A stabilized finite element method for generalized stationary incompressible flows

By Ramon Codina. 2001

285 citations recorded by the source corpus.

Abstract In this paper, we describe a finite element formulation for the numerical solution of the stationary incompressible Navier–Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection–diffusion–reaction equations and then we apply it to the linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.

Subfields: Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/60abca17a1cb78fe16a7e08c8370046ad5850419

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In Computational Mechanics

Citations in the corpus

A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation (2000, methodology): “Similar effects can be obtained using Characteristic Galerkin methods [10,11], Variational Multiscale models [12,13] and analogous Residual-Free Bubbles techniques [14–16].”
A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes (2012, methodology): “K τK(∇qh,f)K + 〈qh,n · ū〉Γ, ∀qh ∈ Qh, − 1 nν (τ h,σh) + 1 n (τh,∇uh)− 〈τ h · n,uh〉Γ = −〈τh · n, ū〉Γ, ∀τh ∈ Sh, and now the stabilization parameter is computed as [5]”
Stabilized finite element schemes with LBB-stable elements for incompressible flows (2005, background): “Moreover, we do not see restriction (30) in the numerical experiments presented in Section 5, which show uniform error estimates even for = 0 and → 0.”
A finite element method for the solution of rotary pumps (2007, unknown): “The method is extensively described in [9].”
Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales (2008, unknown): “(16) A version of this method, including also zero order terms coming from Coriolis forces and permeability effects, is analyzed in [16].”
The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements (2016, methodology): “Therefore, there has been an increasing interest in methods that can enforce boundary and interface conditions without mesh dependent stabilization parameters [9, 23, 44, 48].”