How Do Help “Multigrid Principles based on Numerical Solutions of Partial Differential Equations” for Smoothing Process (Concepts)?
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).
Adams, J. C. (1989). MUDPACK: Multigrid portable FORTRAN software for the efficient solution of linear elliptic partial differential equations. Applied Mathematics and Computation, 34(2), 113-146.
Bank, R. E., & Sherman, A. H. (1981). An adaptive, multi-level method for elliptic boundary value problems. Computing, 26(2), 91-105.
Brandt, A., & Lubrecht, A. A. (1990). Multilevel matrix multiplication and fast solution of integral equations. Journal of Computational Physics, 90(2), 348-370.
Briggs, W. L. (1987). Introduction, multigrid methods. Frontiers in Applied Mathematics.
Frederickson, P. O. (1974). Fast approximate inversion of large elliptic systems. Lakehead University, Department of Mathematical Sciences.
Hackbusch, W. (1977). On the convergence of a multi-grid iteration applied to finite element equations. Rep. 77-8. Institute for Applied Mathematics, University of Cologne, West Germany.
Hemker, P. W. (1990). On the order of prolongations and restrictions in multigrid procedures. Journal of Computational and Applied Mathematics, 32(3), 423-429.
Kettler, R. (1982). Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In Multigrid methods (pp. 502-534). Springer, Berlin, Heidelberg.
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