Fast low-rank solution of the Poisson equation with application to the Stokes problem

TL;DR

This paper introduces a fast low-rank tensor-based algorithm for solving the Poisson equation, which improves efficiency and is effectively integrated into a Stokes problem solver, outperforming standard methods at larger grid sizes.

Contribution

A novel low-rank tensor algorithm using cross approximation in frequency space with better complexity than existing exponential sum methods.

Findings

Outperforms standard methods for grid sizes n ≥ 256

Efficiently integrated into a Stokes problem solver

Demonstrates superior performance in large-scale problems

Abstract

We consider the problem of computing approximate solution of Poisson equation in the low-parametric tensor formats. We propose a new algorithm to compute the solution based on the cross approximation algorithm in the frequency space, and it has better complexity with respect to ranks in comparison with standard algorithms, which are based on the exponential sums approximation. To illustrate the effectiveness of our solver, we incorporate into a Uzawa solver for the Stokes problem on semi-staggered grid as a subsolver. The resulting solver outperforms the standard method for $n \geq 256$.