Multigrid methods are a prime source of important advanced in algorithmic efficiency, finding a rapidly increasing number of solving problems with $N$ unknowns with $O(N)$ work and storage, not just for special cases, but for large classes of problems. In many papers, the convergence of Multigrid algorithms was proved when linear systems are positive definite system and smoother in Multigrid method is Richrdson``s iteration, Jacobi iteration, and Gauss-Seidel iteration by assuming some conditions which is concerned smoothers. In Chapter 2, when the linear systems are positive definite, we show the convergence of Multigrid algorithms with general smoother which satisfy the weaker smoothing assumptions. Also, we show that these weaker smoothing assumptions are satisfied by Richardson``s iteration, Jacobi iteration, Gauss-Seidel iteration, and Kaczmarz iteration.
In Chapter 3, we analyze Jacobi iteration, Gauss-Seidel iteration, and Kaczmarz iteration by using local mode analysis(Fourier modeanalysis) which is a classical method for showing the efficiency of the smoothers.
If linear systems have a unique solution but are not positive definite, we can solve only by using Richardson``s iteration and Kaczmarz iteration. As a example, we analyze Multigrid algorithms for solving the linear system which are generated by discretization of the mixed type formulation of the linear Elasticity in Chapter 4. As a example of overdetermined system, we analyze Multigrid algorithm for the solution of the cell vertex finite volume method for the Cauchy-Rimann equations in Chapter 5. In this chapter, we introduce a new norm and we show that the convergence of Multigrid method with Kaczmarz smoother by using this norm.