# American Institute of Mathematical Sciences

## Journals

DCDS
Discrete & Continuous Dynamical Systems - A 1996, 2(4): 497-524 doi: 10.3934/dcds.1996.2.497
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 $r\geq 0$.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 1997, 3(4): 531-540 doi: 10.3934/dcds.1997.3.531
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.

keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 1996, 2(4): 467-482 doi: 10.3934/dcds.1996.2.467
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.
keywords:
DCDS-B
Discrete & Continuous Dynamical Systems - B 2019, 24(1): 127-147 doi: 10.3934/dcdsb.2018111

A dimension splitting and characteristic projection method is proposed for three-dimensional incompressible flow. First, the characteristics method is adopted to obtain temporal semi-discretization scheme. For the remaining Stokes equations we present a projection method to deal with the incompressibility constraint. In conclusion only independent linear elliptic equations need to be calculated at each step. Furthermore on account of splitting property of dimension splitting method, all the computations are carried out on two-dimensional manifolds, which greatly reduces the difficulty and the computational cost in the mesh generation. And a coarse-grained parallel algorithm can be also constructed, in which the two-dimensional manifold is considered as the computation unit.

keywords:
DCDS-B
Discrete & Continuous Dynamical Systems - B 2004, 4(3): 555-562 doi: 10.3934/dcdsb.2004.4.555
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.
keywords: