A Vanka-based parameter-robust multigrid relaxation for the Stokes-Darcy Brinkman problems

TL;DR

This paper introduces a Vanka-based multigrid relaxation method for efficiently solving the Stokes-Darcy Brinkman equations, demonstrating robustness and high efficiency across various mesh sizes and physical parameters.

Contribution

It develops a novel Vanka-based additive smoother and analyzes its optimal damping, improving multigrid solver robustness for coupled flow problems.

Findings

High efficiency of the proposed relaxation scheme

Robustness to physical parameters and mesh size

Near-optimal performance with damping parameter equal to one

Abstract

We propose a block-structured multigrid relaxation scheme for solving the Stokes-Darcy Brinkman equations discretized by the marker and cell scheme. An element-based additive Vanka smoother is used to solve the corresponding shifted Laplacian operator. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess-Sarazin relaxation for the Stokes-Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same results as these for the optimal damping parameter at a lower computational overhead.