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

Author Affiliation

  • Dr. Hisham Bin Zubair Syed is an Assistant Professor and Chairperson Mathematical Sciences at IBA Karachi, Pakistan.
  • Muhammad Shahid Ashraf is PhD scholar in Department of Mathematical Sciences at the Institute of Business Administration (IBA) Karachi, Pakistan.

Faculty / School

School of Mathematics and Computer Science (SMCS)

Department

Department of Mathematical Sciences

Was this content written or created while at IBA?

Yes

Document Type

Article

Source Publication

Partial Differential Equations in Applied Mathematics

Keywords

Multigrid methods, Local Fourier Analysis, Relaxation, Singularly perturbed convection–diffusion equations

Disciplines

Mathematics

Abstract

In this paper, the authors 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.

Citation/Publisher Attribution

Shahid, M., & bin Zubair Syed, H. (2025). Enhanced multigrid solver for anisotropic equations with non-standard components: 3-Color Jacobi, mesh-tripling, and Fourier Analysis. Partial Differential Equations in Applied Mathematics, 101133.

Publication Status

Published

Rights Information

Note: This article is available under the Creative Commons CC-BY-NC-ND license and permits non-commercial use of the work as published, without adaptation or alteration provided the work is fully attributed.

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