Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations

2001 ◽  
Vol 89 (2) ◽  
pp. 307-339 ◽  
Author(s):  
Mario A. Casarin
Keyword(s):  
Stokes Equations ◽  
Spectral Element ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Element Discretization ◽  
Schwarz Preconditioners

scholarly journals AUTOMATIC INSERTION OF A TURBULENCE MODEL IN THE FINITE ELEMENT DISCRETIZATION OF THE NAVIER–STOKES EQUATIONS

2009 ◽  
Vol 19 (07) ◽  
pp. 1139-1183 ◽  
Author(s):  
CHRISTINE BERNARDI ◽  
TOMÁS CHACÓN REBOLLO ◽  
FRÉDÉRIC HECHT ◽  
ROGER LEWANDOWSKI
Keyword(s):  
Finite Element ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Stokes Equations ◽  
A Posteriori Error Estimates ◽  
Navier Stokes ◽  
Mesh Adaptivity ◽  
Finite Element Discretization ◽  
Navier Stokes Equations ◽  
Element Discretization

We consider the finite element discretization of the Navier–Stokes equations locally coupled with the equation for the turbulent kinetic energy through an eddy viscosity. We prove a posteriori error estimates which allow to automatically determine the zone where the turbulent kinetic energy must be inserted in the Navier–Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.

Numerical calculations of the primary-flow exchange process in the Taylor problem

1988 ◽  
Vol 197 ◽  
pp. 57-79 ◽  
Author(s):  
K. A. Cliffe
Keyword(s):  
Stokes Equations ◽  
Exchange Process ◽  
Numerical Calculations ◽  
Navier Stokes ◽  
Finite Element Discretization ◽  
Primary Flow ◽  
Navier Stokes Equations ◽  
Element Discretization ◽  
Taylor Problem ◽  
Solution Surface

Numerical methods are used to study the way in which the number of cells present in the Taylor experiment changes as the length of the comparatively short annulus varies. The structure of the solution surface is determined by following paths of singular points in a finite-element discretization of the axisymmetric Navier–Stokes equations. The numerical results are compared with the experiments of Benjamin (1978b), Mullin (1982) and Mullin et al. (1982). The calculations are in agreement with the qualitative theory of Benjamin (1978a) and Schaeffer (1980) except that in the interaction involving four- and six-cell flows, the numerical calculations indicate that the six-cell flow can become unstable owing to perturbations that are antisymmetric about the midplane.

Fekete-Gauss Spectral Elements for Incompressible Navier-Stokes Flows: The Two-Dimensional Case

2013 ◽  
Vol 13 (5) ◽  
pp. 1309-1329 ◽  
Author(s):  
Laura Lazar ◽  
Richard Pasquetti ◽  
Francesca Rapetti
Keyword(s):  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Schur Complement ◽  
Spectral Element ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Flow Past A Cylinder ◽  
Accuracy Study

AbstractSpectral element methods on simplicial meshes, say TSEM, show both the advantages of spectral and finite element methods, i.e., spectral accuracy and geometrical flexibility. We present aTSEM solver of the two-dimensional (2D) incompressible Navier-Stokes equations, with possible extension to the 3D case. It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space. The so-called Fekete-Gauss TSEM is employed,i.e., Fekete (resp. Gauss) points of the triangle are used as interpolation (resp. quadrature) points. For the sake of consistency, isoparametric elements are used to approximate curved geometries. The resolution algorithm is based on an efficient Schur complement method, so that one only solves for the element boundary nodes. Moreover, the algebraic system is never assembled, therefore the number of degrees of freedom is not limiting. An accuracy study is carried out and results are provided for classical benchmarks: the driven cavity flow, the flow between eccentric cylinders and the flow past a cylinder.

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