The conjugate gradient approach to solve two dimensions linear elliptic boundary value equations as a prototype of the reaction diffusion system

Abstract

This paper presents a numerical approach to solve the 2-dimensional reaction-diffusion problem, a crucial model in physics and chemistry, with applications ranging from pattern formation to material science. Focusing on addressing a stationary linear elliptic problem within a rectangular domain, boundary conditions are determined through a finite-difference formulation. The Conjugate-Gradient Method is employed for the numerical solution, facilitating efficient computation. Key findings are elucidated: Firstly, the grid size for the symmetric matrix A is intricately linked to a bijective function, enabling the transition of indices to grid points. Notably, the solution to this elliptic problem exhibits a concave-up profile. Secondly, various solvers such as the Conjugate Gradient, Gauss-Seidel, and Jacobi techniques are viable, with the Conjugate Gradient method chosen for its superior accuracy, especially when considering computational efficiency. Moreover, the relationship between grid size and solution accuracy is explored, revealing a proportional dependence. Refinement of the grid leads to increased iteration counts but reduced implementation time, owing to the linearity of the function . The convergence criterion ensures high accuracy in solutions, as demonstrated in the provided figures.