Mathematical Models and Methods in Applied Sciences

This article deals with the long-time behavior of the solution of the two-dimensional Navier–Stokes equations. At each time step, we use finite elements to split the solution into a large-scale component and a small-scale component, and we follow both components in time. Next, considering a mixed finite element approximation of the equations, we prove that many properties that hold for the exact solution extend to the discrete solution as well, uniformly in the discretization parameter.

In this paper we discuss a way to recover a classical residual-based error estimator for elliptic problems by using a finite element space enriched with bubble functions. The advection-dominated case is also discussed.

In this paper numerical problems arising from steady state analysis of nonlinear circuits with quasiperiodic excitation are discussed. The approach we consider is based on the piecewise harmonic balance techniques^8,9 (HB), a methodology which has its theoretical foundations in Galerkin's procedure (see the paper by Urabe¹²).

The original problem, which can be expressed in the form of a system of integro-differential equations in the time domain, is changed into a nonlinear algebraic system through a natural projection technique. Thus, one of the main issues we have investigated consists in the numerical solution of the specific nonlinear algebraic problem.

A theorem by J. M. Ghidaglia and R. Temam shows the existence and finitness of attractors for 2-D Navier–Stokes equations for a sligthly compressible fluid. In this paper we extend their result to the equations of thermohydraulic for the same fluid.

A monitor or controller is smart provided that the device is equipped with local computational resources for analyzing data, detecting changes, and making decisions. The problem for the monitoring function is to design algorithms to flag model shifts for dynamic systems in a context requiring many interacting system components and system reconfigurations. The problem for the control function is to improve system performance by updating control feedbacks after the model shift has been detected. Therefore it is desirable that smart monitors and controllers be adaptive and update with minimal intervention from a central director.

We present here an approach to designing smart monitoring and control devices based on a stochastic linearization of the system whose dynamics is noisy and unknown. This linearization is obtained by factoring the discrete system covariance matrix, estimated from observations, and applying reproducing kernel Hilbert space techniques. The method is nonparametric which allows the smart devices to operate with only a low level logic.

The stationary Stokes equations of a viscous incompressible fluid are considered in domains Ω with m > 1 exits to infinity, which have in some coordinate system the following form:

In the standard approach particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the differential cross-section in the case of the general Boltzmann equation. Recently, construction of an implicit particle scheme for the Boltzmann equation with Maxwellian molecules was shown.

This paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second-order particle method when using an equiweighting of explicit and implicit discretization.