Deflation-accelerated preconditioning of the Poisson–Neumann Schur problem on long domains with a high-order discontinuous element-based collocation method

Abstract

A combination of block-Jacobi and deflation preconditioning is used to solve a high-order discontinuous collocation-based discretization of the Schur complement of the Poisson-Neumann system as arises in the operator splitting of the incompressible Navier-Stokes equations. The preconditioners and deflation vectors are chosen to mitigate the effects of ill-conditioning due to highly-elongated domains and to achieve GMRES convergence independent of the size of the grid. The ill-posedness of the PoissonNeumann system manifests as an inconsistency of the Schur complement problem, but it is shown that this can be accounted for with appropriate projections out of the null space of the Schur matrix without affecting the accuracy of the solution. The combined deflation/block-Jacobi preconditioning is compared with two-level non-overlapping additive Schwarz preconditioning of the Schur problem, and while both methods achieve convergence independent of the grid size, deflation is shown to require half as many GMRES iterations and 25% less wall-clock time for a variety of grid sizes and domain aspect ratios. The deflation methods shown to be effective for the two-dimensional Poisson-Neumann problem are extensible to the three-dimensional problem assuming a Fourier discretization in the third dimension. A Fourier discretization results in a two-dimensional Helmholtz problem for each Fourier component that is solved using deflation/block-Jacobi preconditioning on its Schur complement. Here again deflation is shown to be superior to two-level non-overlapping additive Schwarz preconditioning, requiring about half as many GMRES iterations and 15% less time. While the methods here are demonstrated on a spectral multidomain penalty method discretization, they are readily extensible to any discontinuous elementbased discretization of an elliptic problem, and are particularly well-suited for high-order methods.

Formation of Recirculating Cores in Convectively Breaking Internal Solitary Waves of Depression Shoaling over Gentle Slopes in the South China Sea

“…As the foundation of the numerical tool used in this study, the SMPM, originally developed by Diamessis et al (2005), has been successfully applied to the study of small-scale stratified flow processes (Diamessis et al 2011;Abdilghanie and Diamessis 2012;Zhou andDiamessis 2015, 2016), with minimal artificial dispersion and diffusion, including the propagation of ISWs in a uniform depth waveguide and two-dimensional studies of their interaction with a model no-slip sea floor (Diamessis and Redekopp 2006). Recently, Joshi et al (2016) adapted the method to efficiently account for the nonhydrostatic effects with deformed boundaries while preserving high-order accuracy, thereby allowing the incorporation of gentle bathymetry over long domains.…”

“…Time integration is achieved via a stiffly stable third-order scheme (Karniadakis et al 1991). Nonhydrostatic effects are handled efficiently through a mixed deflation Schurcomplement-based pressure solver (Joshi et al 2016).…”

“…Given the computational resources available nowadays, it is possible to complement the field observations on the Dongsha slope with high accuracy/resolution and fully nonlinear/nonhydrostatic simulations. Thus, an objective of our study is to simulate the formation of a convective instability via a high-order spectral multidomain penalty method (SMPM) (Diamessis et al 2005;Joshi et al 2016), by capturing the wave as it convectively breaks and to reproduce the formation of the subsurface recirculating core. The work can be guided by observations, including the recorded water column properties and the recorded water depth, thereby bridging the gap between localized observations and the full evolution of the shoaling process.…”