scholarly journals Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Erik Burman ◽  
Peter Hansbo ◽  
Mats G. Larson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Large Scale ◽  
Optimal Error Estimates ◽  
Perturbation Equation ◽  
Stability Estimates ◽  
Smagorinsky Model ◽  
Stabilized Finite Element ◽  
High Reynolds ◽  
The Stability

AbstractIn the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $$L^2$$ L 2 -norm for smooth flows in the pre-asymptotic high Reynolds number regime.

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

2011 ◽  
Vol 3 (2) ◽  
pp. 239-258 ◽  
Author(s):  
Ke Zhao ◽  
Yinnian He ◽  
Tong Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimal Error Estimates ◽  
Two Dimensional ◽  
Element Level ◽  
Optimal Error ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
The Stability ◽  
Element Method

AbstractThis paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.

scholarly journals Two-Level Brezzi-Pitkäranta Stabilized Finite Element Methods for the Incompressible Flows

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Rong An ◽  
Xian Wang
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Incompressible Flows ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Linearization Methods ◽  
Correction Scheme ◽  
Iteration Methods ◽  
Level Method ◽  
The Stability

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.

Application of projection-based model reduction to finite-element plate models for two-dimensional traveling waves

2016 ◽  
Vol 28 (14) ◽  
pp. 1886-1904 ◽  
Author(s):  
Vijaya VN Sriram Malladi ◽  
Mohammad I Albakri ◽  
Serkan Gugercin ◽  
Pablo A Tarazaga
Keyword(s):  
Finite Element ◽  
Model Reduction ◽  
Traveling Waves ◽  
Large Scale ◽  
Fe Model ◽  
Plate Model ◽  
Reduction Techniques ◽  
State Frequency ◽  
Scale Down ◽  
The Stability

A finite element (FE) model simulates an unconstrained aluminum thin plate to which four macro-fiber composites are bonded. This plate model is experimentally validated for single and multiple inputs. While a single input excitation results in the frequency response functions and operational deflection shapes, two input excitations under prescribed conditions result in tailored traveling waves. The emphasis of this article is the application of projection-based model reduction techniques to scale-down the large-scale FE plate model. Four model reduction techniques are applied and their performances are studied. This article also discusses the stability issues associated with the rigid-body modes. Furthermore, the reduced-order models are utilized to simulate the steady-state frequency and time response of the plate. The results are in agreement with the experimental and the full-scale FE model results.

A STUDY OF HIGHER-ORDER DISCONTINUOUS GALERKIN AND QUADRATIC LEAST-SQUARES STABILIZED FINITE ELEMENT COMPUTATIONS FOR ACOUSTICS

2006 ◽  
Vol 14 (01) ◽  
pp. 1-19 ◽  
Author(s):  
ISAAC HARARI ◽  
RADEK TEZAUR ◽  
CHARBEL FARHAT
Keyword(s):  
Least Squares ◽  
Discontinuous Galerkin ◽  
Large Scale ◽  
Dispersion Analysis ◽  
Stability Parameter ◽  
New Approach ◽  
One Dimensional ◽  
Stabilized Finite Element ◽  
Quadratic Interpolation ◽  
The Stability

One-dimensional analyses provide novel definitions of the Galerkin/least-squares stability parameter for quadratic interpolation. A new approach to the dispersion analysis of the Lagrange multiplier approximation in discontinuous Galerkin methods is presented. A series of computations comparing the performance of [Formula: see text] Galerkin and GLS methods with Q-8-2 DGM on large-scale problems shows superior DGM results on analogous meshes, both structured and unstructured. The degradation of the [Formula: see text] GLS stabilization on unstructured meshes may be a consequence of inadequate one-dimensional analysis used to derive the stability parameter.

scholarly journals SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 128
Author(s):  
Shahid Hussain ◽  
Afshan Batool ◽  
Md. Al Mahbub ◽  
Nasrin Nasu ◽  
Jiaping Yu
Keyword(s):  
Finite Element ◽  
Fluid Flow ◽  
Viscoelastic Fluid ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Constitutive Law ◽  
Optimal Error Estimates ◽  
Fe Method ◽  
The Stability ◽  
Stability And Accuracy

In this article, a stabilized mixed finite element (FE) method for the Oseen viscoelastic fluid flow (OVFF) obeying an Oldroyd-B type constitutive law is proposed and investigated by using the Streamline Upwind Petrov–Galerkin (SUPG) method. To find the approximate solution of velocity, pressure and stress tensor, we choose lowest-equal order FE triples P 1 - P 1 - P 1 , respectively. However, it is well known that these elements do not fulfill the i n f - s u p condition. Due to the violation of the main stability condition for mixed FE method, the system becomes unstable. To overcome this difficulty, a standard stabilization term is added in finite element variational formulation. The technique is applied herein possesses attractive features, such as parameter-free, flexible in computation and does not require any higher-order derivatives. The stability analysis and optimal error estimates are obtained. Three benchmark numerical tests are carried out to assess the stability and accuracy of the stabilized lowest-equal order feature of the OVFF.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

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