Abstract
As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.
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Lubich, C., Nevanlinna, O. On resolvent conditions and stability estimates. BIT 31, 293–313 (1991). https://doi.org/10.1007/BF01931289
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DOI: https://doi.org/10.1007/BF01931289