A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems (Advances in Applied Mathematics and Mechanics, 2021)
This article extends the finite element method of lines to a parabolicinitial boundary value problem whose diffusion coefficient is discontinuousacross an interface that changes with respect to time. The method presentedhere uses immersed finite element (IFE) functions for the discretization inspatial variables that can be carried out over a fixed mesh (such as aCartesian mesh if desired), and this feature makes it possible to reducethe parabolic equation to a system of ordinary differential equations (ODE)through the usual semi-discretization procedure. Therefore, with a suitablechoice of the ODE solver, this method can reliably and efficiently solve aparabolic moving interface problem over a fixed structured (Cartesian) mesh.Numerical examples are presented to demonstrate features of this new method.