Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements

被引：1
作者：
Javier de Frutos
Bosco García-Archilla
Volker John
Julia Novo
机构：
[1] Universidad de Valladolid,Instituto de Investigación en Matemáticas (IMUVA)
[2] Universidad de Sevilla,Departamento de Matemática Aplicada II
[3] Leibniz Institute in Forschungsverbund Berlin e.V. (WIAS),Weierstrass Institute for Applied Analysis and Stochastics
[4] Free University of Berlin,Department of Mathematics and Computer Science
[5] Universidad Autónoma de Madrid,Departamento de Matemáticas
来源：
Journal of Scientific Computing | 2016年 / 66卷
关键词：
Time-dependent Oseen equations; Inf-sup stable pairs of finite element spaces; Grad-div stabilization; Backward Euler scheme; Two-step backward differentiation scheme (BDF2); Crank–Nicolson scheme; Uniform error estimates;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank–Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.
引用
收藏
页码：991 / 1024
页数：33
相关论文
共 52 条
[1]  
Ayuso B(2005)The postprocessed mixed finite-element method for the Navier–Stokes equations SIAM J. Numer. Anal. 43 1091-1111
[2]  
García-Archilla B(2012)-LU factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems Int. J. Numer. Methods Fluids 68 83-98
[3]  
Novo J(2014)Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization Comput. Methods Appl. Mech. Eng. 275 1-19
[4]  
Börm S(2007)Stabilized finite element methods for the generalized Oseen problem Comput. Methods Appl. Mech. Eng. 196 853-866
[5]  
Le Borne S(2011)A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations SIAM J. Numer. Anal. 49 1461-1481
[6]  
Bowers AL(2002)Stabilized finite element approximation of transient incompressible flows using orthogonal subscales Comput. Methods Appl. Mech. Eng. 191 4295-4321
[7]  
Le Borne S(2008)Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales Appl. Numer. Math. 58 264-283
[8]  
Rebholz LG(2007)Time dependent subscales in the stabilized finite element approximation of incompressible flow problems Comput. Methods Appl. Mech. Eng. 196 2413-2430
[9]  
Braack M(2005)Stabilized finite element schemes with LBB-stable elements for incompressible flows J. Comput. Appl. Math. 177 243-267
[10]  
Burman E(2013)Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization Int. J. Numer. Methods Fluids 71 118-134
123456