arXiv subjects

Raymond Tuminaro

Publications and source records attributed to Raymond Tuminaro.

3 recordsLinked to original sources

arXiv2026

Improving Smoothed Aggregation AMG Robustness on Stretched Mesh Applications

Strength-of-connection algorithms play a key role in algebraic multigrid (AMG). Specifically, they determine which matrix nonzeros are classified as weak and so ignored when coarsening matrix graphs and defining interpolation sparsity patterns. The general goal is to encourage coarsening only in directions where error can be smoothed and to avoid coarsening across sharp problem variations. Unfortunately, developing robust and inexpensive strength-of-connection schemes is challenging. The classification of matrix nonzeros involves four aspects: (a) choosing a strength-of-connection matrix, (b) scaling its values, (c) choosing a criterion to classify scaled values as strong or weak, and (d) dropping weak entries which includes adjusting matrix values to account for dropped terms. Typically, smoothed aggregation AMG uses the linear system being solved as a strength-of-connection matrix. It scales values symmetrically using square-roots of the matrix diagonal. It classifies based on whether scaled values are above or below a threshold. Finally, it adjusts matrix values by modifying the diagonal so that the sum of entries within each row of the dropped matrix matches that of the original. While these procedures can work well, we illustrate failure cases that motivate alternatives. The first alternative uses a distance Laplacian strength-of-connection matrix. The second centers on non-symmetric scaling. We then investigate alternative classification criteria based on identifying gaps in the values of the scaled entries. Finally, an alternative lumping procedure is proposed where row sums are preserved by modifying all retained matrix entries (as opposed to just diagonal entries). A series of numerical results illustrates trade-offs demonstrating in some cases notably more robust convergence on matrices coming from linear finite elements on stretched meshes.

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arXiv2025

A structure preserving H-curl algebraic multigrid method for the eddy current equations

A new algebraic multigrid method (AMG) is presented for solving the linear systems associated with the eddy current approximation to the Maxwell equations. This AMG method extends an idea proposed by Reitzinger and Schoberl. The main feature of the Reitzinger and Schoberl algorithm (RSAMG) is that it maintains null-space properties of the Curl-Curl operator throughout all levels of the AMG hierarchy. It does this by enforcing a commuting relationship involving grid transfers and the discrete gradient operator. This null-space preservation property is critical to the algorithm's success, however enforcing this commuting relationship is non-trivial except in the special case where one leverages a piece-wise constant nodal interpolation operator. For this reason, mesh independent convergence rates are generally not observed for RSAMG due to its reliance on sub-optimal piece-wise constant interpolation. We present a new AMG algorithm that enforces the same commuting relationship. The main advance is that the new structure preserving H-curl algorithm (SpHcurlAMG) does not rely on piece-wise constant interpolation and can leverage fairly general and more sophisticated nodal interpolation operators. The key idea is to employ energy minimization AMG (EAMG) to construct edge interpolation grid transfers and to enforce the commuting relationship by embedding it as constraints within an EAMG procedure. While it might appear that solving such a constrained energy minimization is costly, we illustrate how this is not the case in our context. Numerical results are then given demonstrating mesh independent convergence over a range of test problems.

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arXiv2023

Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations

We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The algorithm is based on the use of the lower-order $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator within a defect-correction setting, in combination with AMG construction of interpolation operators for velocities and pressures. The preconditioning framework is primarily algebraic, though the $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator must be provided. We investigate two relaxation strategies in this setting. Specifically, a novel block factorization approach is devised for Vanka patch systems, which significantly reduces storage requirements and computational overhead, and a Chebyshev adaptation of the LSC-DGS relaxation is developed to improve parallelism. The preconditioner demonstrates robust performance across a variety of 2D and 3D Stokes problems, often matching or exceeding the effectiveness of an inexact block-triangular (or Uzawa) preconditioner, especially in challenging scenarios such as elongated-domain problems.

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