\(H_ 1\)-estimates of optimal orders for Galerkin methods to one-dimensional Stefan problems
The paper is concerned with a Galerkin approximation of a one-phase, one-dimensional Stefan problem transformed into a nonlinear initial-boundary value problem in a fixed domain. The author uses the boundary-fixing transformation \(x=y/s(\tau)\) coupled with the time transformation \(dt/d\tau=s^2(\tau)\), where \(y\), \(\tau\) are the variables in the original problem and \(s\) describes the free boundary. As result one gets a nonlinear initial boundary value problem (without \(s\)) in the space interval \([0,1]\) and two ordinary differential equations. Furthermore, to obtain a weak formulation, the parabolic equation is multiplied with \(w_{xx}\), where \(w\) belongs to \(H^2(0,1)\). A Galerkin semidiscretization in space (line method) using at least cubic splines is formulated and analyzed for this weak formulation. An a priori error estimate is derived and the optimal convergence order is established in the space \(L_\infty(0,\overline t;H^1(0,1))\).
scientific article; zbMATH DE number 902537