How Do Help “Multigrid Principles based on Numerical Solutions of Partial Differential Equations” for Smoothing Process (Concepts)?

Abstract

In Twenty century, Partial Differential Equations (PDE) are solved by Numerical Approach such as Adam – Smith Methods, Numerical Simulation Methods, Finite Difference methods etc., In this article we discuss about difficulties of solve either Differential equations or PDE in 3 – Dimensional cases. So that we discuss only 1 – Dimensional Multigrid Methods (M. M) and we discuss about M. M.  Errors, Corrections, Type of grid such as Coarse Grid (C. G) and Fine Grid (F. G), Smoothing, non – smoothing approximation of C. G. Finally, we explain that M. G. M. works by decomposing problem into separate length scale and also using an iterative method. This method optimizes errors deduction in the length scales globally. In Multigrid Methods (MgM) several sub – routines must be developed to pass the data from C. G to F. G (Interpolation) from F. G to C. G (Reduction) and correction of the error at each grid interval (Smoothing), simply we have results reaction as (Reduction)C.G⇄F.G (Interpolation).