such that u t; u(0) - u t;~(0) u

(t) with u

(0),

(t) 0 as t and

u(0) - u(0) . ~ Constructive conditions on the op erator A and f ensuring that the equation u (t) = Au(t)+ f (u(t)), u(0) = u0 is asymptotically k-dimensional are given in [172, 173]. They concern with the location of eigenvalues of A, i.e., k
+1

- k > 2L, k

+1

> L. 41


Theorem 6.3 ([38]). Assume that conditions (A) and (B ) hold and compact resolvents (I - A)-1 , (In - An )-1 converge: (In - An )-1 (I - A)-1 compactly for some (A). Assume that (i) the functions f and fn are sufficiently smooth, so that there exists an isolated mild solution v (·) of the periodic problem (6.4) with v (0) = x such that the Cauchy problem (6.1) with u(0) = x has a uniformly asymptotical ly stable isolated solution at the point x (in this case, ind v (·) = 1); (ii) fn (t, xn ) f (t, x) uniformly with respect to t [0,T ] as xn x; (iii) the space E is separable. Then, for almost al l n, the problems vn (t) = An vn (t)+ fn t, vn (t) ,vn (t) = vn (t + T ),t R+ , (6.6)

have periodic mild solutions vn (t),t [0,T ], in the neighborhood of pn v (·), where v (·) is a mild periodic solution of (6.4) with v (0) = x . Each sequence {vn (·)} is P -compact and vn (t) v (t) uniformly with

respect to t [0,T ]. We say that a fixed p oint x of the op erator K in Banach lattice E is stable from the ab ove [98] if given > 0, there is > 0 such that Kk x - x for all k N if x x and x - x . Using this

notion, we can reformulate Theorem 6.3 for p ositive semigroups due to the result from [62]. Theorem 6.4. Let the operators An and A from the problems (6.4) and (6.6) be compatible and let E
+ and En be order-one spaces and en D(An ) int En . Assume that the operators An have the POD

property and An en

0 for sufficiently large n and compact resolvents (I - A)-1 , (In - An )-1 converge

(In - An )-1 (I - A)-1 compactly for some (A). Assume that (i) the functions f and fn are sufficiently smooth, bounded and positive, so that there exists a mild solution u (·) of the Cauchy problem (6.3) such that the element u (0) = x is a stable from above and fixed points of operator K with x y, Ky y (in this situation ind x = 1); (ii) fn (t, xn ) f (t, x) uniformly with respect to t [0,T ] as xn x; (iii) the space E is separable.
Then for almost al l n, problems (6.6) have periodic mild solutions vn (t),t [0,T ] in the neighborhood of pn v (·), where v (·) is any mild periodic solution of (6.4) stable from above. Each sequence {vn (·)} is P -compact and vn (t) v (t) uniformly with respect to t [0,T ].

Remark 6.5. The technique which is used here can b e applied to the case of condensing op erators [3]. For example, the resolvent of in L2 (Rd ) is condensing, but it is not compact. In [120], the qualitative b ehavior of spatially semidiscrete finite-element solutions of a semilinear parab olic problem near an unstable hyp erb olic equilibrium was studied. 42


The shadowing approach to the study of the long-time b ehavior of numerical approximations of semilinear parab olic equations was studied in [119]. Many results contained in this survey can b e reformulated for the second-order equation u = Au(t) with the op erator A generating a C0 -cosine op erator function.

Acknowledgments. The authors acknowledge the supp ort of NATO-CP Advanced Fellowship Pro¨ gramme of TUBITAK (Turkish Scientific Research Council), University of Antwerp en, Russian Foundation for Basic Research (01-01-00398) and University of Bologna. We would also like to thank Professor E.H. Twizell for his valuable comments.

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