Stability and Convergence of a Finite Element Method for Solving the Stefan Problem

Abstract

(1-2) «(0,0 =0(0 for 0<t<T.

A Finite Element Method for Solving the Two Phase Stefan Problem in One Space Dimension

“…Finally using Lemma 4, we have the following local maximum principle for the present scheme, the proof of which is exactly the same as that of Lemma 1 in the preceding paper [7]. In view of the boundary condition The other three inequalities can also be verified in the same way with the aid of the following three inequalities:…”

“…In the preceding paper [7] we proposed a finite element scheme for solving the one phase problem and discussed the stability and the convergence of the scheme assuming that the initial data is bounded by a linear function from above. In the one phase problem the free boundary function s(t) is monotone with respect to t, while in the two phase problem s(/) is not monotone in general, so that we need quadratic or some other kind of functions as the bounding functions of the initial data in order that the maximum principle holds with our scheme.…”

“…5) assuming that the initial data are bounded b}-exponential functions under a similar assumption as Assumption D, Various numerical methods have been presented for solving the one phase Stefan problem in one space dimension. See, for example, Douglas and Gallie [4], Meyer [6], Bonnerot and Jamet [1], Nogi [10], Kawarada and Natori [5] and Mori [7]. See also Mori [8] for the numerical solution of the Stefan problem in two space dimension.…”

“…We shall show here only the inequality (1. 20) [7] for the derivatives of (pj (x 9 1) with respect to x and t.…”

“…We expand the approximate solutions tl l and u 2 of and Af l5 A/ 2 are mass matrices, K ly K 2 are stiffness matrices, and N ly N z are velocity matrices [7]. We note that the matrices M lf K^9 N!…”