Convergence Theories of Multigrid Methods Based on the X-z Identity
نویسنده
چکیده
1. X-Z identity 2 1.1. Notation 2 1.2. XZ identities 3 1.3. Some estimates 4 2. Orthogonal Telescope Decomposition 6 2.1. Properties and Assumptions 6 2.2. Convergence 8 3. Approximation and Smoothing Property 8 3.1. High frequency 9 3.2. Approximation property 9 3.3. Smoothing property 9 3.4. Smoothing property of popular smoothers 10 3.5. Convergence 11 4. Stable Decomposition and Quasi-Orthogonality 12 4.1. Assumptions 12 4.2. Convergence 12 4.3. Point-wise smoothers 13 4.4. Decomposition using a stable quasi-interpolation 14 5. Block Smoothers 15 5.1. Smoothers based on micro-decomposition 15 5.2. Point-wise smoother 16 6. Decomposition into leafs 16 6.1. Assumptions 17 6.2. Example: Hierarchical Basis 17 6.3. Smoothing on leaves 17 6.4. Converges 18 7. Smoothing Steps 19 7.1. Smoothing property of m-steps of smoothing 19 7.2. Convergence 20 References 20
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