High-performance implementation of H(div)-conforming elements for incompressible flows

Sammanfattning: In this thesis, evaluation of H(div)-conforming finite elements is implemented in a high-performance setting and used to solve the incompressible Navier-Stokes equation, obtaining an exactly point-wise divergence-free velocity field. In particular, the anisotropic Raviart-Thomas tensor-product polynomial space is considered, where the finite element operators are evaluated with quadrature in a matrix-free fashion using sum-factorization on tensor-product meshes. The implementation includes evaluation over elements and faces in two- and three-dimensional space, supporting non-conforming meshes with hanging nodes, and using the contravariant Piola transformation to preserve normal components on element boundaries. In terms of throughput, the implementation achieves up to an order of magnitude faster evaluation of finite element operators compared to a matrix-based evaluation. Correctness is demonstrated with optimal convergence rates for various polynomial degrees, as well as exactly divergence-free solutions for the velocity field.