1Department de Mathemàtica Aplicada, University de València, 46100 Burjassot, Valencia, Spain 2Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia
Received 31 August 2011; Accepted 15 November 2011
This paper is devoted to the numerical analysis of abstract parabolic problem ;
, with hyperbolic generator . We are developing a general approach to establish
a discrete dichotomy in a very general setting in case of discrete approximation in space and
time. It is a well-known fact that the phase space in the neighborhood of the
hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced
to initial value problems with exponential decaying solutions in opposite time direction. We use
the theory of compact approximation principle and collectively condensing approximation to
show that such a decomposition of the flow persists under rather general approximation schemes.
The main assumption of our results is naturally satisfied, in particular, for operators with
compact resolvents and condensing semigroups and can be verified for finite element as well as
finite difference methods.
1. Introduction
Many problems like approximation of attractors, traveling waves, shadowing e.c. involve the notion of dichotomy. In numerical analysis of such problems, it is very important to know if they keep some kind of exponential estimates uniformly in discretization parameter.
Let denote the Banach algebra of all bounded linear operators on a complex Banach space . The set of all linear-closed densely defined operators in will be denoted by . For , let be its spectrum and , its resolvent set. In the following let be a closed linear operator, such that
Under condition (1.1), the spectrum of is on the left: , so the fractional power operators , (see [1, 2]) associated to and , can be constructed and the corresponding fractional power spaces too, that is, endowed with the graph norm . Define the ball with the center at 0 of radius in space.
To show how the dichotomy problems appear in numerical analysis, we are considering for example the semilinear equation in Banach space
where , is assumed to be continuous, bounded, and continuously Fréchet differentiable function. More precisely, we assume that the following condition holds.(F1) For any , there is such that as for all , where is a hyperbolic equilibrium point of (1.2).
By means of the change of variables in the problem (1.2), where is the hyperbolic equilibrium, we obtain the problem
Such problem can be written in the form
where , . We note that from condition (F1) it follows that the function for small is of order . Since , the operator is the generator of an analytic -semigroup [3]. It can happen that the spectrum of operator can be split into two parts and .
We assume that the part of the spectrum of operator , which is located strictly to the right of the imaginary axis, consists of a finite number of eigenvalues with finite multiplicity. This assumption is satisfied, for instance, if the resolvent of operator is compact. The conditions under which the operator has the dichotomy property were studied say in [4–6]. In case of hyperbolic equilibrium point , there is no spectrum of on . Let be an open connected neighborhood of which has a closed rectifiable curve as a boundary. We decompose using the Riesz projection
defined by . Due to this definition and analyticity of the -semigroup , we have positive constants , such that
for all and . Since for small , the estimates (1.6) are crucial to describe the behavior of solution of the problem (1.2) at the vicinity of the hyperbolic equilibrium point .
If is close to 0, that is, say with small , then the mild solution of (1.4) can stay in the ball , for some time. We denote now the maximal time of staying in by . Now coming back to solution of (1.4) for any two we consider the boundary value problem
A mild solution of problem (1.7) as was shown in [7] satisfies the integral equation
In case one would like to discretize the problem (1.4) in space and time variables it is very important to know what will happen to estimates like (1.6) for approximation solutions. If the estimates like (1.6) hold uniformly in parameter of discretization, then one can expect to get a similar behavior of approximated solutions of (1.8).
So, in this paper we are going to consider general approximation approach for keeping the dichotomy estimates (1.6) for approximations of trajectory .
2. Preliminaries
Let .
Definition 2.1. A -semigroup , defined on a Banach space is called hyperbolic if for all . The generator is called hyperbolic if .
