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Chen, Zhiming; Nochetto, Ricardo H.; Schmidt, Alfred

A characteristic Galerkin method with adaptive error control for the continuous casting problem. (English) Zbl 0960.76045

Comput. Methods Appl. Mech. Eng. 189, No. 1, 249-276 (2000).
Summary: The continuous casting problem can be described by a convection-dominated nonlinearly degenerate diffusion equation. We discretize it implicitly in time via the method of characteristics, and in space via continuous piecewise linear finite elements. A posteriori error estimates are derived for the \(L^1\) norm of temperature which exhibit a mild explicit dependence on velocity. The analysis is based on special properties of a linear dual problem in non-divergent form with vanishing diffusion and strong advection. Several simulations with realistic physical parameters illustrate the reliability of the estimators and the flexibility of the proposed adaptive method.

Cited in 9 Documents

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76T99 Multiphase and multicomponent flows
76R50 Diffusion
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
80A22 Stefan problems, phase changes, etc.

Keywords:

degenerate parabolic equations; a posteriori error estimates; continuous casting problem; convection-dominated nonlinearly degenerate diffusion equation; method of characteristics; continuous piecewise linear finite elements; linear dual problem; vanishing diffusion
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References:

[1] Babuska, I.; Rheinboldt, C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754 (1987) · Zbl 0398.65069
[2] Chen, Z.; Jiang, L., Approximation of a two-phase continuous casting Stefan problem, J. Partial Differential Equations, 11, 59-72 (1998) · Zbl 0907.65128
[3] Z. Chen, R.H. Nochetto, A. Schmidt, A posteriori error control and adaptivity for a phase relaxation model, Math. Model. Numer. Anal. (to appear); Z. Chen, R.H. Nochetto, A. Schmidt, A posteriori error control and adaptivity for a phase relaxation model, Math. Model. Numer. Anal. (to appear) · Zbl 0965.65114
[4] Z. Chen, T.M. Shih, X. Yue, Numerical methods for Stefan problems with prescribed convection and nonlinear flux, IMA J. Numer. Anal. 20 (2000) 81-98; Z. Chen, T.M. Shih, X. Yue, Numerical methods for Stefan problems with prescribed convection and nonlinear flux, IMA J. Numer. Anal. 20 (2000) 81-98 · Zbl 0947.65107
[5] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043
[6] Clément, Ph., Approximation by finite element functions using local regularization, RAIRO Anal. Numer., 9, 77-84 (1975) · Zbl 0368.65008
[7] J. Douglas. Jr., T.F. Russell, Numerical methods for convection-dominated diffusion problem based on combining the method of characteristic with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982) 871-885; J. Douglas. Jr., T.F. Russell, Numerical methods for convection-dominated diffusion problem based on combining the method of characteristic with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982) 871-885 · Zbl 0492.65051
[8] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28, 43-77 (1991) · Zbl 0732.65093
[9] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems IV: nonlinear problems, SIAM J. Numer. Anal., 32, 1729-1749 (1995) · Zbl 0835.65116
[10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983; D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983 · Zbl 0562.35001
[11] Grisvard, P., Elliptic Problems on Non-smooth Domains (1985), Pitman: Pitman Boston · Zbl 0695.35060
[12] Kornhuber, R., Monotone Multigrid Methods for nonlinear Variational Problems (1997), Teubner: Teubner Stuttgart · Zbl 0879.65041
[13] O.A. Ladyzenskaja, V. Solonnikov, N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, vol. TMM 23, AMS, Providence, 1968; O.A. Ladyzenskaja, V. Solonnikov, N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, vol. TMM 23, AMS, Providence, 1968 · Zbl 0174.15403
[14] E.J. Laitinen, On the simulation and control of the continuous casting process, Report 43, Math. Dept. Univ. Jyväskylä, 1989; E.J. Laitinen, On the simulation and control of the continuous casting process, Report 43, Math. Dept. Univ. Jyväskylä, 1989
[15] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications I (1972), Springer: Springer Berlin · Zbl 0223.35039
[16] Louhenkilpi, S.; Laitinen, E.; Nieminen, R., Real-time simulation of heat transfer in continuous casting, Metallurgical Trans. B, 24B, 685-693 (1993)
[17] Morton, K. W.; Süli, E., Evolution Galerkin methods and their supra-convergence, Numer. Math., 64, 1097-1122 (1995)
[18] Nochetto, R. H., Error estimates for the two-phase Stefan problems in several space variables II: nonlinear flux conditions, Calcolo, 22, 501-534 (1985) · Zbl 0606.65085
[19] R.H. Nochetto, G. Savaré, C. Verdi, Error control of nonlinear evolution equations, C. R. Acad. Sci. Paris Sér. I Math 263 (1998) 1437-1442; R.H. Nochetto, G. Savaré, C. Verdi, Error control of nonlinear evolution equations, C. R. Acad. Sci. Paris Sér. I Math 263 (1998) 1437-1442 · Zbl 0944.65077
[20] R.H. Nochetto, A. Schmidt, C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp. 69 (2000) 1-24; R.H. Nochetto, A. Schmidt, C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp. 69 (2000) 1-24 · Zbl 0942.65111
[21] R.H. Nochetto, A. Schmidt and C. Verdi, Adaptive solution of phase change problems over unstructured tetrahedral meshes, M. Bern, J.E. Flaherty, M. Luskin (Eds.), Grid Generation and Adaptive Algorithms, IMA Series 113, Springer, 1999, pp. 163-181; R.H. Nochetto, A. Schmidt and C. Verdi, Adaptive solution of phase change problems over unstructured tetrahedral meshes, M. Bern, J.E. Flaherty, M. Luskin (Eds.), Grid Generation and Adaptive Algorithms, IMA Series 113, Springer, 1999, pp. 163-181 · Zbl 0939.65112
[22] Pironneau, O., On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38, 309-332 (1982) · Zbl 0505.76100
[23] Rodrigues, J. F., Obstacle Problems in Mathematical Physics (1987), North-Holland: North-Holland Amsterdam · Zbl 0606.73017
[24] Rodrigues, J. F.; Yi, F., On a two-phase continuous casting Stefan problem with nonlinear flux, Euro. J. Appl. Math., 1, 259-278 (1990) · Zbl 0723.35086
[25] Rulla, J., Weak solutions to Stefan problems with prescribed convection, SIAM J. Math. Anal., 18, 1784-1800 (1987) · Zbl 0642.35080
[26] Siebert, K. G., An a posteriori error estimator for anisotropic refinement, Numer. Math., 73, 373-398 (1996) · Zbl 0873.65098
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