Applications of approximate gradient schemes for nonlinear parabolic equations

Robert Eymard; Angela Handlovičová; Raphaèle Herbin; Karol Mikula; Olga Stašová

Applications of Mathematics (2015)

  • Volume: 60, Issue: 2, page 135-156
  • ISSN: 0862-7940

Abstract

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We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion equations.

How to cite

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Eymard, Robert, et al. "Applications of approximate gradient schemes for nonlinear parabolic equations." Applications of Mathematics 60.2 (2015): 135-156. <http://eudml.org/doc/269878>.

@article{Eymard2015,
abstract = {We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion equations.},
author = {Eymard, Robert, Handlovičová, Angela, Herbin, Raphaèle, Mikula, Karol, Stašová, Olga},
journal = {Applications of Mathematics},
keywords = {regularized Perona-Malik equation; gradient schemes; nonlinear parabolic equations; Perona-Malik equation; gradient schemes; nonlinear tensor-diffusion equations; convergence; numerical tests},
language = {eng},
number = {2},
pages = {135-156},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Applications of approximate gradient schemes for nonlinear parabolic equations},
url = {http://eudml.org/doc/269878},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Eymard, Robert
AU - Handlovičová, Angela
AU - Herbin, Raphaèle
AU - Mikula, Karol
AU - Stašová, Olga
TI - Applications of approximate gradient schemes for nonlinear parabolic equations
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 2
SP - 135
EP - 156
AB - We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion equations.
LA - eng
KW - regularized Perona-Malik equation; gradient schemes; nonlinear parabolic equations; Perona-Malik equation; gradient schemes; nonlinear tensor-diffusion equations; convergence; numerical tests
UR - http://eudml.org/doc/269878
ER -

References

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