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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 80
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

Full Bibliographic Reference for this paper
, "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", Civil-Comp Press, Stirlingshire, UK, Paper 10, 2004. doi:10.4203/ccp.80.10
Keywords: free boundary, non-linear, multivalued operator, finite elements, adaptive.

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 Crank-Nicholson, 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 obstacle-like 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

where is the energy norm and is the norm; and are discrete solutions on a sufficiently refined mesh.

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 0oC.

The numerical experiment has been developed with the finite element toolbox ALBERT [8], extended with new function basis including bubble functions.

G. Duvaut, "Problèmes a frontiere libre en théorie des milieux continus", Rapport de recherches n^o 185, Laboria I.R.I.A., 1976.
J. L. Lions, "Quelques méthodes de résolution de problemes aux limites non linéaires", Dunod, Paris, 1969.
R. H. Nochetto, A. Schmidt and C. Verdi, "A posteriori error estimation and adaptivity for degenerate parabolic problems", Math. Comp. 69, 1-24, 2000. doi:10.1090/S0025-5718-99-01097-2
E. Bänsch, P.Morin and R.H. Nochetto, "An adaptive Uzawa fem for the Stokes problem: Convergence without the inf-sup condition",SIAM J. Numer. Anal., 40, 1207-1229, 2002. doi:10.1137/S0036142901392134
W. S. B. Paterson, "The physics of glaciers" Butterworth Henemann, Oxford, 103-132, 1994.
U. Spring and K. Hutter, "Conduit flow of a fluid through its solid phase and its application to intraglacial channel flow", Int. J. Engng. Sci., 20, 327-363, 1981. doi:10.1016/0020-7225(82)90029-5
I. N. Smith and W. F. Budd, "The derivation of past climate changes from observed changes of glaciers", IAHS, 131, 31-52, 1981.
A. Schmidt and K.G. Siebert, "ALBERT: An adaptive hierarchical finite element toolbox", Preprint 06/2000, Freiburg, 2000.

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