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Mathematics of Computation

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Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Authors: Daniele A. Di Pietro, Martin Vohralík and Soleiman Yousef
Journal: Math. Comp. 84 (2015), 153-186
MSC (2010): Primary 65N08, 65N15, 65N50, 80A22
Published electronically: June 18, 2014
MathSciNet review: 3266956
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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 us 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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Additional Information

Daniele A. Di Pietro
Affiliation: Université Montpellier 2, I3M, 34057 Montpellier, France

Martin Vohralík
Affiliation: UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France – and – CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France
Address at time of publication: INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France

Soleiman Yousef
Affiliation: UPMC Université, Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France – and – IFP Energies nouvelles, Department of Applied Mathematics, 1 & 4 avenue Bois Préau, 92852 Rueil-Malmaison, France

Received by editor(s): April 24, 2012
Received by editor(s) in revised form: April 27, 2013
Published electronically: June 18, 2014
Additional Notes: This work was supported by the ERT project “Enhanced oil recovery and geological sequestration of $\mathrm{CO}_{2}$: mesh adaptivity, a posteriori error control, and other advanced techniques” (LJLL/IFPEN)
The third author is the corresponding author
Article copyright: © Copyright 2014 American Mathematical Society