Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems

TL;DR

This work develops a robust reduction-based AMG method, called constrained $ ext{ell}$-AIR (C$ ext{l}$AIR), for nonsymmetric linear systems arising from advection–diffusion problems. It combines the local ideal-restriction philosophy of $ ext{ell}$-AIR with energy-minimization mode constraints and aggressive root-node coarsening to achieve fast convergence across advective and diffusive regimes while maintaining low operator complexity. The method introduces a global constraint framework using a matrix $Q$ and near-nullspace modes to enforce desired interpolation constraints, formulated via constrained minimization and solved either directly (KKT) or iteratively with projected Krylov methods. Numerical results across Poisson, anisotropic diffusion, coefficient jumps, and advection–diffusion tests show that C$ ext{l}$AIR matches or exceeds the performance of existing solvers (including $ ext{ell}$-AIR and root-node) with significantly lower operator complexity and greater robustness to parameter settings. The approach provides a practical, scalable preconditioner for nonsymmetric PDE discretizations, with potential for theoretical two-grid convergence analysis and compatibility-relaxation-based coarsening in future work.

Abstract

This paper focuses on developing a reduction-based algebraic multigrid method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based algebraic multigrid (AMG) approach, $\ell$AIR (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion dominated problems in two or three dimensions. Motivated by the success of $\ell$AIR in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy minimization AMG methods with the local approximation of ideal operators used in $\ell$AIR. The resulting constrained $\ell$AIR (C$\ell$AIR) algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that has been previously difficult for reduction-based methods.

Figures (7)

• Figure 1: Sawtooth coefficient jump domain, $d(x,y)=10^4$ in shaded region and $d(x,y)=1$ outside.
• Figure 1: WAP and SAP constants for the restriction operators of $\ell$AIR and C$\ell$AIR for the constant advection with zero diffusion ($\alpha=0$) problem. Five (5) iterations of CFF-weighted-Jacobi relaxation has been used to improve the mode constraint vector $B=\mathbf{1}$ in C$\ell$AIR. Singular values are shown in the dotted blue line and are associated with the right vertical axis, and the dot-dashed lines show the approximation constant for each of the left singular vectors of $A$. Horizontal dashed lines show the approximation constant for $\ell$AIR and C$\ell$AIR that holds for all vectors.
• Figure 2: AMR throughout different stages for the Laplace problem.
• Figure 2: WAP and SAP constants for the restriction operators of $\ell$AIR and C$\ell$AIR for the constant advection with added diffusion ($\alpha=10$) problem. Different number of iterations (5 and then 25) of CFF-weighted-Jacobi relaxation has been used to improve the mode constraint vector $B=\mathbf{1}$ in C$\ell$AIR. Singular values are shown in the dotted blue line and are associated with the right vertical axis, and the dot-dashed lines show the approximation constant for each of the left singular vectors of $A$. Horizontal dashed lines show the approximation constant for $\ell$AIR and C$\ell$AIR that holds for all vectors.
• Figure 3: 2D and 3D Poisson, comparison of iterations and work-per-digit of accuracy for $\ell$AIR, C$\ell$AIR, and root-node.

Theorems & Definitions (5)

• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Remark 4.3