Numerical Solution of the 2D Poisson Equation Using Finite Difference Method and Iterative Solvers

Abstract

Background:
The two-dimensional Poisson equation is a fundamental equation to model many physical phenomena including electrostatics, heat conduction and fluid flow. The finite difference method (FDM) is a reliable and popular numerical technique, particularly on structured domains as in the unit square, although analytical solutions are confined to idealized geometries.

Materials and Methods:

Implemented standard five-point FDM discretization of 2D Poisson equation with Dirichlet boundary conditions, and systematically compared four iterative solvers (Jacobi, Gauss–Seidel, Successive Over-Relaxation (SOR), Preconditioned Conjugate Gradient (PCG) with Jacobi and Incomplete Cholesky (IC(0)) preconditioners) at three grid resolutions (n = 64, 128, 256). We assess performance in terms of iteration count, runtime, convergence rate, scalability, and statistical significance (ANOVA + Tukey HSD, p < 10⁻¹⁵).

Results:
PCG with IC(0) preconditioning required as few as 99% fewer iterations than Jacobi and hence yielded O(n) scaling very close to optimal. When tuned well (ω ≈ 1.92), SOR achieved equally good results but SOR was more sensitive to ω. All solvers achieved discretization-limited accuracy (∼10⁻⁵), and parallel (OpenMP) tests demonstrated strong scaling for Jacobi and the PCG method, but limited parallel efficiency for the Gauss–Seidel solver due to data dependencies.

Conclusion:
In the case of 2D Poisson problems on regular grids that are of general relevance, IC-preconditioned PCG achieves by far the best balance of speed, robustness and ease of implementation. While SOR excels with brass knobs for ADI when considered in a two-dimensional, rectangular lens, SOR remains viable for educational or reduced discrete unit use if selectively tuned. Scope of Statistical Validation & Open Source ReproducibilityNumerical benchmarks are strengthened by statistical validation and open source replicability.