# An AMG saddle point preconditioner with application to mixed Poisson problems on adaptive quad/cube meshes

## TL;DR

This paper introduces a specialized algebraic multigrid (AMG) preconditioner for saddle point problems in mixed Poisson discretizations, demonstrating near mesh-independent convergence on adaptive meshes with heterogeneous coefficients.

## Contribution

The paper develops a novel AMG saddle point preconditioner with a stabilized prolongation operator tailored for mixed Poisson problems on adaptive meshes.

## Key findings

- SPAMG preconditioner achieves mesh-independent iteration counts.
- Effective on both 2D and 3D adaptive meshes.
- Handles heterogeneous coefficients efficiently.

## Abstract

We investigate various block preconditioners for a low-order Raviart-Thomas discretization of the mixed Poisson problem on adaptive quadrilateral meshes. In addition to standard diagonal and Schur complement preconditioners, we present a dedicated AMG solver for saddle point problems (SPAMG). A key element is a stabilized prolongation operator that couples the flux and scalar components. Our numerical experiments in 2D and 3D show that the SPAMG preconditioner displays nearly mesh-independent iteration counts for adaptive meshes and heterogeneous coefficients.