Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities

Abstract

AbstractThe change of variable for the temperature Θ in the one‐phase Stefan problem leads to the evolution inequality, (ut – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space‐time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two‐sided weak maximum principle for um – um‐1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of UΔt and ∇UΔt, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates xΔt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.