Tracking the Interface of the Diffusion-Absorption Equation: Numerical Analysis

Abstract

Our concern in this article is the Cauchy problem for the degenerate parabolic equation of the diffusion-absorption type $u_t=\varDelta u^m-au^q$ with the exponents $m>1$, $q>0$, $m+q\ge 2$ and constant $a>0$. In a previous article (Khedr W.S. Tracking the Interface of the Diffusion-Absorption Equation: Theoretical Analysis) we investigated an algorithm for tracking the moving interface of the above model based on the idea of Shmarev (Nonlinear Anal 53:791–828, 2003; Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, 2005). By means of domain reduction, introduction of local system of Lagrangian coordinates, utilization of the modified Newton method, linearization and regularization we managed to transform the nonlinear problem into a system of linearized equations, and we proved the convergence of the approximated solution of the regularized problem to the solution of the original problem in the weak sense. The introduction of the regularization parameters provided the necessary requirement for a stable numerical implementation of the algorithm. In this article we investigate the numerical error at each Newtonian iteration in terms of the discretization and the regularization parameters. We also try to deduce the minimum order of the finite element’s interpolating polynomials to be used in order to maintain the stability of the algorithm. Finally, we present a number of numerical experiments to validate the algorithm and to investigate its advantages and disadvantages. We will also illustrate the preference of adaptive implementation of the algorithm over direct implementation.