Abstract
We examine single step time discrete approximations to an abstract Cauchy problem considered in a pair of Banach spaces, of which one (space of solutions) is densely embedded in the other (space of initial data). The stability and error analysis of such discretizations is carried out. Our approach is applicable to the analysis of approximations to parabolic PDE problems in various pairs of function spaces arising from practical needs. In the final part of the paper, we present a possible application of our results for studying a semi-discrete version of a model initial-boundary value problem of heat conduction.
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