Last week we studied the key concepts of Finite Element Methods, in the context of solving the Laplace equation. We saw the …nite element method arise naturally as an instance of the Galerkin method, a discretization procedure for the corresponding weak formulation. Finally, we learned about error estimates, examples of elements and implementation practices. This week, our main objective is to ...

The main purpose of this work is the efficient implementation of a multigrid algorithm for solving Navier-Stokes problems at low Reynolds numbers in different triangular geometries. In particular, a finite element formulation of the Navier-Stokes equations, using quadratic finite elements for the velocities and linear finite elements to approximate the pressure, is used to solve the problem of ...

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth H, Schwartz, or mild H categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in H for the periodic Navier-Stokes problem wit...

Cascaded coherent anti-Stokes Raman scattering (CARS) signals can be efficiently generated from CARS when propagating collinearly with the pump/Stokes and probe beams. This effect seen as beam acting being inelastically scattered a ‘second time’ coherence induced along focus of axis. Although much weaker, this additionally signal co-propagates may complicate analysis spectrum used for diagnosti...

This paper is aimed to present the combination of the parallel computing and the multigrid method on the Navier-Stokes solver. The combination is based on the concept of the object-oriented programming (OOP), which consists of 4 independent modules: Grid Generation, Navier-Stokes Solver, Multigrid Method and Parallel Computing modules. The multigrid method is implemented by employing the full a...

We investigate the long-term dynamics of the three-dimensional NavierStokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the N...

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data a bounded domain containing finite number of subdomains $C^{1,\rm{Dini}}$ boundaries. prove that if $(u, p)$ is weak solution the system, then $(Du, continuous. The corresponding results for Navier-Stokes are also established, from which Lipschitz regularity $H^1$-weak dimensions $d=...