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Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem
Prof. Daniele di Pietro, University of Montpellier
Sala Conferenze IMATI-CNR - Martedì 23 Febbraio 2016 h.15:00
Abstract. We consider in this paper 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.
Our analysis is quite general and is not focused on a specific choice
of the space discretization and of the linearization. As an example,
we apply it to the vertex-centered finite volume (finite element with
mass lumping and quadrature) and Newton methods. Numerical
results illustrate the effectiveness of our estimates and the
performance of the adaptive algorithm.
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