scholarly journals Error analysis of a fully discrete finite element method for incompressible flow with variable density

2020 ◽  
Author(s):  
Wentao Cai ◽  
Buyang Li ◽  
Ying Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Incompressible Flow ◽  
Stokes Equations ◽  
Coupled System ◽  
Variable Density ◽  
Spatial Error ◽  
Fully Discrete ◽  
Stabilized Finite Element Method ◽  
Element Method

An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density. In particular, the error of the numerical solution is split into the temporal and spatial components, separately. The temporal error is estimated by applying discrete maximal L^p-regularity of time-dependent Stokes equations, and the spatial error is estimated by using energy techniques based on the uniform regularity of the solutions given by semi-discretization in time.

CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

2007 ◽  
Vol 17 (03) ◽  
pp. 439-459 ◽  
Author(s):  
LEOPOLDO P. FRANCA ◽  
SAULO P. OLIVEIRA ◽  
MARCUS SARKIS
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Least Squares Method ◽  
Approximation Properties ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Bubble Functions ◽  
Velocity Pressure ◽  
Element Method

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

scholarly journals An Oseen Two-Level Stabilized Mixed Finite-Element Method for the 2D/3D Stationary Navier-Stokes Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Aiwen Wang ◽  
Xin Zhao ◽  
Peihua Qin ◽  
Dongxiu Xie
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mesh Size ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Element Method

We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e.,Q1−P0andP1−P0). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh sizeH, a large general Stokes equation on the fine mesh with mesh sizeh=O(H)2. The Oseen two-level stabilized finite-element method provides an approximate solution (uh,ph) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh sizeh. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

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