Journal ArticleDOI
On the multi-grid method applied to difference equations
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A multi-grid method is applied to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.Abstract:
Multi-grid methods are characterized by the simultaneous use of additional auxiliary grids corresponding to coarser step widths. Contrary to usual iterative methods the speed of convergence is very fast and does not tend to one if the step size approaches zero. The computational amount of one iteration is proportional toN, the number of grid points. Thus, a solution with accuracy ɛ requires 0 (|log ɛ|N) operations. In this paper we apply a multi-grid method to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.read more
Citations
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Journal ArticleDOI
High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method
U Ghia,K.N Ghia,C. T. Shin +2 more
TL;DR: The vorticity-stream function formulation of the two-dimensional incompressible NavierStokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions.
Journal ArticleDOI
Black Box Multigrid
TL;DR: The end result is code, BOXMG, in which one need only specify the (logically rectangular, positive definite) matrix problem; BOXMG does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution.
Journal ArticleDOI
Metric‐based upscaling
Houman Owhadi,Lei Zhang +1 more
TL;DR: In this paper, divergence form elliptic operators in dimension n ge; 2 with L∞ coefficients were considered and it was shown that they are differentiable (C1, α) with respect to harmonic coordinates.
Journal ArticleDOI
Multigrid Methods for PDE Optimization
Alfio Borzì,Volker Schulz +1 more
TL;DR: Research on multigrid methods for optimization problems is reviewed and problems considered include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type.
Book ChapterDOI
The analysis of multigrid methods
James H. Bramble,Xuejun Zhang +1 more
TL;DR: This chapter discusses the analysis of multigrid methods and details the construction of two classes of commonly used smoothing operators (smoothers): additive and multiplicative smoothers.
References
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Journal ArticleDOI
Multi-level adaptive solutions to boundary-value problems
TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.