Let us denote by any of the spaces or Stepanov’s space . We consider in the Banach space (we call this space as Palmer’s space, see [8], where Fredholm property was mentioned for the first time) the linear differential operator
where generates -semigroup, domain of is assumed to consist of the functions such that for some function one has , and . Let us note [9] that the operator is the generator of -semigroup on a Banach space , which is defined for all by formula
Definition 2.2. A -semigroup , has an exponential dichotomy on with exponential dichotomy data if there exists projector such that(i) for all ,(ii)the restriction , is invertible on and
Theorem 2.3 (see [10]). The operator in the Banach space is invertible if and only if the condition
holds. If condition (2.4) is satisfied, then
where the Green’s function
where
and æ .
We note that the constant is defined as if or and if or .
Denote by the Banach space of -valued sequences with corresponding discrete norm which is consistent with the norm of . For any , that is, , and , we define an operator by formula
Now we define an operator as
Proposition 2.4 (see [10]). Let an operator be invertible. Then, an operator is invertible too and
Conversely, if is invertible, then is invertible and
Theorem 2.5 (see [11]). The difference operator
is invertible if and only if
If condition (2.14) is satisfied, then the inverse operator has the form
where , and the function is defined by
where is the restriction of on .
Definition 2.6. The operator has an exponential discrete dichotomy with data if is a projector in and are constants with such that the following properties hold(i) for all ,(ii) for all ,(iii) is a homeomorphism that satisfies .
The following result relates the constants in an exponential discrete dichotomy to a bound on the resolvent for .
Theorem 2.7. For , the following conditions are equivalent:(i) for all and ;(ii) has an exponential dichotomy with data . More precisely, one shows that (i) implies (ii) with and . Conversely, (ii) implies (i) with .
Proof. First assume (i) and without loss of generality let . For , we have
Hence, if , the classical perturbation estimate shows that and
We define as the Riesz projector defined by the formula
Since commutes with the resolvent, condition (i) of Definition 2.6 holds. Further, using (2.18) and Cauchy’s theorem, we can shift the contour
Now we claim
For this follows from (2.20). If (2.21) holds for some , then we obtain
and thus the assertion holds for . Equations (2.21) and (2.18) immediately lead to the first dichotomy estimate for
For the second dichotomy estimate, we use the resolvent equation
For , (2.20) and (2.24) lead to
This shows that the following equality holds for
If (2.26) is known for some , then use (2.24) and find
We apply (2.26) to , using that commutes with as well as the estimate (2.18)
Summarizing, we have shown for all
For , this estimate shows that is one-to-one with a bound for the inverse. To show that is onto, we take and set
From this equation, we have , and using (2.24), we find
Therefore, is a linear homeomorphism on satisfying
This proves exponential dichotomy. To arrive at the specific constants, we choose in (2.23) and obtain the bound . In order to have the same rate in the opposite direction, we apply (2.32), with . In (2.32) we then find the upper bound with the constant . Since , our assertion follows. Now we assume exponential dichotomy and prove condition (i). For the equation is equivalent to the system
which we can rewrite as
Both equations have a unique solution given by a geometric series
The exponential dichotomy then implies the estimates
By the triangle inequality, we obtain condition (i) with .
3. Discretization of Operators and Semigroups
In the papers [12–15], a general framework was developed that allows to analyze convergence properties of numerical discretizations in a unifying way. This approach is able to cover such seemingly different methods as (conforming and nonconforming) finite elements, finite differences, or collocation methods. It is the purpose of this paper to show that it is also possible to handle dichotomy properties with discretization in space and time on general approximation scheme. Moreover, we also consider the case when resolvent of operator is not necessarily compact.
3.1. General Approximation Scheme
Let and be Banach spaces and a sequence of linear bounded operators , with the property:
Definition 3.1. The sequence of elements , is said to be -convergent to if and only if as , and we write this .
Definition 3.2. The sequence of bounded linear operators , , is said to be -convergent to the bounded linear operator if for every and for every sequence , such that one has . We write then .
In the case of unbounded operators as it occurs for general infinitesimal generators of PDE’s, the notion of compatibility turns out to be useful.
Definition 3.3. The sequence of closed linear operators , is called compatible with a closed linear operator if and only if for each there is a sequence , such that and . We write that are compatible.
