An algebraic multigrid method for quadratic finite element equations of elliptic and saddle point systems in 3D

TL;DR

This paper introduces a robust algebraic multigrid method tailored for quadratic finite element equations in 3D, effectively solving various elliptic and saddle point systems with mesh-independent convergence.

Contribution

It presents a new heuristic coarsening strategy and demonstrates the method's robustness and ease of implementation for complex 3D finite element problems.

Findings

Mesh-independent convergence demonstrated

Effective as a stand-alone solver or preconditioner

Applicable to multiple 3D PDE systems

Abstract

In this work, we propose a robust and easily implemented algebraic multigrid method as a stand-alone solver or a preconditioner in Krylov subspace methods for solving either symmetric and positive definite or saddle point linear systems of equations arising from the finite element discretization of the vector Laplacian problem, linear elasticity problem in pure displacement and mixed displacement-pressure form, and Stokes problem in mixed velocity-pressure form in 3D, respectively. We use hierarchical quadratic basis functions to construct the finite element spaces. A new heuristic algebraic coarsening strategy is introduced for construction of the hierarchical coarse system matrices. We focus on numerical study of the mesh-independence robustness of the algebraic multigrid and the algebraic multigrid preconditioned Krylov subspace methods.