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Error estimates for two-phase stefan problems in several space variables, II: Non-linear flux conditions

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Abstract

In this paper we derive L2-error estimates for multidimensional two-phase Stefan problems involving further non-linearities such as non-linear flux conditions. We approximate the enthalpy formulation by a regularization procedure combined with a C0-piecewise linear finite element scheme in space, and the implicit Euler scheme in time.

Under some restrictions on the initial datum and on the finite element mesh, we obtain an L2-rate of convergence of order ε1/2 for regularized problems, and an L2-rate of convergence of order (h|log h|k+τ+ε)1/2 for the discrete problems (k=0,1). These estimates lead to the choice h∼ε∼τ and yeild a global L2-rate essentially of order h1/2.

The a priori relationship between the approximation parameters allows the discrete problem to satisfy a maximum principle (without a lumping procedure); from which a priori bounds and monotonicity properties of the discrete solutions are shown.

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Authors and Affiliations

  1. Instituto de Desarrollo Tecnológico para la Industria Química, Programa Especial de Matemática Aplicada, Santa Fe, (Argentina)

    Ricardo H. Nochetto

  2. Istituto di Analisi Numerica del C.N.R., Pavia, (Italy)

    Ricardo H. Nochetto

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  1. Ricardo H. Nochetto
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This work was supported by the “Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina”.

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Nochetto, R.H. Error estimates for two-phase stefan problems in several space variables, II: Non-linear flux conditions. Calcolo 22, 501–534 (1985). https://doi.org/10.1007/BF02575899

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