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101.
We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.
102.
103.
104.
B. Cano A. M. Stuart E. Süli J. O. Warren 《Foundations of Computational Mathematics》2001,1(1):69-100
Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that
the forcing term approximates a delta function as and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators
is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step satisfies The convergence of the algorithms is studied in this case in the limit and For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically
and proved theoretically.
August 25, 1999. Final version received: May 3, 2000. 相似文献
105.
In this paper we discuss the global behaviour of some connected sets of solutions of a broad class of second order quasilinear elliptic equations
for where is a real parameter and the function u is required to satisfy the condition
The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier.
To use this degree the problem must be expressed in the form where J is an interval, X and Y are Banach spaces and F is a map which is Fredholm and proper on closed bounded subsets. We use the usual spaces and . Then the main difficulty involves finding general conditions on and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of
solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form
our results cover situations outside the scope of other methods in the literature.
Received March 30, 1999; in final form January 17, 2000 / Published online February 5, 2001 相似文献
106.
Stuart A. Steinberg 《Czechoslovak Mathematical Journal》2001,51(2):387-394
In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets 1 and 2 and the corresponding -value sets
and
. If R is a unital -ring, then each unital -module over R is an f-module and has
exactly when R is an f-ring in which 1 is a strong order unit. 相似文献
107.
C. A. Stuart 《Milan Journal of Mathematics》2011,79(1):327-341
For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C 1–functional on H01(W){H_0^1(Omega)}. One solution is a local minimum and the other is of mountain pass type. 相似文献
108.
Stuart S. Antman 《Journal of Elasticity》2003,70(1-3):55-64
The introduction of artificial viscosity into the partial differential equations of mechanics is often useful for both analytic and numerical studies. The traditional forms of artificial viscosity, originally designed to treat problems for fluids, when applied to problems for solids often lead to equations describing material properties that are not invariant under rigid motions. Consequently, for rapidly rotating bodies, artificial viscosity could produce serious errors. In this paper it is shown how to introduce artificial viscosity in a properly invariant way, and that the resulting systems have a rich and attractive structure, which beckons analysis. 相似文献
109.
Positive Eigenfunctions of a Schrodinger Operator 总被引:1,自引:0,他引:1
The paper considers the eigenvalue problem
where , and
for some bounded open set RN. Given >0, does there exist a value of >0 for which theproblem has a positive solution? It is shown that this occursif and only if lies in a certain interval (,1) and that inthis case the value of is unique, =(). The properties of thefunction () are also discussed. 相似文献
110.
Christopher E. Stuart Paul Abraham 《Journal of Mathematical Analysis and Applications》2004,300(2):224-361
Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems for scalar-valued measures on rings of sets that are in general not σ-rings are presented. As a consequence, the rings of subsets of N with density zero and uniform density zero are shown to have the Nikodym property. In addition, vector measure generalizations of the Vitali–Hahn–Saks theorem are given. 相似文献