Approximation Solution For Nonlinear Poisson Equation By Finite Element_Homotopy Method

Abstract

The main aim to this research is to find the approximation solution of the nonlinear Poisson equation by combining the finite element method FEM with homotopy analysis method HAM in a single approximation method because the finite element method needs when applied to nonlinear partial differential equations either for iterative methods such as ( Newton_Gauss Method¹⁴, Picard's iterative method³, Newton_Galarkin Method⁴) or for other approximation methods such as(B_Splain²⁴, Homotopy Analysis⁶).

In this article, the finite element method was combined with the Homotopy analysis method with one method called Homotopy _Finite Element Method FE_HM, to convert the nonlinear matter into a linear matter through the HAM method, and to overcome the engineering complexity of the region by a grid of finite elements through the FEM method, We obtained good results when applying this method to the nonlinear Poisson equation on Dirichlet boundary condition, and this method was programmed through the Matlab program.