A $P^1-P^0$ finite element method for a model of polymer crystallization

Comput. Methods Appl. Mech. Engrg., 125 (1995), 303-317.

Xun Jiang
Department of Mathematics
University of California at Davis
Davis, CA 95616, USA


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


Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy



We consider a practical finite element approximation of a 3D model for the crystallization of polymers. The model is a system consisting of a parabolic PDE for the thermal balance coupled with several nonlinear ODEs for the crystallization kinetics. The isokinetic assumption implies a non-Lipschitz continuous dependence of the kinetic equations on the crystalline volume fraction. Piecewise linear elements are used for temperature and piecewise constants for the kinetic variables. The numerical algorithm is simple, easy to implement on a computer, and a linear system with the same symmetric positive definite matrix has to be solved per time step. We prove optimal linear $L^\infty L^1$ a priori error estimates in terms of both discretization parameters, using monotonicity and $L^1$ techniques. A relevant simulation in 3D with axial symmetry shows qualitative agreement of the mathematical model with experimental results.