Nodal AMG Coarsening and Interpolation for PDE Systems

TL;DR

This paper introduces a novel algebraic multigrid coarsening and interpolation method specifically designed for PDE systems with large near-kernels, improving efficiency and applicability to complex systems like Stokes.

Contribution

It develops a practical coarsening algorithm and interpolation operator tailored for PDE systems, incorporating modifications to compatible relaxation and ideal interpolation within the GAMG framework.

Findings

Effective interpolation scheme demonstrated on PDE systems

Reproduces re-discretization on unstructured meshes

Numerical results show improved multigrid performance

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.