The Parallel Full Approximation Scheme in Space and Time for a Parabolic Finite Element Problem

TL;DR

This paper extends the PFASST parallel-in-time integrator to finite element spatial discretizations, demonstrating its effectiveness and accuracy improvements for parabolic PDEs.

Contribution

It develops a PFASST algorithm compatible with finite element discretizations, addressing mass matrix challenges and demonstrating enhanced convergence.

Findings

PFASST achieves two orders of accuracy per iteration after initial steps.

The algorithm effectively integrates with finite element methods for parabolic problems.

Numerical results confirm the method's convergence and accuracy improvements.

Abstract

The parallel full approximation scheme in space and time (PFASST) is a parallel-in-time integrator that allows to integrate multiple time-steps simultaneously. It has been shown to extend scaling limits of spatial parallelization strategies when coupled with finite differences, spectral discretizations, or particle methods. In this paper we show how to use PFASST together with a finite element discretization in space. While seemingly straightforward, the appearance of the mass matrix and the need to restrict iterates as well as residuals in space makes this task slightly more intricate. We derive the PFASST algorithm with mass matrices and appropriate prolongation and restriction operators and show numerically that PFASST can, after some initial iterations, gain two orders of accuracy per iteration.