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Error estimates for a semi-explicit numerical scheme for Stefan-type problems. (English) Zbl 0617.65125

A parabolic problem of the following form is considered \[ (1)\quad \partial /\partial t[a\vartheta +w]-\Delta \vartheta =f \]
\[ (2)\quad w\in \Lambda (\vartheta), \] where a is a positive constant, f is a datum and \(\Lambda\) is a maximal monotone graph. This system contains the (weak formulation of the) Stefan problem as a particular case. Here the problem (1), (2) is approximated by coupling (1) with the relaxed equation \[ (3)\quad \epsilon (\partial w/\partial t)+\Lambda ^{-1}(w)\ni \vartheta \quad (\epsilon: cons\tan t>0). \] The problem (1), (3) is then discretized in time by the semi-explicit scheme \[ (4)\quad a(\vartheta ^ n-\vartheta ^{n-1})/\tau +(w^ n-w^{n-1})/\tau -\Delta \vartheta ^ n=f^ n \]
\[ (5)\quad \epsilon (w^ n-w^{n-1})/\tau +\Lambda ^{-1}(w^ n)\ni \vartheta ^{n-1}; \] a finite element space discretization and quadrature formulae are then introduced. Thus at each time-step (5) is replaced by a finite number of independent algebraic equations, which can be solved with respect to the barycentral values of \(w^ n\); then (4) is reduced to a linear system of algebraic equations having as unknowns the nodal values of \(\vartheta ^ n\). Assuming the condition \(\tau\) /\(\epsilon\leq a\), the fully discrete scheme is stable and its solution converges to that of (1), (2). Error estimates are proved. The results of some numerical experiments are discussed; they show that the present method is faster than other classical procedures.

MSC:

65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R35 Free boundary problems for PDEs
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References:

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