A characteristic Galerkin method with adaptive error control for the continuous casting problem

Z. Chen
Institute of Mathematics, Academia Sinica, Beijing 100080, PR China

R.H. Nochetto
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

A. Schmidt
Institut für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104 Freiburg, Germany

CNS1_1.gif, 17 kB
Temperature graph from a 2D continuous casting simulation

CNS1_2.gif, 5 kB
Corresponding finite element mesh (vertical zoom by factor 16)

Abstract:
The continuous casting problem is a convection-dominated nonlinearly degenerate diffusion problem. It is discretized 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 $L^1(L^1)$ 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-divergence 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.

Keywords: a posteriori error estimates, continuous casting, method of characteristics, convection dominated diffusion, degenerate parabolic equations
1991 Mathematics subject classification: 65N15, 65N30, 35K60

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