Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T03:07:04.340Z Has data issue: false hasContentIssue false
  • English
  • Français

Error Control and Andaptivity for a Phase Relaxation Model

Published online by Cambridge University Press:  15 April 2002

Zhiming Chen ,
Ricardo H. Nochetto  and
Alfred Schmidt
Zhiming Chen
Affiliation:
Institute of Mathematics, Academia Sinica, Beijing 100080, PR China. The first author was partially supported by the National Natural Science Foundation of China under the grant No. 19771080 and China National Key Project "Large Scale Scientific and Engineering Computing" .
Ricardo H. Nochetto
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. Partially supported by NSF Grant DMS-9623394 and NSF SCREMS 9628467.
Alfred Schmidt
Affiliation:
Institut für Angewandte Mathematik, Universität Freiburg, 79104 Freiburg, Germany. Partially supported by DFG and EU Grant HCM "Phase Transitions and Surface Tension" . ([email protected])
Get access

Abstract

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperatureθ and an ODE with double obstaclesfor phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requiresthe stability constraint τ ≤ ε. Conforming piecewiselinear finite elements over highly graded simplicial mesheswith parameter h are further employed for space discretization.A posteriori errorestimates are derived for both unknowns θ and χ, whichexhibit the correct asymptotic order in terms of ε, h andτ. This result circumvents the use of duality, which does noteven apply in this context.Several numerical experiments illustrate the reliability of theestimators and document the excellent performance of the ensuingadaptive method.

Keywords

Phase relaxationdiffuse interfacesubdifferential operatorfinite elementsa posteriori estimatesadaptivity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 775 - 797
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chen, Z. and Nochetto, R.H., Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527-548. CrossRef
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).
Clément, Ph., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84.
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43-77. CrossRef
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729-1749. CrossRef
Eriksson, K., Johnson, C. and Larsson, S., Adaptive finite element methods for parabolic problems. VI. Analytic semigroups. SIAM J. Numer. Anal. 35 (1998) 1315-1325. CrossRef
P. Grisvard, Elliptic Problems on Non-smooth Domains. Pitman, Boston (1985).
Jiang, X. and Nochetto, R.H., Optimal error estimates for semidiscrete phase relaxation models. RAIRO Modél. Math. Anal. Numér. 31 (1997) 91-120. CrossRef
Jiang, X. and Nochetto, R.H., A P 1-P 1 finite element method for a phase relaxation model. I. Quasi uniform mesh. SIAM J. Numer. Anal. 35 (1998) 1176-1190. CrossRef
Jiang, X., Nochetto, R.H. and Verdi, C., A P 1-P 1 finite element method for a phase relaxation model. II. Adaptively refined meshes. SIAM J. Numér. Anal. 36 (1999) 974-999. CrossRef
Nochetto, R.H., Paolini, M. and Verdi, C., Continuous and semidiscrete traveling waves for a phase relaxation model. European J. Appl. Math. 5 (1994) 177-199. CrossRef
Nochetto, R.H., Savaré, G. and Verdi, C., Error control for nonlinear evolution equations. C.R. Acad. Sci. Paris Sér. I 326 (1998) 1437-1442. CrossRef
Nochetto, R.H., Savaré, G. and Verdi, C., A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 529-589. 3.0.CO;2-M>CrossRef
Nochetto, R.H., Schmidt, A. and Verdi, C., A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp. 69 (2000) 1-24. CrossRef
Verdi, C. and Visintin, A., Numerical analysis of the multidimensional Stefan problem with supercooling and superheating. Boll. Un. Mat. Ital. B 7 (1987) 795-814.
Verdi, C. and Visintin, A., Error estimates for a semi-explicit numerical scheme for Stefan-type problems. Numer. Math. 52 (1988) 165-185. CrossRef
Visintin, A., Stefan problem with phase relaxation. IMA J. Appl. Math. 34 (1985) 225-245. CrossRef
Visintin, A., Supercooling and superheating effects in phase transitions. IMA J. Appl. Math. 35 (1986) 233-256. CrossRef