Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

Sergey Korotov

Applications of Mathematics (2007)

  • Volume: 52, Issue: 3, page 235-249
  • ISSN: 0862-7940

Abstract

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The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow.

How to cite

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Korotov, Sergey. "Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions." Applications of Mathematics 52.3 (2007): 235-249. <http://eudml.org/doc/33286>.

@article{Korotov2007,
abstract = {The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow.},
author = {Korotov, Sergey},
journal = {Applications of Mathematics},
keywords = {a posteriori error estimation; error control in energy norm; two-sided error estimation; differential equation of elliptic type; mixed boundary conditions; error control in energy norm; two-sided error estimation; result verification; linear elliptic (reaction-diffusion) equation; mesh refinement},
language = {eng},
number = {3},
pages = {235-249},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions},
url = {http://eudml.org/doc/33286},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Korotov, Sergey
TI - Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 235
EP - 249
AB - The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow.
LA - eng
KW - a posteriori error estimation; error control in energy norm; two-sided error estimation; differential equation of elliptic type; mixed boundary conditions; error control in energy norm; two-sided error estimation; result verification; linear elliptic (reaction-diffusion) equation; mesh refinement
UR - http://eudml.org/doc/33286
ER -

References

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Citations in EuDML Documents

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  1. Michal Křížek, Hans-Goerg Roos, Wei Chen, Two-sided bounds of the discretization error for finite elements
  2. Michal Křížek, Hans-Goerg Roos, Wei Chen, Two-sided bounds of the discretization error for finite elements
  3. János Karátson, Sergey Korotov, Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems
  4. Ibrahim Cheddadi, Radek Fučík, Mariana I. Prieto, Martin Vohralík, Guaranteed and robust error estimates for singularly perturbed reaction–diffusion problems

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