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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Error estimates for the multidimensional two-phase Stefan problem
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by Joseph W. Jerome and Michael E. Rose PDF
Math. Comp. 39 (1982), 377-414 Request permission


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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 377-414
  • MSC: Primary 65M60; Secondary 65M05, 65M10
  • DOI:
  • MathSciNet review: 669635