Numerical solution of elliptic PDEs using 9-point modified relaxation method (MRM)

Abstract

Elliptic partial differential equations (PDEs) frequently arise in steady-state physical models such as heat conduction, electrostatics, and incompressible fluid flow. Analytical solutions for such equations are often infeasible for complex geometries or boundary conditions, necessitating efficient numerical approaches. This study presents the development and application of a 9-point modified relaxation method for solving second-order elliptic PDEs. Unlike the traditional 5-point finite difference scheme, the 9-point stencil enhances accuracy by incorporating diagonal neighbors, allowing better approximation of the Laplacian operator. The method is coupled with an iterative relaxation strategy that accelerates convergence through optimal weighting and boundary adjustment techniques. Numerical experiments conducted on benchmark Dirichlet problems demonstrate improved accuracy and stability over standard Gauss-Seidel and Jacobi schemes. The proposed method also exhibits faster convergence and reduced discretization error on refined grids. This research underscores the effectiveness of high-order stencil relaxation methods for accurate and computationally efficient solutions of elliptic boundary value problems. The 9-point Relaxation Method (MRM) derived by averaging the Standard 5-point and Diagonal 5-point Relaxation Method. The newly modified method was implemented on two-dimensional PDEs of Laplace and Poisson type. For the sake of accuracy, the MRM was compared with the Standard 5-point Method.