Nodal AMG Coarsening and Interpolation for PDE Systems

TL;DR

This work develops an algebraic multigrid framework tailored for PDE systems with large near-kernel components, such as $H(\mathrm{curl})$ and $H(\mathrm{div})$, by combining compatible relaxation (CR) with generalized ideal interpolation in the GAMG setting. A key contribution is a nodal dual coarsening strategy that, together with carefully oriented averaging, preserves near-kernel structures and yields a sparse, effective interpolation operator; the coarse space is constructed algebraically via path-averaging and local orientation, reproducing rediscretization on refined meshes when aligned with geometric refinement. The authors also automate smoother construction to identify local near kernels and integrate this into a practical two-grid method, validated on curl-curl and Stokes problems; results show improved convergence over classical ideal interpolation, including stable performance on unstructured meshes. Overall, the paper provides a robust, practical AMG approach for challenging PDE systems, highlighting the role of near-kernel preservation in coarse-grid design and offering directions for boundary-aware refinements and broader applicability.

Abstract

We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) and H(div). Our method builds on compatible relaxation (CR) and the ideal interpolation model within the generalized AMG (GAMG) framework but introduces several modifications to define an AMG method for PDE systems. We construct an interpolation operator through a coarsening process that first coarsens a nodal dual problem and then builds the coarse and fine variables using a matching algorithm. Our interpolation follows the ideal formulation; however, we enhance the sparsity of ideal interpolation by decoupling the fine and coarse variables completely. When the coarse variables align with the geometric refinement, our method reproduces re-discretization on unstructured meshes. Together with an automatic smoother construction scheme that identifies the local near kernels, our approach forms a complete two-grid method. Finally, we also show numerical results that demonstrate the effectiveness of this interpolation scheme by applying it to targeted problems and the Stokes system.

Figures (6)

• Figure 1: Curl-curl on uniform quadrilateral mesh.
• Figure 1: uniform quad.
• Figure 2: Uniform triangular mesh with Dirichlet boundary conditions
• Figure 3: pcg.
• Figure 4: Stokes.