## 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
Full Text:

### References:

 [1] Berger, A.E., Brezis, H., Rogers, J.C.W.: A numerical method for solving the problemu t ??f(u)=0. RAIRO Anal. Num?r.13, 297-312 (1979) [2] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam. North-Holland 1978 · Zbl 0383.65058 [3] Ciavaldini, J.F.: Analyse num?rique d’un probl?me de Stefan ? deux phases par une m?thode d’?l?ments finis. SIAM J. Numer. Anal.12, 464-487 (1975) · Zbl 0272.65101 · doi:10.1137/0712037 [4] Damlamian, A.: Homogenization for Eddy currents. Delft Prog. Rep.6, 268-275 (1981) · Zbl 0483.35004 [5] Elliott, C.M.: Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal.7, 61-71 (1987) · Zbl 0638.65088 · doi:10.1093/imanum/7.1.61 [6] Jerome, J.W., Rose, M.E.: Error estimates for the multidimensional two-phase Stefan problem. Math. Comput.39, 377-414 (1982) · Zbl 0505.65060 · doi:10.1090/S0025-5718-1982-0669635-2 [7] Magenes, E.: Problemi di Stefan bifase in pi? variabili spaziali. Catania, Le Matematiche36, 65-108 (1981) · Zbl 0545.35096 [8] Meyer, G.H.: Multidimensional Stefan problems. SIAM J. Numer. Anal.10, 522-538 (1973) · Zbl 0256.65054 · doi:10.1137/0710047 [9] Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. (to appear) · Zbl 0655.65131 [10] Raviart, P.A.: The use of numerical integration in finite element methods for solving parabolic equations. In: Topics in numerical analysis. Miller, J.J.H. (ed.), pp. 233-264. New York: Academic Press 1973 · Zbl 0293.65086 [11] Shamir, E.: Regularization of mixed second-order elliptic problems. Isr. J. Math.6, 150-168 (1968) · Zbl 0157.18202 · doi:10.1007/BF02760180 [12] Verdi, C., Visintin, A.: Numerical analysis of the multidimensional Stefan problem with supercooling and superheating. Boll. Unione Mat. Ital. I-B7, 795-814 (1987) · Zbl 0629.65130 [13] Visintin, A.: Stefan problem with phase relaxation. IMA J. Appl. Math.34, 225-245 (1985) · Zbl 0585.35053 · doi:10.1093/imamat/34.3.225 [14] White, R.E.: An enthalpy formulation of the Stefan problem. SIAM J. Numer. Anal.19, 1129-1157 (1982) · Zbl 0501.65058 · doi:10.1137/0719082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.