Massively Parallel Solution Techniques for Higher-order Finite-element Discretizations in CFD

The purpose of this paper is to present techniques to solve higher-order finite element discretizations on massively parallel architectures. Implicit schemes are considered as a means of achieving mesh independent convergence rates for both time dependent problems and steady state solutions obtained through pseudo-transient continuation. Domain decomposition preconditioners are presented for the scalable parallel solution of the linear system arising at each iteration of a Newton-Krylov approach. Basic domain decomposition methods are presented along with theoretical results for simple model problems. Practical extensions of these algorithms for simulations of the Euler and Navier-Stokes equations are reviewed in reference to the theoretical results from the model problems. Extensions of some recently developed iterative substructuring algorithms are also proposed for the Euler and Navier-Stokes equations. Numerical examples using several domain decomposition algorithms are presented for a higher-order simulation of a convection-diffusion model problem.