Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries
Abstract
Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order P, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to P=5 on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of 2.87×106 mixed prismatic and tetrahedral elements giving 32.2×106 pressure degrees of freedom at P=3. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.
Related
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BibTeX
@misc{khurana2026loworder,
title = {{Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries}},
author = {Parv Khurana and Henrik Wüstenberg and David Moxey and Athanasios Chatzopoulos and Julien Hoessler and Spencer J. Sherwin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31443},
eprint = {2609.31443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31443},
}
RIS
TY - UNPB
TI - Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries
AU - Parv Khurana
AU - Henrik Wüstenberg
AU - David Moxey
AU - Athanasios Chatzopoulos
AU - Julien Hoessler
AU - Spencer J. Sherwin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.31443
UR - https://arxiv.org/abs/2609.31443
AB - Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order P, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to P=5 on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of 2.87×10^6 mixed prismatic and tetrahedral elements giving 32.2×10^6 pressure degrees of freedom at P=3. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.
ER -