UNIFORM STABILITY OF LINEAR MULTISTEP METHODS IN GALERKIN PROCEDURES FOR PARABOLIC PROBLEMS

Linear multistep methods are considered which have a stability region S and are D-stable on the whole boundary S c S of S. Error estimates are derived which hold uniformly for the class of initial value problems Y' AY + B(t), t > 0, Y(0) Y with normal matrix A satisfying the spectral condition Sp(AtA) S At O time step, Sp(A) spectrum of A. Because of this property, the result can be applied to semidiscrete systems arising in the Galerkin approximation of parabolic problems. Using known results of the Ritz theory in elliptic boundary value problems error bounds for Galerkin multistep procedures are then obtained in this


INTRODUCTION
If a linear parabolic initial boundary value problem is descretized in the space dimensions by the finite element method then the resulting semidiscrete problem is an initial value problem for a system of ordinary differential equa-  [!9]. In the nomenclature of matrix structural analysis, P(t) is the external load vector, M(Ax) the mass matrix, and K(Ax) the stiffness matrix; cf. e.g. Bathe and Wilson [2] and Przemieniecki [!5]. In the present communication both matrices are supposed to be independent of the time t, real symmetric and positive definite. They depend together with their dimension on the small parameter Ax which is, in general, the maximum diameter of all elements in the finite element subspace; see e.g. [2, !5, !9]. The  -I/2 we find cond(L(Ax)) !/Ax j with j 2 or j 4 respectively if the elliptic operator in the analytic problem is of order two or four, cf.e.g.
Strang and Fix [19, ch. 5]. These properties follow in a natural way since the finite-dimensional operator L must be an approximation to the analytical operator As a consequence the problem (!) becomes very stiff if a small mesh width Ax is chosen.
Apart from the drawback to need a special starting procedure, (one-stage) multistep methods have two advantageous properties in comparison with multistage (one-step) methods (io e. Runge-Kutta methods etc. The order of the discretization error is not negatively affected if the mesh width Ax becomes small. Ao-stable (definition below).
In the last section we apply our estimates to Galerkin multistep discretizations of parabolic initial boundary problems and show that the order of consistence is the order of convergence in this class of numerical approximations.
The results improve some of our error bounds derived in [6] where exponential stability was not yet obtained.
An other goal of the present contribution was to obtain error estimates in a form which is applicable to multistep methods in systems of second order. In a subsequent paper we use our results and consider Galerkin multistep discretizations of hyperbolic initial boundary value problems. The error estimates deduced there correspond to a high degree to those established here for parabolic problems.
2. UNIFORM STABILITY.  Especially, every consistent and strongly D-stable method <0,> is AG-Stable with G including the negative real line in a neighborhood of the origin since the essential root I has the growth parameter X I.
If a method <0,> is AG-Stable then the numerical approximation of (2) obtained by (3)  small At then G must contain the negative real line in a neighborhood of the origin, i.e., 0 E G.) Therefore we must require that <0,o> is D-stable on Gand not only in the origin. The next definition was also used by Nevanlinna [12]. Obviously, an AG-Stable method <0,(I> has the stability region S G. With this notation the below needed spactral condition can now be written as follows. <0,> has a stability region S # @.
S is closed in .
where c() depends only on and the dimension k. Therefore the assertion follows for every fixed N S. eigenvalues. Now, we have by (i) that c(N*) < for every N*6 S N but by construction of the norm II.
(cf. [18] and the explicit Jordan decomposition of Fz(N) given below) we find lim_D.c(N) for N*6 S N . Thus we must prove the assertion of the lemma for N -* in a separated way. For  >( ) (7) "k-I "k-I 1 k where the argument N is omitted. Consequently, using Cramer's rule we obtain after some simple calculations f(n) (n) ij () det(Qij (N))/Het(Q(N)) (8) (n) Here Qij (N) is obtained from Q(N) by replacing the th row by (gl()n+i-l'''''gk(B)n+i-l) (9) The denominator of (8) satisfies by (7) limD+N,det(Q(N)) 0 with exactly the rate of convergence m(m-l)/2m (m-l)/2 since by assumption exactly m roots v coalesce in N*. (We assume that I * 0 in (5) otherwise the proof follows in a slightly modified way.) (v) By (6) and (9) the assertion of the lemma is now proved for when we show that the numerator in (8) where UA(-)O,...,UA(-)k_ are assumed to be known. It is easily shown that (17) and (18) are equivalent to (3') and (16).  (19) directly. However, it is not the aim of the present contribution to discuss the vast field of error bounds in the Ritz theory of elliptic boundary value problems. Therefore we make the following assumption on the bilinear form a, the boundary of , and the subspace , cf. Ciarlet [3,Th. [16], and Zlamal [21,22]. (el(t) 'gm(t)) we obtain (u R uA)(',t)ll 0 llE(t)ll, t nat II" Euclid norm.
In order to deduce an estimation of u R u A we first observe that by (8) and Def. But (U-UR) t u t (ut) R as (UR) t (ut) R in the present case of a time-homogeneous bilinear form a. We therefore obtain under Assumption A liD < lid< (At u(-))II + <O<RAtAxllII utllll,n+k n 00 > nO (22) Ill u i111,n maxo_t_nAtllu(.,t II1 since u t can be viewed as solution of an el-666 E. GEKELER liptic problem, too.
Finally we can apply Theorem 2 to the error equation (2!) and estimate the defect D by (22). Then, by means of (20) we obtain an error bound for the second n term on the right side of (19). We summarize our result in the following theorem. If the solution u of (!5) is sufficiently smooth, Assumption A is fulfilled, and the method <0,o> is consistent of order q then IIDII < col(<c At q+I + <O<RAtAx

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
March 1, 2009 First Round of Reviews June 1, 2009