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CivilComp Proceedings
ISSN 17593433 CCP: 80
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 10
Adaptive Method for the Stefan Problem and its Application to Endoglacial Conduits F.A. Pérez, L. Ferragut and J.M. Cascón
Departement of Applied Mathematics, University of Salamanca, Spain , "Adaptive Method for the Stefan Problem and its Application to Endoglacial Conduits", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Fourth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 10, 2004. doi:10.4203/ccp.80.10
Keywords: free boundary, nonlinear, multivalued operator, finite elements, adaptive.
Summary
This paper concerns an adaptive finite element method for the
Stefan one phase problem. We derive a parabolic variational
inequality using the Duvaut transformation [1]. This
formulation is an alternative to the considered one in
[2], which is used in the derived a posteriori estimation
in [3]. As an application of a time integration
method, implicit Euler or CrankNicholson, we obtain in each time
step a steady variational inequality. From a computational and
numerical point of view we focus our attention in an adaptive
nonlinear algorithm for solving the obstaclelike problem in each
time step. The adaptive algorithm we construct is based on a
combination of the Uzawa method associated with the corresponding
multivalued operator and a convergent adaptive method for the
linear problem. As it is well known, the Uzawa algorithm consists
in solving in each iteration a linear problem and a nonlinear
adaptation of the Lagrange multiplier associated with the
multivalued operator.As our main result we show that if the
adaptive method for the linear problem is convergent, then the
adaptive modified Uzawa method is convergent as well.
The convergence is proved with respect to a discrete solution in the space corresponding to a sufficiently refined mesh. This is needed to obtain the convergence of the Lagrange multiplier (in the Uzawa method) in the norm as the equivalence of norms in finite dimension is used. In order to assure this result the discrete space of Lagrange multiplier, piecewise constant finite element functions,is extended with bubble functions. We get the following convergence result (the proof is inspired in [4]):
Main result: Let the sequence of finite element solutions of the linear problem and the corresponding Lagrange multiplier produced by the adaptive modified Uzawa algorithm. There exist positive constants and such that As an application of the method described above, we model an endoglacial conduit in which takes place a phase change phenomena. The determination of free boundary is essential to measure the size of the conduit which is related with mass ice loss(see [5,6]). The importance of the endoglacial drainage to describe glacial dynamics is justified by the contribution of changes of glaciers in the study of the climatical evolution (see [7]). Water flowing in a conduit enlarges it by melting ice from the walls. Viscous dissipation in the water and friction of water against the walls produce the necessary heat. In addition, water from the surface may be warmer than 0^{o}C. The numerical experiment has been developed with the finite element toolbox ALBERT [8], extended with new function basis including bubble functions. References
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