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

Comput. Methods Appl. Mech. Engrg., 189 (2000), 249-276.

Zhiming Chen
Institute of Mathematics
Academia Sinica
Beijing 100080, P.R. China


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


Alfred Schmidt
Institut fur Angewandte Mathematik
Universitat Freiburg
7800 Freiburg, Germany



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 $L^1L^1$ 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.