For analytic -semigroups, the following ABC Theorem holds.
Theorem 3.4 (see [16]). Let operators and generate analytic -semigroups. The following conditions and are equivalent to condition .() Compatibility. There exists such that the resolvents converge .Stability. There are some constants and such that Convergence. For any finite and some , one hasHere one used the sector of angle and radius given by , and .
It is natural to assume in semidiscretization that conditions like (A) and () are satisfied.
Definition 3.5. The region of stability , is defined as the set of all such that for almost all and such that the sequence is bounded. The region of convergence , is defined as the set of all such that and such that the sequence of operators is -convergent to some operator .
Definition 3.6. A sequence of operators , is said to be stably convergent to an operator if and only if and , . We will write this as: stably.
Definition 3.7. A sequence of operators , , is called regularly convergent to the operator if and only if and the following implication holds
We write this as: regularly.
Theorem 3.8 (see [15]). Let , and . Assume also that compactly and stably. Then, converge regularly.
Theorem 3.9 (see [15]). For and , the following conditions are equivalent:(i) regularly, are Fredholm operators of index 0 and ;(ii) stably and ;(iii) stably and regularly; (iv)if one of conditions (i)–(iii) holds, then there exist , , and regularly and stably.
Theorem 3.10. Let operators be Fredholm operators with for any . Assume also that has the property and regularly for any . Then stably for any and .
Proof. Assume that there are some sequences , and , such that and as . Since is compact, one can find such that as . In the meantime, regularly for such , and for . Therefore, there is such that as . But in such case as , which contradicts our assumption .
Definition 3.11. The operator has a uniform exponential discrete dichotomy with data if is a projector in and are constants with such that the following properties hold(i) and for all ;(ii) for all ;(iii) is a homeomorphism that satisfies
Theorem 3.12. The following conditions are equivalent:(i) stably and for any ;(ii)operator is invertible and stably, where ;(iii) and be invertible for any and the operators have an exponential discrete dichotomy with data uniformly in .
Proof. The equivalence (i)⇔(iii) follows from Theorem 2.7. Indeed, by formula (2.19), one gets from (i) that and . By Theorem 3.10, one has , and by Theorem 2.7 (ii) we have (iii). Conversely, from condition (iii), it follows by Theorem 2.7 (i) that stably for all . To prove (ii)⇒(i), we note that (ii) means that for any one has as , that is, . Assume now that is not uniformly invertible in , that is, for some sequence , one has as for some . This means that for stationary sequence , one has for any and . But, stably, that is, which contradicts as . Now we show that . For any , consider . The solution of is a sequence , which is stationary too, that is, , where . To show (i)(ii), we note that . Now for any and for any , one can find such that . In the meantime, as , since . So we have . The convergence follows from formula (2.15). The theorem is proved.
3.2. Dichotomy for Compact Resolvents in Semidiscretization
In the case of operators which have compact resolvent, it is natural to consider approximate operators that “preserve” the property of compactness.
Definition 3.13. A sequence of operators , converges compactly to an operator if and the following compactness condition holds
Definition 3.14. The region of compact convergence of resolvents, , where and is defined as the set of all such that compactly.
Proposition 3.15. Assume that operators are compact, and . Then, stably for any and .
