Iterative Solution of Laplace's Equation on Rectangular Domains Using Finite Difference 5-Point Stencil Method

In this article, we explore the implementation of the finite difference 5-point stencil iterative method for solving Laplace's equation on rectangular domains. The method involves discretizing the domain into a grid mesh and approximating Laplace's equation at each grid point, resulting in a large linear system. The implementation typically requires defining grid spacing parameters (Δx and Δy) and establishing boundary conditions programmatically. The core algorithm computes the finite difference approximation using the 5-point stencil formula: ∇²u ≈ (u(i+1,j) + u(i-1,j) - 2u(i,j))/Δx² + (u(i,j+1) + u(i,j-1) - 2u(i,j))/Δy². This formulation generates a system of equations that can be solved iteratively using methods like Jacobi or Gauss-Seidel iteration. During the iterative process, critical implementation considerations include selecting appropriate convergence criteria (such as tolerance thresholds for residual norms) and initial guess strategies to ensure algorithm convergence and numerical accuracy. The code implementation typically involves nested loops for grid traversal and convergence checking mechanisms. Key programming aspects include handling boundary conditions through conditional statements, implementing efficient data structures for storing grid values, and optimizing computational performance through potential parallelization techniques. The finite difference 5-point stencil iterative method proves to be an effective approach for solving Laplace's equation with broad applications in computational physics and engineering simulations.