Babuška, Ivo; Feistauer, Miloslav; Šolín, Pavel On one approach to a posteriori error estimates for evolution problems solved by the method of lines. (English) Zbl 0993.65103 Numer. Math. 89, No. 2, 225-256 (2001). The authors describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of the author’s goals is to apply known estimates derived for elliptic problems to evolution equations. The new technique is applied to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second-order hyperbolic problem. The error is measured with the aid of the \(L^{2}\)-norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented. Reviewer: Violeta A.Kostova (Russe) Cited in 13 Documents MSC: 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35L15 Initial value problems for second-order hyperbolic equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 35K05 Heat equation Keywords:a posteriori error estimates; finite elements; method of lines; convection-diffusion problem; nonlinear parabolic problem; second-order hyperbolic problem; numerical examples; evolution equations Software:DASSL PDFBibTeX XMLCite \textit{I. Babuška} et al., Numer. Math. 89, No. 2, 225--256 (2001; Zbl 0993.65103) Full Text: DOI