2017/11/03 by Jaime Mora, Mora, Jaime, Leszek Demkowicz +1 · 1 citation
Engineering · #65D30 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1711.00984
openalex publication_date 2017/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H1, H(curl), H(div), and L2 inner products, have the O(p7) computational complexity. Additionally, a modified version of the algorithms is proposed when the element map can be simplified, resulting in the reduced O(p6) complexity. Use of Legendre polynomials for shape functions is critical in this implementation. Computational experiments performed with H1, H(div) and H(curl) test shape functions show good correspondence with the expected rates.