scholarly journals The implementation of the mortar spectral element discretization of the heat equation with discontinuous diffusion coefficient

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mohamed Abdelwahed ◽  
Nejmeddine Chorfi
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Spectral Element ◽  
Element Discretization ◽  
Discontinuous Diffusion

scholarly journals Uniform null controllability for a parabolic equations with discontinuous diffusion coefficient

2021 ◽  
Author(s):  
Jérémi Dardé ◽  
Sylvain Ervedoza ◽  
Roberto Morales
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Parabolic Equations ◽  
Null Controllability ◽  
Wave Operators ◽  
Conductive Medium ◽  
Geometric Conditions ◽  
Discontinuous Diffusion ◽  
Controllability Result

In this article, we study the null-controllability of a heat equation in a domain composed of two media of different constant conductivities. In particular, we are interested in the behavior of the system when the conductivity of the medium on which the control does not act goes to infinity, corresponding at the limit to a perfectly conductive medium. In that case, and under suitable geometric conditions, we obtain a uniform null-controllability result. Our strategy is based on   the analysis of the controllability of the corresponding wave operators and the transmutation technique, which explains the geometric conditions.

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

2014 ◽  
Vol 22 (3) ◽  
pp. 13-36
Author(s):  
Nejmeddine Chorfi ◽  
Mohamed Abdelwahed ◽  
Ines Ben Omrane
Keyword(s):  
Heat Equation ◽  
Spectral Method ◽  
Spectral Element Method ◽  
Spectral Element ◽  
A Posteriori ◽  
Element Discretization ◽  
Optimal Estimates ◽  
Posteriori Analysis ◽  
Error Indicators ◽  
A Posteriori Analysis

AbstractIn this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler’s implicit scheme in time and spectral method in space. We propose two families of error indicators both of them are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

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.

scholarly journals Fractional stochastic heat equation with piecewise constant coefficients

2020 ◽  
Vol 21 (01) ◽  
pp. 2150002
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Mounir Zili ◽  
Eya Zougar
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Divergence Form ◽  
Stochastic Heat Equation ◽  
Piecewise Constant ◽  
Constant Diffusion Coefficient ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Diffusion ◽  
Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

A Space-Time Discontinuous Galerkin Spectral Element Method for Nonlinear Hyperbolic Problems

2018 ◽  
Vol 16 (01) ◽  
pp. 1850093 ◽  
Author(s):  
Chaoxu Pei ◽  
Mark Sussman ◽  
M. Yousuff Hussaini
Keyword(s):  
Discontinuous Galerkin ◽  
Spectral Element Method ◽  
Burgers Equation ◽  
Spectral Element ◽  
Space Time ◽  
Picard Iteration ◽  
Spectral Accuracy ◽  
Element Discretization ◽  
Space And Time ◽  
Element Method

A space-time discontinuous Galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions: (i) adding a space-time dependent artificial viscosity, and (ii) tracking the discontinuity with space-time spectral accuracy. A Picard iteration method is employed to solve nonlinear system of equations derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is demonstrated for the Burgers’ equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. The spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers’ equation obtained by the tracking method.

scholarly journals Fully discrete finite element data assimilation method for the heat equation

2018 ◽  
Vol 52 (5) ◽  
pp. 2065-2082 ◽  
Author(s):  
Erik Burman ◽  
Jonathan Ish-Horowicz ◽  
Lauri Oksanen
Keyword(s):  
Finite Element ◽  
Initial Data ◽  
Heat Equation ◽  
Time Derivative ◽  
Classical Problem ◽  
Classical Case ◽  
Space Time ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Discretization

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty on the H2-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen [Numer. Math. 139 (2018) 505–528], combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from t = 0, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the L2-norm at the final time.

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