An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

Abstract

Abstract.This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of 0(h1/2) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown. Introduction.The aim of this paper is to analyze from both a theoretical and computational viewpoint the performance of a linear scheme to approximate the following multidimensional parabolic problem:

Error Estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

“…see, e.g [22,23]. It is easy to see that θ(s) := L α s + α(s) is Lipschitz continuous and monotone, which follows from…”

The use of linear approximation scheme for solving the Stefan problem

“…The scheme (8)-(13) (Scheme 1) was also compared with the linear approximation schemes introduced in [4] (Scheme 2) and in [7] (Scheme 3). We set…”

“…Nochetto and Verdi in [7] used a linear approximation scheme to approximate the singular parabolic problem by a linear discrete problem. Kačur, Handlovičová and Kačurová in [4] replaced the parameter µ in the linear approximation scheme by a function µ(x).…”

A linear scheme to approximate nonlinear cross-diffusion systems

“…This scheme consists of solving a linear elliptic equation and updating Z. Many authors investigated this scheme because of its effectiveness (e.g., [10,11,14,18,19,21,26]). This scheme can be implemented very easily.…”