scholarly journals An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier–Stokes Equations

1997 ◽  
Vol 133 (1) ◽  
pp. 84-101 ◽  
Author(s):  
Paul F Fischer
Keyword(s):  
Stokes Equations ◽  
Spectral Element ◽  
Navier Stokes ◽  
Element Solution ◽  
Navier Stokes Equations ◽  
Schwarz Method ◽  
Overlapping Schwarz

scholarly journals The inf-sup stability of the lowest order Taylor–Hood pair on affine anisotropic meshes

2019 ◽  
Vol 40 (4) ◽  
pp. 2377-2398
Author(s):  
Gabriel R Barrenechea ◽  
Andreas Wachtel
Keyword(s):  
Finite Element ◽  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Element Solution ◽  
Navier Stokes Equations ◽  
Anisotropic Meshes ◽  
Theoretical Results

Abstract Uniform inf-sup conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier–Stokes equations. In this work we prove a uniform inf-sup condition for the lowest-order Taylor–Hood pairs $\mathbb{Q}_2\times \mathbb{Q}_1$ and $\mathbb{P}_2\times \mathbb{P}_1$ on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalize Verfürth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.

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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