scholarly journals A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems

2013 ◽  
Vol 51 (2) ◽  
pp. 773-793 ◽  
Author(s):  
Vít Dolejší ◽  
Alexandre Ern ◽  
Martin Vohralík
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Advection Diffusion ◽  
Nonlinear Advection ◽  
Diffusion Problems

scholarly journals Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

2016 ◽  
Vol 16 (4) ◽  
pp. 609-631 ◽  
Author(s):  
Immanuel Anjam ◽  
Dirk Pauly
Keyword(s):  
Error Control ◽  
A Posteriori Error Estimates ◽  
Reaction Diffusion ◽  
Posteriori Error ◽  
Diffusion Problem ◽  
Mixed Formulation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Linear Partial Differential Equations ◽  
Primal And Dual

AbstractThe results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation ${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$, where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$where ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation ${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 1119-1139 ◽  
Author(s):  
RODOLFO ARAYA ◽  
ABNER H. POZA ◽  
ERNST P. STEPHAN
Keyword(s):  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Diffusion Reaction ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Advection Diffusion ◽  
Norm Of The Error ◽  
Reaction Problem

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

Pointwise a posteriori error control for elliptic obstacle problems

2003 ◽  
Vol 95 (1) ◽  
pp. 163-195 ◽  
Author(s):  
Ricardo H. Nochetto ◽  
Kunibert G. Siebert ◽  
Andreas Veeser
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
Obstacle Problems ◽  
A Posteriori ◽  
A Posteriori Error

scholarly journals A Posteriori Error Control for Fully Discrete Crank--Nicolson Schemes

2012 ◽  
Vol 50 (6) ◽  
pp. 2845-2872 ◽  
Author(s):  
E. Bänsch ◽  
F. Karakatsani ◽  
Ch. Makridakis
Keyword(s):  
Error Control ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Fully Discrete ◽  
Crank Nicolson
Sign in / Sign up

Export Citation Format

Share Document