A relaxation method for solving elliptic difference equations

Fedorenko, R.P.

doi:10.1016/0041-5553(62)90031-9

Cited by 243 publications

(139 citation statements)

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“…• The remaining low-frequency components in the error can be well represented on a coarser grid, 2 as Federenko points out in his 1961 paper presenting the first complete multigrid method [28]:…”

Section: Multigrid Methodsmentioning

confidence: 99%

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Ernst

Gander

2011

Lecture Notes in Computational Science and Engineering

252

262

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Section: Multigrid Methodsmentioning

confidence: 99%

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Ernst

Gander

2011

Lecture Notes in Computational Science and Engineering

252

262

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“…Since the pioneering work of R.P. Fedorenko [15], multigrid literature has grown at an astonishing rate.…”

Section: Future Development Of Discrete-continual Finite Element Methodsmentioning

confidence: 99%

Semianalytical analysis of shear walls with the use of discrete-continual finite element method. Part 2: Numerical examples, future development

2016

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Abstract. The distinctive paper is devoted to the two-dimensional semianalytical solution of boundary problems of analysis of shear walls with the use of discrete-continual finite element method (DCFEM). This approach allows obtaining the exact analytical solution in one direction (so-called "basic" direction), also decrease the size of the problem to onedimensional common finite element analysis. Two numerical examples of structural analysis with the use of DCFEM are considered, conventional finite element method (FEM) is used for verification purposes. The presented examples show some of the advantages of the suggested approach to semianalytical analysis of the shear wall. Future development of DCFEM, particularly associated with multigrid approach, is under consideration as well.

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“…Stiefel [15], Fedorenko [5], Ahamed [19], Wachspress [17], de la Vallée Poussin [16] and Settari and Aziz [24]. Southwell called his technique "block" and more generally "group relaxation", described it as "almost essential to practical success", and gave heuristic explanation as well as practical implementation methods based on variational considerations ("the aim being to reduce the total energy by as great an amount as possible").…”

Section: "K+l Hkmentioning

confidence: 99%

Untitled

1977

Math. Comp.

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Abstract.The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to 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. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed.

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A relaxation method for solving elliptic difference equations

Cited by 243 publications

References 1 publication

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Semianalytical analysis of shear walls with the use of discrete-continual finite element method. Part 2: Numerical examples, future development

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