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Considering the space-time adaptive method for parabolic evolution equationsintroduced in [arXiv:2101.03956 [math.NA]], this work discusses animplementation of the method in which every step is of linear complexity.Exploiting the product structure of the space-time cylinder, the method allowsfor a family of trial spaces given as the spans of wavelets-in-time tensorizedwith (locally refined) finite element spaces-in-space. On spaces whose basesare indexed by double-trees, we derive an algorithm that applies the resultingbilinear forms in linear complexity. We provide extensive numerical experimentsto demonstrate the linear runtime of the resulting adaptive loop.

Efficient space-time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space