Adaptive Regularization, Linearization, and Discretization and a Posteriori Error Control for the Two-Phase Stefan Problem

Abstract eng:
We consider in this paper the time-dependent two-phase Stefan problem. We derive aposteriori error estimates and complete 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 is considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on a L2 (L2) error of the temperature and L2 (H−1) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. A complete adaptive algorithm is proposed, which ensures computational savings through online choice of a sufficient regularization parameter, a stopping criterion on 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. Our analysis is quite general and independent of the numerical method and linearization chosen. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) method and Newton linearization. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.

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