Parallel matching-based AMG preconditioners for elliptic equations discretized by IgA

TL;DR

This paper evaluates parallel algebraic multigrid preconditioners for large, ill-conditioned linear systems from isogeometric analysis of elliptic problems, emphasizing scalability and efficiency in high-performance computing environments.

Contribution

It introduces and assesses parallel AMG preconditioners specifically designed for IgA discretizations, demonstrating their robustness and scalability on modern HPC architectures.

Findings

AMG preconditioners improve convergence for IgA systems.

Parallel implementation achieves good scalability on HPC systems.

Numerical tests confirm robustness across various problem sizes.

Abstract

Isogeometric analysis (IgA) offers enhanced approximation capabilities for the discretization of elliptic boundary-value problems, yet it results in large, sparse, and increasingly ill-conditioned linear systems due to higher interconnectivity among degrees of freedom. In particular, the discretization with tensor-product B-splines or NURBS of degree $p$ on a mesh with $n$ elements per parametric direction leads to symmetric positive-definite systems of the form $K\mathbf{u} = \mathbf{F}$, where the matrix bandwidth and condition number scale unfavorably with both $p$ and spatial dimension $d$. To address the computational challenges posed by such systems, especially in three-dimensional or high-order scenarios, Krylov subspace methods with specialized preconditioners become essential. This paper investigates the efficacy of algebraic multigrid (AMG) preconditioners tailored for IgA-based discretizations, with a focus on performance in modern high-performance computing (HPC) environments. Leveraging the Parallel Sparse Computation Toolkit (PSCToolkit), we explore distributed-memory and GPU-accelerated strategies for solving large-scale problems. The study assesses algorithmic efficiency and scalability across a range of benchmark tests. The results demonstrate that AMG preconditioners can achieve robust and scalable performance, confirming their potential as practical solvers for large IgA systems in engineering and scientific applications.