scholarly journals Error Estimation of Euler Method for the Instationary Stokes–Biot Coupled Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Koffi Wilfrid Houédanou ◽  
Jamal Adetola
Keyword(s):  
Finite Element ◽  
Computational Model ◽  
Error Estimation ◽  
Stokes Equations ◽  
Euler Method ◽  
Error Estimators ◽  
Fully Discrete ◽  
Coupled Problem ◽  
System Equilibrium ◽  
Biot System

In this paper, we study a finite element computational model for solving the interaction between a fluid and a poroelastic structure that couples the Stokes equations with the Biot system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is used to impose weakly this condition. With the obtained finite element solutions, the error estimators are performed for the fully discrete formulations.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

2014 ◽  
Vol 6 (5) ◽  
pp. 615-636 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Stokes Equations ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
Stationary Navier Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

scholarly journals Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-21
Author(s):  
Jae-Hong Pyo
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Boussinesq Equations ◽  
Finite Element Approximation ◽  
Navier Stokes ◽  
Element Approximation ◽  
Uzawa Method ◽  
Element Space ◽  
Fully Discrete ◽  
Discrete Finite Element

The stabilized Gauge-Uzawa method (SGUM), which is a 2nd-order projection type algorithm used to solve Navier-Stokes equations, has been newly constructed in the work of Pyo, 2013. In this paper, we apply the SGUM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We prove that SGUM is unconditionally stable, and we perform error estimations on the fully discrete finite element space via variational approach for the velocity, pressure, and temperature, the three physical unknowns. We conclude with numerical tests to check accuracy and physically relevant numerical simulations, the Bénard convection problem and the thermal driven cavity flow.

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.

Assessment of two a posteriori error estimators for elasticity problems

2008 ◽  
Vol 35 (11) ◽  
pp. 1239-1250
Author(s):  
A. H. ElSheikh ◽  
S. E. Chidiac ◽  
S. Smith
Keyword(s):  
Finite Element ◽  
Error Estimation ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error Estimators ◽  
A Posteriori Error ◽  
Estimation Techniques ◽  
Error Estimators ◽  
Elasticity Problems ◽  
Posteriori Error Estimators

The main focus of this paper is on the evaluation of local a posteriori error estimation techniques for the finite element method (FEM). The standard error estimation techniques are presented for the coupled displacement fields appearing in elasticity problems. The two error estimators, the element residual method (ERM) and Zienkiewicz–Zhu (ZZ) patch recovery technique, are evaluated numerically and then used as drivers for a mesh adaptation process. The results demonstrate the advantages of employing a posteriori error estimators for obtaining finite element solutions with a pre-specified error tolerance. Of the two methods, the ERM is shown to produce adapted meshes that are similar to those adapted by the exact error. Furthermore, the ERM provides higher quality estimates of the error in the global energy norm when compared to the ZZ estimator.

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