A Family of Compact Finite Difference Formulations for Three-Space Dimensional Nonlinear Poisson’s Equations in Cartesian Coordinates

Abstract

In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in three-space dimensions has been developed. The discretization formula is developed on a non-uniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of \(l^\infty \)- and \(l^2\)-error of the exact and approximate solution values.