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

Considering the space-time adaptive method for parabolic evolution equations we introduced in Stevenson et al., this work discusses an implementation of the method in which every step is of linear complexity. Exploiting the tensor-product structure of the space-time cylinder, the method allows for a family of trial spaces given as spans of wavelets-in-time tensorized with finite element spaces-in-space. On spaces whose bases are indexed by double-trees, we derive an algorithm that applies the resulting bilinear forms in linear complexity. We provide extensive numerical experiments to demonstrate the linear runtime of the resulting adaptive loop.