Finite Element Scheme with Crank–Nicolson Method for Parabolic Nonlocal Problems Involving the Dirichlet Energy

2016 ◽  
Vol 14 (05) ◽  
pp. 1750053
Author(s):  
Sudhakar Chaudhary ◽  
Vimal Srivastava ◽  
V. V. K. Srinivas Kumar
Keyword(s):  
Finite Element ◽  
A Priori ◽  
Parabolic Problems ◽  
Dirichlet Energy ◽  
Weak Formulation ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Crank Nicolson

In this paper, we present a finite element scheme with Crank–Nicolson method for solving nonlocal parabolic problems involving the Dirichlet energy. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

scholarly journals Crank-Nicolson Fully DiscreteH1-Galerkin Mixed Finite Element Approximation of One Nonlinear Integrodifferential Model

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Fengxin Chen
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Time Derivative ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Crank Nicolson

We consider a fully discreteH1-Galerkin mixed finite element approximation of one nonlinear integrodifferential model which often arises in mathematical modeling of the process of a magnetic field penetrating into a substance. We adopt the Crank-Nicolson discretization for time derivative. Optimal order a priori error estimates for the unknown function inL2andH1norm and its gradient function inL2norm are presented. A numerical example is given to verify the theoretical results.

Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

2015 ◽  
Vol 53 (1-2) ◽  
pp. 413-443 ◽  
Author(s):  
Vimal Srivastava ◽  
Sudhakar Chaudhary ◽  
V. V. K. Srinivas Kumar ◽  
Balaji Srinivasan
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Dirichlet Energy ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Discrete Finite Element

scholarly journals Convergence Analysis of H(div)-Conforming Finite Element Methods for a Nonlinear Poroelasticity Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yuping Zeng ◽  
Zhifeng Weng ◽  
Fen Liang
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
A Priori ◽  
Mixed Finite Elements ◽  
Elastic Displacement ◽  
Fully Discrete ◽  
Conforming Finite Element ◽  
Priori Error Estimates ◽  
Galerkin Formulation ◽  
Flow Variables

In this paper, we introduce and analyze H(div)-conforming finite element methods for a nonlinear model in poroelasticity. More precisely, the flow variables are discretized by H(div)-conforming mixed finite elements, while the elastic displacement is approximated by the H(div)-conforming finite element with the interior penalty discontinuous Galerkin formulation. Optimal a priori error estimates are derived for both semidiscrete and fully discrete schemes.

scholarly journals AN H1 -GALERKIN MIXED FINITE ELEMENT APPROXIMATION OF A NONLOCAL HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (5) ◽  
pp. 643-653
Author(s):  
Fengxin Chen ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Hyperbolic Equation ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Nonlinear Hyperbolic Equation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

2017 ◽  
Vol 08 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Yaxin Hou ◽  
Ruihan Feng ◽  
Yang Liu ◽  
Hong Li ◽  
Wei Gao
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Diffusion System ◽  
Fully Discrete ◽  
Coupled Diffusion ◽  
L1 Approximation ◽  
Priori Error Estimates ◽  
The Stability ◽  
Discrete Finite Element

In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

scholarly journals Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Kening Wang ◽  
Shuang Li
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Approximate Solution ◽  
Smooth Boundary ◽  
Model Problem ◽  
Parabolic Problems ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Two Space Dimensions ◽  
Superconvergence Results

We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems on , where is a bounded domain in with piecewise smooth boundary. We establish the global two order superconvergence results for the error between the approximate solution and the Ritz projection of the exact solution of our model problem in and with and the almost two order superconvergence in and . Results of the case are also included in two space dimensions ( or 2). By applying the interpolated postprocessing technique, similar results are also obtained on the error between the interpolation of the approximate solution and the exact solution.

scholarly journals Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mouhamadou Samsidy Goudiaby ◽  
Ababacar Diagne ◽  
Leon Matar Tine
Keyword(s):  
Finite Element ◽  
Liquid Crystal ◽  
Type Inequality ◽  
Mesh Size ◽  
Time Step ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Fully Discrete ◽  
Discrete Finite Element ◽  
Longtime Behavior

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

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