Fast integration of DPG matrices based on tensorization

TL;DR

This paper introduces fast algorithms for integrating DPG matrices in high-order finite element methods using tensorization and sum factorization, significantly reducing computational complexity.

Contribution

It develops tensorization-based algorithms for exact sequence FE spaces on hexahedra, achieving reduced complexity for DPG matrix assembly.

Findings

Algorithms achieve $O(p^7)$ complexity for general cases.

Modified algorithms reduce complexity to $O(p^6)$ when element maps are simplified.

Computational experiments confirm the expected convergence rates.

Abstract

Numerical integration of the stiffness matrix in higher order finite element (FE) methods is recognized as one of the heaviest computational tasks in a FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of $H^1$, $H(\text{curl})$, $H(\text{div})$, and $L^2$ inner products, have the $O(p^7)$ computational complexity. Additionally, a modified version of the algorithms is proposed when the element map can be simplified, resulting in the reduced $O(p^6)$ complexity. Use of Legendre polynomials for shape functions is critical in this implementation. Computational experiments performed with $H^1$, $H(\text{div})$ and $H(\text{curl})$ test shape functions show good correspondence with the expected rates.