# 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

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topChen, 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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- R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp.69 (2000) 1-24. Zbl0942.65111
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