A Fully Discrete Adaptive Nonlinear Chernoff Formula

Abstract

An adaptive piecewise linear finite element approximation to a linear method, the so-called nonlinear Chernoif formula, for the simplest two-phase Stefan problem in two dimensions are discussed. The full discretization amounts to solving linear positive definite symmetric systems, followed by an explicit nodewise algebraic correction. Consecutive meshes are highly graded and noncompatible. The pointwise information needed to generate a new mesh is extracted from both the discrete enthalpy $U^n $ and temperature $\Theta ^n $. For well-behaved discrete transition layers $\mathcal{T}^n = \{ 0 < U^n - \Theta ^n < 1\} $, triangles are expected to be $\mathcal{O}(\tau )$ within a thin strip of width $\mathcal{O}(\tau ^{{1 / 2}} )$, the so-called refined region, which in turn must contain the transition layer. The local meshsize increases up to $\mathcal {O}(\tau ^{1/2} )$ elsewhere; here $\tau $ stands for the uniform time-step. The expected number of spatial degrees of freedom becomes $\mathcal {O}(\tau ^{ - 3/2} )$, which compares quite favorably with that required without adaptivity, namely $\mathcal {O}(\tau ^{ - 2} )$. Stability is examined in a number of Sobolev norms and then is used to derive a quasi-optimal rate of convergence of order essentially $\mathcal {O}(\tau ^{1/2} )$ in the natural energy spaces. Numerical experiments illustrate the performance of the proposed method.