Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Comparison of Two Time-Integration Algorithms for an Anisotropic Damage Model Coupled with Plasticity

2015 ◽  
Vol 784 ◽  
pp. 292-299 ◽  
Author(s):  
Stephan Wulfinghoff ◽  
Marek Fassin ◽  
Stefanie Reese
Keyword(s):  
Evolution Equations ◽  
Damage Model ◽  
Time Integration ◽  
Three Dimensional ◽  
Anisotropic Damage ◽  
Euler Scheme ◽  
The Finite Element Method ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Integration Algorithms

In this work, two time integration algorithms for the anisotropic damage model proposed by Lemaitre et al. (2000) are compared. Specifically, the standard implicit Euler scheme is compared to an algorithm which implicitly solves the elasto-plastic evolution equations and explicitly computes the damage update. To this end, a three dimensional bending example is solved using the finite element method and the results of the two algorithms are compared for different time step sizes.

scholarly journals A posteriori analysis of the spectral element discretization of a non linear heat equation

2020 ◽  
Vol 10 (1) ◽  
pp. 477-493
Author(s):  
Mohamed Abdelwahed ◽  
Nejmeddine Chorfi
Keyword(s):  
Heat Equation ◽  
Spectral Element ◽  
A Posteriori ◽  
Element Discretization ◽  
Linear Heat Equation ◽  
Linear Heat ◽  
Non Linear ◽  
Posteriori Analysis ◽  
Error Indicators ◽  
A Posteriori Analysis

Abstract The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Implicit Euler scheme for an abstract evolution inequality

2011 ◽  
Vol 47 (8) ◽  
pp. 1130-1138 ◽  
Author(s):  
R. Z. Dautov ◽  
A. I. Mikheeva
Keyword(s):  
Euler Scheme ◽  
Implicit Euler Scheme ◽  
Evolution Inequality

scholarly journals Null controllability of the heat equation in pseudo-cylinders by an internal control

2020 ◽  
Vol 26 ◽  
pp. 122
Author(s):  
Jon Asier Bárcena-Petisco
Keyword(s):  
Heat Equation ◽  
Internal Control ◽  
Cylindrical Part ◽  
Null Controllability ◽  
Carleman Estimate ◽  
Main Difficulty ◽  
Absorption Properties ◽  
Weighted Function ◽  
Cylindrical Structure ◽  
Linear Heat

In this paper we prove the null controllability of the heat equation in domains with a cylindrical part and limited by a Lipschitz graph. The proof consists mainly on getting a Carleman estimate which presents the usual absorption properties. The main difficulty we face is the loss of existence of the usual weighted function in C2 smooth domains. In order to deal with this, we use its cylindrical structure and approximate the system by the same system stated in regular domains. Finally, we show some applications like the controllability of the semi-linear heat equation in those domains.

Controllability Property of a Superlinear Climate System

2013 ◽  
Vol 367 ◽  
pp. 264-269
Author(s):  
Liang Zhang ◽  
Yang Liu
Keyword(s):  
Fixed Point ◽  
Parabolic Equation ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Point Of View ◽  
Climate System ◽  
Linear Parabolic Equation ◽  
Controllability Property

This work concerns a climate system in the point of view of controllability. We obtain by the Kakutani’s fixed point theorem and the controllability property of the linear parabolic equation that the superlinear climate system is null controllable in the case with interior control.

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