Martin Vohralík
INRIA Paris-Rocquencourt

Adaptive regularization, linearization, and discretization for the two-phase Stefan problem

joint work with Daniele A. Di Pietro and Soleiman Yousef
We consider the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L^2(L^2) error of the temperature and the L^2(H^{-1}) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Numerical results illustrate the performance of the adaptive algorithm.