Efficient finite element assembly of high order Whitney forms

Abstract

This paper presents an efficient method for the finite element assembly of high order Whitney elements. We start by reviewing the classical assembly technique and by highlighting the most time consuming part. Then, we show how this classical approach can be reformulated into a computationally efficient matrix -matrix product. We also address the problem of the basis orientation by considering more than one reference space. We conclude by presenting numerical results on a wave guide problem.

Modal analysis of the ultrahigh finesse Haroche QED cavity

“…In this work, we used a homemade high-order FE code 4 , which takes advantage of an efficient assembly technique, as described in [28]. Indeed, quite often with high-order techniques, the computational bottleneck is found to be the assembly of the discrete operator itself.…”

Comparison of 2.5D finite element formulations with perfectly matched layers for solving open axisymmetric electromagnetic cavity problems

“…This necessitates either (i) a large number of quadrature points when using a classical Gaussian quadrature or (ii) specialized quadrature rules 2 ,. Section 5.1 Since the latter differ from element to element, the use of fast assembly techniques 5 is prevented. Therefore, the former approach is followed throughout this paper.…”

GMSH-FEM: An Efficient Finite Element Library Based on GMSH

“…While high-order finite elements naturally lead to increased arithmetic intensity, since the local elementwise matrices become larger and more dense, the number of quadrature points also dramatically increases. The best way to achieve good performance in such cases is to reformulate all the quadratures as dense matrix-matrix products [7,20], and by pre-computing as many of the underlying matrices as possible. While a natural decomposition resides in the separation of the metric-dependent and metricindependent parts in the Galerkin terms [20], many codes trade-off accuracy and generality for performance, by assuming for example that all parts of the integrands are interpolated using the same bases as the unknown fields [7].…”

“…The cost of pre-computing and storing local matrices is exacerbated when using hierarchical bases, or vector-valued basis functions for e.g. H (curl) or H (div), where the basis functions depend on the orientation of the elements [16,20]. Storing all unassembled matrices in such cases rapidly leads to prohibitive memory requirements, even in cases where the matrix is eventually factorized by direct linear solvers.…”