Abstract
Multigrid methods belong to the class of fastest linear iterations, since their convergence rate is bounded independently of the step size h. Furthermore, their applicability does not require symmetry or positive definiteness. Books devoted to multigrid are Hackbusch [183], Wesseling [395], Trottenberg–Oosterlee–Schuller [367], Shaidurov [338], and Vassilevski [378]; see also [205, pp. 1–312]. The ‘smoothing step’ and the ‘coarse-grid correction’ together with the involved restrictions and prolongations are typical ingredients of the multigrid iteration. They are introduced in Section 11.1 for the Poisson model problem. The two-grid iteration explained in Section 11.2 is the first step towards the multigrid method. The iteration matrix is provided in §11.2.3. First numerical examples are presented in §11.2.4. Before a more general proof of convergence is presented, Section 11.3 investigates the one-dimensional model problem. The proof demonstrates the complementary roles of the smoothing part and the coarse-grid correction. Moreover, the dependence of the convergence rate on the number of smoothing steps is determined. The multigrid iteration is defined in Section 11.4. Its computational work is discussed and numerical examples are presented. The iteration matrix is described in §11.4.4. The nested iteration presented in Section 11.5 is a typical technique combined with the multigrid iteration. In principle, it can be combined with any iteration, provided that a hierarchy of discretisations is given. Besides a reduction of the computational work, the nested iteration technique allows us to adjust the iteration error to the discretisation error. A general convergence analysis of the W-cycle is presented in Section 11.6. Stronger statements are possible in the positive definite case which is studied in Section 11.7. Here, also the V-cycle convergence is proved. As long as lower order terms are responsible for the nonsymmetric structure, the symmetric convergence results can be transferred as shown in §11.7.6. This includes the case of the V-cycle. Possible combinations with semi-iterative methods are discussed in Section 11.8. Concluding comments are given in Section 11.9.
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Hackbusch, W. (2016). Multigrid Iterations. In: Iterative Solution of Large Sparse Systems of Equations. Applied Mathematical Sciences, vol 95 . Springer, Cham. https://doi.org/10.1007/978-3-319-28483-5_11
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DOI: https://doi.org/10.1007/978-3-319-28483-5_11
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