On multigrid methods for the solution of least-squares finite element models for viscous flows

Abstract

There is a vast literature on least-squares finite element models (LSFEM) applied to fluid dynamics problems. The hp version of the least-squares models is computationally expensive, which necessitates the usage of elegant methods for solving resulting systems of equations. Amongst some of the schemes used for solving large systems of equations is the element-by-element (EBE) solution technique, which has found widespread use in least-squares applications. However, the use of EBE techniques with Jacobi preconditioning leads to very little performance gains as compared to solving a non-preconditioned system. Because of such considerations, the hp version LSFEM solutions are computationally intensive. In this study, we propose to solve the LSFEM systems using the multigrid method, which offers superior convergence rates compared to the EBE–JCG. We demonstrate the superior convergence of the Multigrid solver compared to Jacobi preconditioning for the wall-driven cavity and backward facing step problems using the full Navier–Stokes equations. Load balancing issues encountered with multigrid solvers in a parallel environment are resolved elegantly with an element-by-element solution of the coarse grid problem with Jacobi preconditioning.