Error Control and Andaptivity for a Phase Relaxation Model

Zhiming Chen; Ricardo H. Nochetto; Alfred Schmidt

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 4, page 775-797
  • ISSN: 0764-583X

Abstract

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The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperature θ and an ODE with double obstacles for phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requires the stability constraint τ ≤ ε. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori error estimates are derived for both unknowns θ and χ, which exhibit the correct asymptotic order in terms of ε, h and τ. This result circumvents the use of duality, which does not even apply in this context. Several numerical experiments illustrate the reliability of the estimators and document the excellent performance of the ensuing adaptive method.

How to cite

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Chen, Zhiming, Nochetto, Ricardo H., and Schmidt, Alfred. "Error Control and Andaptivity for a Phase Relaxation Model." ESAIM: Mathematical Modelling and Numerical Analysis 34.4 (2010): 775-797. <http://eudml.org/doc/197455>.

@article{Chen2010,
abstract = { The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperature θ and an ODE with double obstacles for phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requires the stability constraint τ ≤ ε. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori error estimates are derived for both unknowns θ and χ, which exhibit the correct asymptotic order in terms of ε, h and τ. This result circumvents the use of duality, which does not even apply in this context. Several numerical experiments illustrate the reliability of the estimators and document the excellent performance of the ensuing adaptive method. },
author = {Chen, Zhiming, Nochetto, Ricardo H., Schmidt, Alfred},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Phase relaxation; diffuse interface; subdifferential operator; finite elements; a posteriori estimates; adaptivity.; phase relaxation; finite elements; semi-explicit Euler method; adaptivity; variable step-size; stability; a posteriori error estimates; numerical experiments; performance},
language = {eng},
month = {3},
number = {4},
pages = {775-797},
publisher = {EDP Sciences},
title = {Error Control and Andaptivity for a Phase Relaxation Model},
url = {http://eudml.org/doc/197455},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Chen, Zhiming
AU - Nochetto, Ricardo H.
AU - Schmidt, Alfred
TI - Error Control and Andaptivity for a Phase Relaxation Model
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 4
SP - 775
EP - 797
AB - The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperature θ and an ODE with double obstacles for phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requires the stability constraint τ ≤ ε. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori error estimates are derived for both unknowns θ and χ, which exhibit the correct asymptotic order in terms of ε, h and τ. This result circumvents the use of duality, which does not even apply in this context. Several numerical experiments illustrate the reliability of the estimators and document the excellent performance of the ensuing adaptive method.
LA - eng
KW - Phase relaxation; diffuse interface; subdifferential operator; finite elements; a posteriori estimates; adaptivity.; phase relaxation; finite elements; semi-explicit Euler method; adaptivity; variable step-size; stability; a posteriori error estimates; numerical experiments; performance
UR - http://eudml.org/doc/197455
ER -

References

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  15. R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp.69 (2000) 1-24.  
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