Enhanced multigrid solver for anisotropic equations with non-standard components: 3-Color Jacobi, mesh-tripling, and Fourier Analysis

Abstract

In this paper, we introduce an enhanced multigrid solver that offers an efficient solution method which is quite robust across a variety of boundary value problems. The solver’s theoretical foundation includes a framework for deriving optimal relaxation parameters, and features an auto-tuned, customizable meshing approach. It employs a hierarchical structure capable of handling various grid configurations, optimized through Local Fourier Analysis (LFA). Although primarily developed for the anisotropic diffusion equation, we extend the investigation to include the singularly perturbed convection diffusion equation; where we fine-tune meshing parameters, refine discretization techniques, and implement customized multigrid operators to address its unique challenges. Numerical experiments are included that demonstrate the solver’s robustness and efficiency, thereby making a strong case for its use across a wide range of second order elliptic problems.