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An adaptive finite element method for two-phase Stefan problems in two space dimensions. I: Stability and error estimates. (English) Zbl 0733.65087

Math. Comput. 57, No. 195, 73-108 (1991); Supplement S1-S11 (1991).
[For part II see SIAM J. Sci. Stat. Comput. 12, No.5, 1207-1244 (1991; Zbl 0733.65088.]
The authors describes an adaptive finite element method for two-phase Stefan problems in two space dimensions. A coarse triangulation is chosen away from the interface, and the mesh is refined near the interface. This is the first part of an analysis, which in great detail analyzes the title problem. A large portion of the paper is devoted to stability and error estimates. The paper offers a local refinement strategy, interpolation estimates for noncompatible meshes, a stability analysis, and energy error estimates. Finally, computational issues are discussed. The paper is theoretical in nature.
Reviewer: E.Krause (Aachen)

MSC:

65Z05 Applications to the sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35R35 Free boundary problems for PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
80A22 Stefan problems, phase changes, etc.

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Zbl 0733.65088
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