Optimal High-Order Solvers

Project Overview

The purpose of this LDRD project was to develop new numerical methods and algorithms for the solution of large linear systems resulting from high-order finite element discretizations, in particular when the associated system matrix is not available. Such methods, often termed matrix-free methods, are of increasing importance on GPU-based supercomputers such as LLNL's Sierra and the upcoming exascale machine El Capitan. The main approach pursued in this project is known as "low-order-refined," whereby a low-order discretization is posed on a refined mesh. Using specialized basis constructions developed during this project, the low-order-refined matrix can be shown to be spectrally equivalent to the high-order operator, even for problems posed in H(curl), H(div), and L2 spaces, allowing for the construction of effective, scalable algebraic multigrid preconditioners. The new methods and technologies developed in this LDRD enable the construction of efficient, memory-lean matrix-free solvers for problems posed in all the spaces of the high-order finite element de Rham complex.

Mission Impact

The methods, solvers, preconditioners, and algorithms researched in this LDRD allow for the efficient solution of challenging computational physics problems (including electromagnetic diffusion, radiation diffusion, and radiative transfer) at high orders (and thus resulting in high accuracy), making effective use of the computational resources provided by the latest GPU-based supercomputers. These technologies fill an important gap in the laboratory's computational physics capabilities, with application to stockpile stewardship and the DOE's energy mission.

Publications, Presentations, and Patents

Barker, A. 2021. "Matrix-free Algebraic Multigrid." SIAM CSE 2021.

Barker, A. 2019. "Matrix-free Preconditioning in H(curl)." SIAM CSE 2019.

Barker, A., and Tz Kolev. 2020. "Matrix-free Preconditioning for High-order H(curl) Discretizations." Numerical Linear Algebra with Applications 1-17.

Pazner, W., and Tz Kolev. 2021. "Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement." Communications on Applied Mathematics and Computation 1-31.

Pazner, W. 2022. "Low-order Tools for High-order Finite Elements." North American High-Order Methods Conference (Invited Plenary Presentation). San Diego State University.