# 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

top- Z. Chen and R.H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math.84 (2000) 527-548.
- Z. Chen, R.H. Nochetto and A. Schmidt, Adaptive finite element methods for diffuse interface models (in preparation).
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
- Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77-84.
- K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal.28 (1991) 43-77.
- K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal.32 (1995) 1729-1749.
- K. Eriksson, C. Johnson and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups. SIAM J. Numer. Anal.35 (1998) 1315-1325.
- P. Grisvard, Elliptic Problems on Non-smooth Domains. Pitman, Boston (1985).
- X. Jiang and R.H. Nochetto, Optimal error estimates for semidiscrete phase relaxation models. RAIRO Modél. Math. Anal. Numér.31 (1997) 91-120.
- X. Jiang and R.H. Nochetto, A P1-P1 finite element method for a phase relaxation model. I. Quasi uniform mesh. SIAM J. Numer. Anal.35 (1998) 1176-1190.
- X. Jiang, R.H. Nochetto and C. Verdi, A P1-P1 finite element method for a phase relaxation model. II. Adaptively refined meshes. SIAM J. Numér. Anal.36 (1999) 974-999.
- R.H. Nochetto, M. Paolini and C. Verdi, Continuous and semidiscrete traveling waves for a phase relaxation model. European J. Appl. Math.5 (1994) 177-199.
- R.H. Nochetto, G. Savaré and C. Verdi, Error control for nonlinear evolution equations. C.R. Acad. Sci. Paris Sér. I326 (1998) 1437-1442.
- R.H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math.53 (2000) 529-589.
- R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp.69 (2000) 1-24.
- C. Verdi and A. Visintin, Numerical analysis of the multidimensional Stefan problem with supercooling and superheating. Boll. Un. Mat. Ital. B7 (1987) 795-814.
- C. Verdi and A. Visintin, Error estimates for a semi-explicit numerical scheme for Stefan-type problems. Numer. Math.52 (1988) 165-185.
- A. Visintin, Stefan problem with phase relaxation. IMA J. Appl. Math.34 (1985) 225-245.
- A. Visintin, Supercooling and superheating effects in phase transitions. IMA J. Appl. Math.35 (1986) 233-256.

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