A parallel algorithm for solving linear parabolic evolution equations

TL;DR

This paper introduces a parallel algorithm for efficiently solving linear parabolic evolution equations using space-time discretization, wavelets, and finite elements, achieving linear and polylogarithmic complexity with demonstrated large-scale parallel performance.

Contribution

The paper presents a novel parallel algorithm that recasts the discretization into a Schur-complement system, enabling efficient solution with wavelets and finite elements, and demonstrates its scalability.

Findings

Linear complexity solution on a single processor.

Polylogarithmic complexity with parallelization in space and time.

Successful large-scale parallel computations validating the method.

Abstract

We present an algorithm for the solution of a simultaneous space-time discretization of linear parabolic evolution equations with a symmetric differential operator in space. Building on earlier work, we recast this discretization into a Schur-complement equation whose solution is a quasi-optimal approximation to the weak solution of the equation at hand. Choosing a tensor-product discretization, we arrive at a remarkably simple linear system. Using wavelets in time and standard finite elements in space, we solve the resulting system in linear complexity on a single processor, and in polylogarithmic complexity when parallelized in both space and time. We complement these theoretical findings with large-scale parallel computations showing the effectiveness of the method.