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

TL;DR

This paper presents a linear complexity implementation of a space-time adaptive method for parabolic equations, combining wavelets in time with finite elements in space, enabling efficient computations.

Contribution

It introduces a new algorithm exploiting product structure and double-tree bases for linear complexity application of bilinear forms in adaptive space-time methods.

Findings

Algorithm achieves linear runtime in numerical experiments.

Extensive experiments confirm efficiency of the adaptive loop.

Method effectively combines wavelets and finite elements for parabolic equations.

Abstract

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.