Nodal Coarsening and Sparse Ideal Interpolation for H(curl) Problems in Algebraic Multigrid

TL;DR

This paper introduces a novel algebraic multigrid approach for H(curl) problems using sparse interpolation and coarsening techniques, demonstrating improved robustness and performance over traditional methods.

Contribution

It presents a new coarsening algorithm and interpolation construction specifically designed for H(curl) problems within algebraic multigrid frameworks.

Findings

Outperforms standard geometric multigrid in robustness and efficiency.

Maintains weak approximation property and commuting relations.

Effective with strong coefficient jumps.

Abstract

We propose a sparse interpolation construction and a practical coarsening algorithm for the algebraic multigrid (AMG) method, tailored towards H(curl). Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. We establish the weak approximation property and the commuting relation under certain assumptions. Combined with matching block smoothers, the proposed interpolation yields an effective algebraic multilevel method. Numerical experiments show robustness under strong coefficient jumps, where the proposed methods substantially outperform standard geometric multigrid.