Efficient Implementations of High-Order Finite Elements with the deal.II Library

Abstract

My talk will present experiences and insights from the development of high-performance matrix-free operator evaluation algorithms with the deal.II finite element library. Originally, our efforts have concentrated on hexahedral finite elements, where integrals can be computed very efficiently with sum factorization techniques, but we have also looked into other element shapes recently, motivated by encouraging results from the Nektar++ team. The resulting implementations come with an arithmetic intensity of one to five Flop/byte, with memory transfer primarily due to the access into input and output vectors as well as some geometry or variable coefficient data. This enables a cost per degree of freedom that is almost constant for a broad range of polynomial degrees of cell and face integrals of high order continuous and discontinuous finite elements, and allows to select the polynomial degree as a parameter to trade against mesh generation limitations. From a hardware perspective, the Flop/byte ratio suggests that memory bandwidth is the main performance limit on many CPU and GPU architectures. Furthermore, many downstream solvers, such as multigrid smoothers, explicit time stepping, or conjugate gradient solvers, are now no longer dominated by the matrix-vector products, and vector operations need to be explicitly taken into consideration as well. My talk will show the effect of algorithmic optimizations on fluid dynamics simulations on massively parallel computers.