A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators

TL;DR

This paper introduces a Hermite-like basis for discontinuous Galerkin methods that reduces data access and enhances computational efficiency on modern processors, especially in multigrid solvers.

Contribution

It develops a novel basis based on Hermite polynomials that minimizes neighbor data access and integrates seamlessly with sum factorization for faster matrix-free evaluations.

Findings

Reduces neighbor data access to one point for function value and derivative.

Enables efficient sum factorization with the new basis.

Improves multigrid solver performance with basis change techniques.

Abstract

This work proposes a basis for improved throughput of matrix-free evaluation of discontinuous Galerkin symmetric interior penalty discretizations on hexahedral elements. The basis relies on ideas of Hermite polynomials. It is used in a fully discontinuous setting not for higher order continuity but to minimize the effective stencil width, namely to limit the neighbor access of an element to one data point for the function value and one for the derivative. The basis is extended to higher orders with nodal contributions derived from roots of Jacobi polynomials and extended to multiple dimensions with tensor products, which enable the use of sum factorization. The beneficial effect of the reduced data access on modern processors is shown. Furthermore, the viability of the basis in the context of multigrid solvers is analyzed. While a plain point-Jacobi approach is less efficient than with the best nodal polynomials, a basis change via sum-factorization techniques enables the combination of the fast matrix-vector products with effective multigrid constituents. The basis change is essentially for free on modern hardware because these computations can be hidden behind the cost of the data access.