Approximation of a second order Stefan-like problem by means of a finite element method
For the equation \(U_{yy}-U_{\tau}=0\) a Stefan-like problem is treated in which the time derivative of the free boundary depends on its position, on time, on U and on \(U_{yy}\) (the latter two taken at the position of the free boundary). The problem is transformed into one with fixed boundary, and then existence and uniqueness are established (under appropriate assumptions and after introduction of suitable function spaces) by a further transformation into a fixed point equation with contractive operator. The iteration operator is approximated by finite element discretization of the space variable, convergence for refinement of the grid is shown, and an error estimate is given. Finally the author demonstrates the applicability of his method by displaying results of a numerical case study (in which, of course, the time variable too is discretized).
scientific article; zbMATH DE number 3887053