scholarly journals A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epée
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Product Formula ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Nicolson Scheme ◽  
Crank Nicolson

The aim of this work is to study a semidiscrete Crank-Nicolson type scheme in order to approximate numerically the Dirichlet-to-Neumann semigroup. We construct an approximating family of operators for the Dirichlet-to-Neumann semigroup, which satisfies the assumptions of Chernoff’s product formula, and consequently the Crank-Nicolson scheme converges to the exact solution. Finally, we write aP1finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.

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.

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.

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