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.

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 mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms 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.

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 A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jinfeng Wang ◽  
Hong Li ◽  
Siriguleng He ◽  
Wei Gao ◽  
Yang Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Order Equation ◽  
Second Order ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Element Method

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient∇ubelongs to the weaker(L2(Ω))2space taking the place of the classicalH(div;Ω)space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates inL2andH1-norm for both the scalar unknownuand the diffusion termw=−Δuand a priori error estimates in(L2)2-norm for its gradientχ=∇ufor both semi-discrete and fully discrete schemes.

scholarly journals The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement

2017 ◽  
Vol 6 (2) ◽  
pp. 44
Author(s):  
Pongui ngoma Diogene Vianney ◽  
Nguimbi Germain ◽  
Likibi Pellat Rhoss Beaunheur
Keyword(s):  
Finite Element ◽  
Numerical Integration ◽  
Mixed Finite Element ◽  
Sufficient Conditions ◽  
Finite Element Approximation ◽  
Miscible Displacement ◽  
Optimal Order ◽  
Element Approximation ◽  
Quadrature Scheme ◽  
Effect Of Numerical Integration

We consider the effect of numerical integration in finite element  procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index  and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

2014 ◽  
Vol 6 (5) ◽  
pp. 552-569 ◽  
Author(s):  
Wanfang Shen ◽  
Liang Ge ◽  
Danping Yang ◽  
Wenbin Liu
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Mathematical Formulation ◽  
Finite Element Approximation ◽  
A Priori Error Estimates ◽  
Element Approximation ◽  
Weak Formulation ◽  
Priori Error Estimates

AbstractIn this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L2 norms. Furthermore some numerical tests are presented to verify the theoretical results.

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