Convergence Analysis Of Multigrid Method For Shifted Laplace At Various Levels Using Fourier Modes

In this paper, the convergence analysis of Multigrid solver is discussed for shifted Laplace equation. Multigrid is considered best choice for elliptic type partial differential equations, so is for Laplace equation. However inclusion of shift in Laplace equation disturbs spectral properties, which are usually not favorable for basic iterative methods. An analysis is inevitable to know the reasons for bad convergence of Multigrid for shifted Laplace equation. Multigrid components are separately analyzed and spectral expressions are derived. Their graphical interpretation is presented. The analysis of components of Multigrid is combined, in order to derive the closed-form of convergence factor of Multigrid method in two-grid fashion. The graphical interpretation of analysis is given, with recommendations of best and optimal parameters. This helps to recognize components of Multigrid causing slow convergence. Recommendation for fine tuning such components is given in order to obtain better convergence for shifted Laplace problem.