Existence, Uniqueness, Stability, and Monotone Dependence in a Stefan Problem for the Heat Equation

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

“…In this way we have a system of linear equations with respect to a(kAt) corresponding to (2)(3)(4)(5)(6)(7)(8). If we employ the values at t=(k~ Y)At in a(£) of the right hand side of (2-8), we have a forward difference scheme.…”

“…For the later convenience we introduce the following operator P L : (1)(2)(3)(4)(5)(6) then the scheme …”

“…What is left to be proved is that u(x, t) , which is the solution of (1-1) ^(1-3) with the boundary s^^t), satisfies the Stefan condition (1)(2)(3)(4). For that purpose we define the second difference of a*:…”

“…(2-1) 0 Although s(t) is an unknown function of t which should be determined simultaneously together with u(x, £), we compute 5(2^) and n(x, t) alternatively in the actual process of computation by means of a similar technique to the idea of "retarding the argument" by Cannon and Hill [2], and hence we write s n (t) instead of s(t) in order to show explicitly that it is an approximation.…”

“…Then we substitute (2-6) into u of (1-1), multiply 0 f , and integrate over 0<.r<O n (£). Then we have a system of ordinary differential equations (2 -8) at where a,(f) is an (w + 1) -dimensional vector defined by (2)(3)(4)(5)(6)(7)(8)(9) o(0= I…”