Considering the space-time adaptive method for parabolic evolution equations introduced in [arXiv:2101.03956 [math.NA]], this work discusses an implementation of the method in which every step is of linear complexity. Exploiting the product structure of the space-time cylinder, the method allows for a family of trial spaces given as the spans of wavelets-in-time tensorized with (locally refined) 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.

翻译：考虑到[arXiv:2101.03956[math.NA]]中引入的抛物线进化方程式的时空适应方法,本项工作讨论了每个步骤都具有线性复杂度的方法的实施情况。利用时空圆筒的产品结构,该方法允许以(当地改进的)有限元素空间空间作为时波子拉长的试验空间组合。在以双树为基准的空格上,我们得出一种算法,在线性复杂度中应用由此形成的双线形式。我们提供了广泛的数字实验,以证明由此形成的适应环的线性运行时间。