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On one approach to a posteriori error estimates for evolution problems solved by the method of lines. (English) Zbl 0993.65103

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.

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

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