Nochetto, R. H.; Paolini, M.; Verdi, C. 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) Cited in 1 ReviewCited in 23 Documents 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. Keywords:mesh refinement; adaptive finite element method; two-phase Stefan problems; triangulation; stability; local refinement strategy; energy error estimates Citations:Zbl 0733.65088 PDFBibTeX XMLCite \textit{R. H. Nochetto} et al., Math. Comput. 57, No. 195, 73--108 (1991; Zbl 0733.65087) Full Text: DOI