An algebraic multigrid method for $Q_2-Q_1$ mixed discretizations of the Navier-Stokes equations

TL;DR

This paper develops an algebraic multigrid method tailored for the $Q_2-Q_1$ mixed finite element discretization of Navier-Stokes equations, effectively handling non-co-located velocity and pressure unknowns.

Contribution

It introduces an automatic AMG coarsening strategy that mimics pressure/velocity relationships in $Q_2-Q_1$ discretizations, enabling efficient solvers for complex PDE systems.

Findings

Effective solver performance on Navier-Stokes problems

Automatic coarsening mimics physical dof relationships

Applicable to Stokes and incompressible flow simulations

Abstract

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.