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

A nonlinear two-phase Stefan problem with melting point gradient: a constructive approach

“…Example 1 (see Fig. 1) was taken from Mori [4]. In our notation, the values used are m(t) = 2, 4,(x) = +2(x) = 1 -x, f(t) = 1, h(t) = -+(cos(at) + +) for t E [O,l], h(t) = -+ for larger t, b = 1, c = 2, pi = p2 = 1, y1 = 0.1, y2 = 0.3, & = p2 = 0 (that is, the melting temperature is zero independent of the pressure), kf = 0.3, kz = 0.1.…”

“…Concerning the large literature on the Stefan problem, we mention only some papers which we consider particularly relevant for our purpose, namely those of Friedman [2], Rubinstein [6], MacDonald and Ness [3], O'Connell and Wasserburg [5], and Mori [4]. Rubinstein provides a broadly based review of the history of the Stefan problem.…”

“…Papers [3] and [5] elucidate the geophysical importance of the problem: in them, the set of partial differential equations has been integrated numerically. Mori [4] treats the numerical solution of a similar problem in great detail. His numerical example provides a convenient test problem for our method.…”

Moving Free Surface Heat Transfer Analysis by Continuously Deforming Finite Elements

“…Continuously deforming finite-element mesh systems have been used [9] two-dimensional Stefan problems [12], the time derivatives of the dependent variables in a deforming mesh system produce additional (convective) terms arising from the deformation of the mesh system itself. This approach was further developed and applied to the shallow water problem [I31 and to the one-dimensional Stefan problem [141.…”