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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Error estimates for the multidimensional two-phase Stefan problem

Authors: Joseph W. Jerome and Michael E. Rose
Journal: Math. Comp. 39 (1982), 377-414
MSC: Primary 65M60; Secondary 65M05, 65M10
MathSciNet review: 669635
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Abstract: In this paper we derive rates of convergence for regularizations of the multidimensional two-phase Stefan problem and use the regularized problems to define backward-difference in time and $ {C^0}$ piecewise-linear in space Galerkin approximations. We find an $ {L^2}$ rate of convergence of order $ \sqrt \varepsilon $ in the $ \varepsilon $-regularization and an $ {L^2}$ rate of convergence of order $ ({h^2}/\varepsilon + \Delta t/\sqrt \varepsilon )$ in the Galerkin estimates which leads to the natural choices $ \varepsilon \sim {h^{4/3}}$, $ \Delta t \sim {h^{4/3}}$, and a resulting $ O({h^{2/3}})\;{L^2}$ rate of convergence of the numerical scheme to the solution of the differential equation. An essentially $ O(h)$ rate is demonstrated when $ \varepsilon = 0$ and $ \Delta t \sim {h^2}$ in our Galerkin scheme under a boundedness hypothesis on the Galerkin approximations. The latter result is consistent with computational experience.

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Article copyright: © Copyright 1982 American Mathematical Society