scholarly journals A fully discrete finite element scheme for the Kelvin-Voigt model

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5813-5827 ◽  
Author(s):  
Xiaoli Lu ◽  
Lei Zhang ◽  
Pengzhan Huang
Keyword(s):  
Finite Element ◽  
Time Discretization ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Step Method ◽  
Fully Discrete ◽  
Space Discretization ◽  
Voigt Model ◽  
Discrete Finite Element ◽  
Present Algorithm

In this paper, we study convergence of a fully discrete scheme for the two-dimensional nonstationary Kelvin-Voigt model. This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization, which is a two step method. Moreover, we obtain error estimates of velocity and pressure. At last, the applicability and effectiveness of the present algorithm are illustrated by numerical experiments.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 273-292 ◽  
Author(s):  
Thirupathi Gudi ◽  
Papri Majumder
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Time Discretization ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Backward Euler ◽  
Space Discretization ◽  
Speed Of Propagation

AbstractWe introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

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 Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Cheng Fang ◽  
Yuan Li
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Two Dimensional ◽  
Stabilized Finite Element ◽  
Fully Discrete ◽  
Stabilized Finite Element Method ◽  
Discrete Finite Element

This paper presents fully discrete stabilized finite element methods for two-dimensional Bingham fluid flow based on the method of regularization. Motivated by the Brezzi-Pitkäranta stabilized finite element method, the equal-order piecewise linear finite element approximation is used for both the velocity and the pressure. Based on Euler semi-implicit scheme, a fully discrete scheme is introduced. It is shown that the proposed fully discrete stabilized finite element scheme results in the h1/2 error order for the velocity in the discrete norms corresponding to L2(0,T;H1(Ω)2)∩L∞(0,T;L2(Ω)2).

Well-posedness of an electric interface model and its finite element approximation

2016 ◽  
Vol 26 (03) ◽  
pp. 601-625 ◽  
Author(s):  
Habib Ammari ◽  
Dehan Chen ◽  
Jun Zou
Keyword(s):  
Finite Element ◽  
Electric Fields ◽  
Finite Element Approximation ◽  
Interface Problem ◽  
Element Approximation ◽  
Well Posedness ◽  
Fully Discrete ◽  
Potential Applications ◽  
Shaped Pulse ◽  
Discrete Finite Element

This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on the physical media that have a heterogeneous permittivity and a heterogeneous conductivity. Well-posedness of the model interface problem and the regularity of its solutions are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and space simultaneously for an arbitrary-shaped pulse, and it is demonstrated to enjoy the optimal convergence order in both space and time. The new results and numerical scheme have potential applications in the fields of electromagnetism, medicine, food sciences, and biotechnology.

Fully discrete finite element approximation of the stochastic Cahn–Hilliard–Navier–Stokes system

2020 ◽  
Author(s):  
G Deugoué ◽  
B Jidjou Moghomye ◽  
T Tachim Medjo
Keyword(s):  
Finite Element ◽  
Stokes System ◽  
Finite Element Approximation ◽  
Navier Stokes ◽  
Element Approximation ◽  
Polygonal Domain ◽  
Martingale Solution ◽  
Fully Discrete ◽  
Discretization Step ◽  
Discrete Finite Element

Abstract In this paper we study the numerical approximation of the stochastic Cahn–Hilliard–Navier–Stokes system on a bounded polygonal domain of $\mathbb{R}^{d}$, $d=2,3$. We propose and analyze an algorithm based on the finite element method and a semiimplicit Euler scheme in time for a fully discretization. We prove that the proposed numerical scheme satisfies the discrete mass conservative law, has finite energies and constructs a weak martingale solution of the stochastic Cahn–Hilliard–Navier–Stokes system when the discretization step (both in time and in space) tends to zero.

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