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DCDS

Nonlinear Galerkin Methods (NGMs) are numerical schemes for evolutionary partial
differential equations based on the theory of Inertial Manifolds
(IMs) [1] and Approximate
Inertial Manifolds (AIMs) [2]. In this paper,
we focus our attention on the 2-D Navier-
Stokes equations with periodic boundary conditions, and use Fourier methods
to study its nonlinear Galerkin approximation which we call Fourier Nonlinear
Galerkin Methods (FNGMs) here. The first part is contributed to the semidiscrete case.
In this part, we derive the well-posedness of the nonlinear Galerkin form and
the
distance between the nonlinear Galerkin approximation
and the genuine solution in Sobolev spaces of any orders. The second part
is concerned about the full discrete case, in which, for a given numerical scheme
based on NGMs, we investigate the stability and error estimate respectively.
We derive the stability conditions for the scheme in any fractional Sobolev spaces.
Finally, we give its error estimate in $H^r$ for any $r\geq 0$.

DCDS

In this article, we obtain the existence of inertial
manifolds under time
discretization based on their invariant property.
In [1], the authors gave
their
existence by finding the fixed point of some
inertial mapping defined by a sum
of infinite series:

$ T_h^0\Phi(p)=\sum_{k=1}^{\infty}R(h)^kQF(p^{-k}+\Phi(p^{-k})) $

where $p^{-k}=(S^h_\Phi)^{-k}(p)$, see [1] for detailed definition. Here we get the existence by solving the following equation about $\Phi$:

$\Phi(S_\Phi^h(p))=R(h)[\Phi(p)+hQF(p+\Phi(p))] \mbox{ for }\forall p\in PH.$

See section 1 for further explanation which describes just the invariant property of inertial manifolds. Finally we prove the $C^1$ smoothness of inertial manifolds.

DCDS

In this paper, we present an Oseen coupling problem to approximate the two
dimensional exterior unsteady Navier-Stokes problem with the nonhomogeneous
boundary conditions. The Oseen coupling problem consists of the
Navier-Stokes equations in a bounded region and the Oseen equations in an
unbounded region. Then we derive the reduced Oseen coupling problem by use of
the integral representations of the solution of the Oseen equations in an
unbounded region. Moreover, we present the Galerkin approximation and the
nonlinear Galerkin approximation for the reduced Oseen coupling problem. By
analysing their convergence rates, we find that the nonlinear Galerkin
approximation provides the same convergence order as the classical Galerkin
approximation if we choose the space discrete parameter $H=O(h^{1/2})$.
However, in this approximation, the nonlinearity is treated on the coarse grid
finite element space and only the linear problem needs to be solved on the fine
grid finite element space.

DCDS-B

In this paper, a monotone-iterative scheme is established for finding
positive periodic solutions of a competition
model of tumor-normal cell interaction. The model describes the
evolution of a population with normal and tumor cells in a
periodically changing environment. This population is under
periodical chemotherapeutic treatment. Competition among the two
kinds of cells is considered. The mathematical problem involves a
coupled system of Lotka-Volterra together with periodically pulsed
conditions. The existence of positive periodic solutions is proved
by the monotone iterative technique and in a special case, the
uniqueness of a periodic solution is obtained by proving that any
two periodic solutions have the same average. Moreover, we also
show that the system is permanent under the conditions which
guarantee the existence of the periodic solution. Some computer
simulations are carried out to demonstrate the main
results.

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