A cell structure implementation of the multigrid method for the two-dimensional diffusion equation

To solve the two-dimensional diffusion equation using the finite difference method, we propose a simple MATLAB implementation of the multigrid method. The diffusion equation plays a fundamental role in modeling many significant physical phenomena and is ubiquitous in many governing equations. Some examples include the reaction-diffusion equations, the convection-diffusion equations, and others. These equations often lack analytical solutions or pose extreme challenges in finding them. Therefore, numerical techniques are indispensable for obtaining practical and accurate approximations for these equations. The multigrid method is known for its computational efficiency and effectiveness as an iterative technique for solving the discretized diffusion equation. Due to its popularity, the multigrid method has been implemented in several programming languages, such as Python, Java, C++, C, Fortran, and others. However, it is not easy for beginners to understand the implementation of the multigrid method due to its complex data structures and recursive routines. To resolve these difficulties, we develop a straightforward MATLAB implementation of the two-dimensional diffusion equation using a cell structure in MATLAB. This work provides an accessible and efficient framework for understanding and applying the multigrid method, thereby simplifying its implementation for researchers and practitioners. (c) 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/).https://doi.org/10.1063/5.0247042