scholarly journals A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

2016 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epee
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Time Discretization ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Dynamical Boundary Conditions ◽  
Error Indicators ◽  
Crank Nicolson

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

scholarly journals A posteriori error estimates for the Crank--Nicolson method for parabolic equations

2005 ◽  
Vol 75 (254) ◽  
pp. 511-532 ◽  
Author(s):  
Georgios Akrivis ◽  
Charalambos Makridakis ◽  
Ricardo H. Nochetto
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Crank Nicolson

scholarly journals CONVERGENCE AND ERROR ESTIMATES OF TWO ITERATIVE METHODS FOR THE STRONG SOLUTION OF THE INCOMPRESSIBLE KORTEWEG MODEL

2009 ◽  
Vol 19 (09) ◽  
pp. 1713-1742
Author(s):  
F. GUILLÉN-GONZÁLEZ ◽  
M. A. RODRÍGUEZ-BELLIDO
Keyword(s):  
Strong Solution ◽  
Error Estimates ◽  
Fluid Model ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Strong Solutions ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Banach’S Fixed Point Theorem

We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of Banach's fixed point theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.

A posteriori error estimates for Darcy’s problem coupled with the heat equation

2019 ◽  
Vol 53 (6) ◽  
pp. 2121-2159 ◽  
Author(s):  
Séréna Dib ◽  
Vivette Girault ◽  
Frédéric Hecht ◽  
Toni Sayah
Keyword(s):  
Heat Equation ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Three Dimensions ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Nonlinear Viscosity ◽  
Error Indicators ◽  
Variational Formulations

This work derives a posteriori error estimates, in two and three dimensions, for the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We introduce two variational formulations and discretize them by finite element methods. We prove optimal a posteriori errors with two types of computable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we prove upper and lower error bounds under regularity assumptions on the solutions. Finally, numerical computations are performed to show the effectiveness of the error indicators.

scholarly journals A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yuping Zeng ◽  
Kunwen Wen ◽  
Fen Liang ◽  
Huijian Zhu
Keyword(s):  
Heat Equation ◽  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Fully Discrete ◽  
Backward Euler ◽  
Elliptic Reconstruction

We introduce and analyze a weakly overpenalized symmetric interior penalty method for solving the heat equation. We first provide optimal a priori error estimates in the energy norm for the fully discrete scheme with backward Euler time-stepping. In addition, we apply elliptic reconstruction techniques to derive a posteriori error estimators, which can be used to design adaptive algorithms. Finally, we present two numerical experiments to validate our theoretical analysis.

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