Error Estimates of a Fully Discrete Linear Approximation Scheme for Stefan Problem

where u : (0, T ) × Ω → < is unknown function, Ω ⊂ < is a polygonal convex domain with the boundary Γ, 0 < T <∞, ν is the outward normal to Γ, f(t, x, s) and ac(t, x) are Lipschitz continuous functions and β : < → < is a nondecreasing Lipschitz continuous function. Finally cg is a real number and cg ≥ 0. There are several linear approximation schemes, deal with the Stefan like problems or with the problems concerning non linear diffusion. Among them linear approximation scheme based on so-called nonlinear Chernoff’s formula with constant relaxation parameter μ have been studied especially in [1], [12], [14], [15], where also some energy error estimates have been investigated. Another linear approximation schemes have been proposed in [5], [6], [7] and [3]. [8] investigates problems with elliptic operator and a nonlinearity also on the boundary (function g(t, x, s)). Jäger-Kačur approximation scheme [5] is of the type