Spectral-element preconditioners for the uzawa pressure operator applied to incompressible flows

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

“…This solution method was introduced by Lynch, Rice, and Thomas [20] and used in spectral element preconditioning applications by Couzy and Deville [6], Couzy [5], and Fischer et al [15].…”

“…Our original intent was to base the multigrid smoother, M , on the additive overlapping Schwarz method of Dryja and Widlund [11], with local subdomain problems discretized by finite elements having nodes coincident with the GLL nodes, as considered in [3,13,28]. Following Couzy and Deville [6] and Fischer et al [15], one can solve the subdomain problems by using a fast diagonalization method similar to (18), guaranteeing that the cost of M is essentially equivalent to the cost of applying A. However, the use of the FDM frees one from the constraint of using FE-based preconditioners, since the FDM solution cost depends only on the use of the tensor-product form and not on the sparsity of the originating operator.…”

High accuracy flow simulations: Advances and challenges for future needs in aeronautics

“…Motivated by the prospect of obtaining high accuracy with the greater more general, curved geometries, the variant p-type finite element method is introduced, known as the Spectral Element method [47,48]. A common bottleneck for numericists is the complexity of the implementation programs.…”

Computational performance of a parallelized three-dimensional high-order spectral element toolbox

“…The first preconditioner, referred to as the Couzy preconditioner [11], holds a full matrix representation of the preconditioning matrix. Applying this preconditioner therefore requires a number of operations proportional to N 6 .…”