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

TL;DR

The paper tackles the computational challenge posed by large, ill-conditioned linear systems from IgA discretizations of elliptic problems. It develops algebraic multigrid preconditioners based on compatible weighted graph matching to construct aggregation and interpolation operators, embedded in a Galerkin framework and implemented in PSCToolkit for distributed and GPU architectures. The proposed V‑cycle AMG with l1‑Jacobi and Chebyshev acceleration demonstrates robust convergence and good parallel scalability across single patches, multi‑patch geometries, and non‑isoparametric configurations, with substantial speedups from GPU acceleration. The results indicate practical, scalable solution strategies for large IgA systems in engineering applications and point to future enhancements such as hybrid geometric–algebraic multigrid and broader OpenACC support.

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.

Figures (12)

• Figure 1.1: Example of a B-spline (left) and NURBS 2D geometry (right) descriptions. The dashed lines represents the control polygon passing through the control points $\{\mathbf{P}_i\}_i$, while the curves are depicted with a bold line.
• Figure 1.2: Spy patterns of an IgA stiffnes matrix for the Poisson equation on a cube with B-Spine basis of degree 3 and regularity 2, and of the Poisson equation on a thick $L$-shaped domain---with the same degree and regularity---but built with three patches.
• Figure 1.3: Depiction of the solution for the three test problem considered in the following sections.
• Figure 1.4: Operator complexity for the Poisson cube case for increasing values of $k$ and $p$, and number of MPI tasks running from $1$ to $512$.
• Figure 1.5: Strong scaling for the Poisson on a 3D Cube problem for a number of task running from $1$ to $512$.

Theorems & Definitions (1)

• remark thmcounterremark: Assembly cost