A comparative study of efficient multigrid solvers for high-order local discontinuous Galerkin methods: Poisson, elliptic interface, and multiphase Stokes problems

TL;DR

This paper compares various multigrid solvers, including novel sparse approximate inverse smoothers, for high-order local discontinuous Galerkin methods applied to complex elliptic and Stokes problems, demonstrating effective performance across diverse test cases.

Contribution

It introduces and evaluates new SAI-based multigrid smoothers tailored for high-order DG methods on challenging elliptic and multiphase Stokes problems, showing comparable efficiency to classical methods.

Findings

At least one multigrid approach matches classical geometric multigrid performance.

SAI smoothers effectively handle high-contrast viscosity in multiphase problems.

Multiple solver configurations achieve significant residual reduction within few iterations.

Abstract

We design and investigate a variety of multigrid solvers for high-order local discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes problems. Using the template of a standard multigrid V-cycle, we consider a variety of element-wise block smoothers, including Jacobi, multi-coloured Gauss-Seidel, processor-block Gauss-Seidel, and with special interest, smoothers based on sparse approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i) balance the smoothing of velocity and pressure variables in Stokes problems; and (ii) robustly handles high-contrast viscosity coefficients in multiphase problems. Across a broad range of two- and three-dimensional test cases, including Poisson, elliptic interface, steady-state Stokes, and unsteady Stokes problems, we examine a multitude of multigrid smoother and solver combinations. In every case, there is at least one approach that matches the performance of classical geometric multigrid algorithms, e.g., 4 to 8 iterations reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.