Error estimates for two-phase Stefan problems in several space variables. I: Linear boundary conditions
The enthalpy formulation of two-phase Stefan problems, with linear boundary conditions, is approximated by \(C^ 0\)-piecewise linear finite elements in space and backward-differences in time combined with a regularization procedure. Error estimates of \(L^ 2\)-type are obtained. For general regularized problems an order \(\epsilon^{1/2}\) is proved, while the order is shown to be \(\epsilon\) for non-degenerate cases. For discrete problems an order \(h^ 2\epsilon^{-1}+h+\tau \epsilon^{- }+\tau^{2/3}\) is obtained. These orders impose the relations \(\epsilon \sim \tau \sim h^{4/3}\) for the general case and \(\epsilon \sim h\sim \tau^{2/3}\) for non-degenerate problems, in order to obtain rates of convergence \(h^{2/3}\) or h respectively. Besides, an order \(h| \log h| +\tau^{1/2}\) is shown for finite element meshes with certain approximation property. Also continuous dependence of discrete solutions upon the data is proved.
scientific article; zbMATH DE number 3980429
scientific article