Proof. The proof follows from Theorems 3.8, 3.9, and 3.10. Here we continue considering an example of discretization of problem (1.2). To this end we will also use the operators which satisfy property (3.1), but for the spaces . The operators and are supposed to be related by condition (1.1), and conditions and of Theorem 3.4 are assumed to hold. So we say that if and only if as . One can see that and for any and . Consider in Banach spaces the family of parabolic problems
where , operators are compatible, are globally bounded and globally Lipschitz continuous both uniformly in , and continuously Fréchet differentiable. Under the above assumptions, the mild solution of (3.7) is defined for all (see [1, 17]), and we define it by . The nonlinear semigroup satisfies the variation of constants formula Recall that a hyperbolic equilibrium point is a solution of equation , or equivalently . Since the operator has a compact resolvent, the operator is compact. In case of , the operators compactly. From [15], it follows that equations have solutions , such that . Now, consider the problems (3.7) near hyperbolic equilibrium points . In this case one gets
where . From now on, we consider hyperbolic point and hyperbolic points . We decompose using the projection
defined by the set , which is enclosed in a contour consisting of a part of and the contour from condition for operators . There are some positive , because of analyticity of -semigroup and condition , which is applied to operator , such that [7, 18]
for all and . One has to note that the neighborhood of any part of does not intersect . Moreover, the condition implies that compactly, and therefore, as was shown in [14], for . One can consider for the with the problem
The mild solution of problem (3.12) is given by the formula (for )
Now we are in a position to state our main result on uniform in index estimates for the terms of discrete solutions (3.13).
Theorem 3.16. Let operators be generators of analytic -semigroups and let condition be satisfied. Assume also that the semigroup is hyperbolic, , and for such that , operators are Fredholm operators of ind 0, and operators are regularly consistent for any . Then, compactly and
where .
Proof. The condition implies that as for big enough. For the other , the convergence follows from analogy of Theorem 3.9 for closed operators. Now the compact convergence can be obtained in the same way as in [3] and the estimates (3.14) follow say as in [7, 18]. The theorem is proved.
Remark 3.17. Of course, Theorem 3.16 holds for the case of any operator which generates analytic hyperbolic -semigroup with condition and corresponding conditions on approximation of operators. The structure of operator like is not necessary.
Theorem 3.18. Let the operators be generators of analytic -semigroups and let the condition be satisfied. Assume also that the -semigroup is hyperbolic, . Assume also that and resolvents of are compact operators. Then, (3.14) holds.
Proof. We set and and apply Proposition 3.15. It is known [3] that is equivalent to compact convergence . Then, the condition (i) of Theorem 3.12 is satisfied and one gets discrete dichotomy for . From the other side, since we have exactly the form , this means by Theorem 2.7 and Proposition 2.4 that operators are invertible, and thus by Theorem 2.3 the estimates (3.14) are followed. The theorem is proved.
3.3. Dichotomy for Condensing Operators in Semidiscretization
Let be some open connected set, and let . For an isolated point , the corresponding maximal invariant space (or generalized eigenspace) will be denoted by , where and is small enough so that there are no points of in the disc different from . The isolated point is a Riesz point of if is a Fredholm operator of index zero and is of finite rank. Denote
It is clear that , where
Definition 3.19. The function is said to be the measure of noncompactness if for any bounded sequence , one has
Definition 3.20. We say that the operators are jointly condensing with constant with respect to measure if for any bounded sequence , one has
It is known [19, page 82], that outside a closed disc of radius centered at zero each operator has only isolated points of spectrum, each of which can only be an eigenvalue of finite multiplicity.
Proposition 3.21. Let for and , for any bounded sequence . Assume that , where bounded closed set and consists only of discrete spectrum. Then, there is a constant such that .
Proof. Any point belongs to . This means that one gets for a sequence , two cases: or with some . We are going to show that in reality we have . Assume in contradiction that there are sequences , such that
Then, . One has for
which means because of , that , that is, is -compact. Now and , that is, with , which contradicts our assumption . The proposition is proved.
Proposition 3.22. Let and for any bounded sequence . Assume that any , is an isolated eigenvalue with the finite dimensional projector . Then, there are sequence , and sequence of projectors such that and converge compactly.
Proof. Note first that for , where can be taken small enough, we have by Proposition 3.21
Therefore, . To show compact convergence of these projectors one can note that
where . This means that
By functional calculus
From this representation using (3.23), one has
It is clear that from it follows that as . This means that compactly. The proposition is proved.
Theorem 3.23. Let the conditions and be satisfied, and let the analytic -semigroup , be hyperbolic such that the set
