Linearizing the hybridizable discontinuous Galerkin method: A linearly scaling operator

TL;DR

This paper introduces a matrix-free, linearly scaling residual evaluation technique for the hybridizable discontinuous Galerkin method, enabling extremely fast solutions on standard CPU cores by combining tensor-product bases, a specific penalty parameter, and a face-wise preconditioner.

Contribution

It presents a novel, linearly scaling operator for the HDG method that significantly improves computational efficiency using tensor-product bases and a specialized preconditioner.

Findings

Achieves solutions in 1 microsecond per unknown on a single CPU core.

Demonstrates linear scaling of iteration time with degrees of freedom.

Provides a practical, fast residual evaluation technique for HDG methods.

Abstract

This paper proposes a matrix-free residual evaluation technique for the hybridizable discontinuous Galerkin method requiring a number of operations scaling only linearly with the number of degrees of freedom. The method results from application of tensor-product bases on cuboidal Cartesian elements, a specific choice for the penalty parameter, and the fast diagonalization technique. In combination with a linearly scaling, face-wise preconditioner, a linearly scaling iteration time for a conjugate gradient method is attained. This allows for solutions in 1 $\mu s$ per unknown on one CPU core - a number typically associated with low-order methods.