# Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

## TL;DR

This paper introduces a new tensor-product preconditioner for high-order discontinuous Galerkin methods that significantly reduces computational costs and storage, enabling efficient solutions for high polynomial degrees in 2D and 3D.

## Contribution

The paper presents an algebraic, SVD-based tensor-product preconditioner that approximates the block Jacobi preconditioner with lower complexity, suitable for very high polynomial degrees.

## Key findings

- Reduces storage from $ ext{O}(p^{2d})$ to $ ext{O}(p^{d+1})$
- Achieves near-linear time solution per degree of freedom in 2D
- Reduces computational complexity from $ ext{O}(p^9)$ to $ ext{O}(p^5)$ in 3D

## Abstract

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.