A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations

TL;DR

This paper introduces an adaptive solver for parabolic evolution equations that combines wavelet-in-time and finite element-in-space methods, achieving linear convergence and demonstrated effectiveness through numerical experiments.

Contribution

The work develops a novel space-time adaptive method using wavelets and finite elements with proven linear convergence for parabolic equations.

Findings

Achieves r-linear convergence in adaptive solving.

Numerical results confirm theoretical convergence rates.

Combines wavelet-in-time with finite element-in-space for efficiency.

Abstract

In this work, an $r$-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our theoretical findings.