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Introductory Model Problem
General TwoGrid Method
General MultiGrid Iteration
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analogous analysis applied approximation property Assume boundary conditions boundary value problem bounded Brandt choice coarse-grid correction computed contraction number convergence rate damped Jacobi iteration defect correction defined denoted difference schemes discrete discretisation error eigenvalue problem eigenvectors equivalent estimate Euclidean norm Example Exercise finite element method formula GauB-Seidel iteration grid function grid points Hackbusch Hence Hilbert space holds implies inequality integer integral equations interpolation iteration matrix K-cycle Lemma linear M-matrix Math matrix norm multi-grid algorithm multi-grid convergence multi-grid iteration multi-grid method nested iteration nine-point nonlinear norm Note obtain operator piecewise Poisson Poisson's equation positive definite post-smoothing prolongation Proof Proposition prove replaced restriction rtnj satisfies scalar product smoothing iteration smoothing process smoothing property solving spectral norm spectral radius step symmetric Table Theorem tion two-grid iteration variant vector x-line yields