## Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity (1990)

Citations: | 40 - 1 self |

### BibTeX

@TECHREPORT{Smith90domaindecomposition,

author = {Barry F. Smith and Barry F. Smith},

title = {Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity},

institution = {},

year = {1990}

}

### OpenURL

### Abstract

The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often ill-conditioned and hence require good preconditioners if they are to be solved iteratively. We propose and analyze three new, parallel iterative domain decomposition algorithms for the solution of these linear systems. The algorithms are also useful for other elliptic partial differential equations. Domain decomposition algorithms are designed to take advantage of a new generation of parallel computers. The domain is decomposed into overlapping or non-overlapping subdomains. The discrete approximation to a partial differential equation is then obtained iteratively by solving problems associated with each subdomain. The algorithms are often accelerated using the conjugate gradient method. The first new algorithm presented here borrows heavily from multi-level type a...

### Citations

103 | The Hierarchical Basis Multigrid Method
- BANK, DUPONT, et al.
- 1988
(Show Context)
Citation Context ...techniques of this chapter to the schemes of Bramble, Pasciak, and Xu as well. Remark: We note that an accelerated version of Yserentant's algorithm has been developed by Bank, Dupont, and Yserentant =-=[3]-=-. It can be viewed as a symmetric multiplicative scheme while the original formulation is an additive scheme. The algorithm proposed in this chapter is an additive scheme. Multiplicative versions can ... |

73 |
Iterative methods for the solution of elliptic problems on regions partitioned into substructures
- rstad, Widlund
- 1986
(Show Context)
Citation Context ...ating algorithm which has some similarities to the algorithm presented in Chapter 4. Much work using domain decomposition has focused on the scalar elliptic problem, including Bjrstad and Widlund [8],=-=[9]-=-; Bramble, Pasciak, and Schatz [12],[15]; Chan and Resasco [23],[24]; Dryja [33],[34]; Dryja and Widlund [35],[36]; and Widlund [84]. Experimental work on the relative effectiveness of various domain ... |

40 |
Efficient preconditioning for the p-version finite element method in two dimensions
- Babuˇska, Guo
- 1991
(Show Context)
Citation Context ...analyzed iterative substructuring type algorithms for elasticity [60],[61],[62],[63]. Other work for the p-version finite element method has been carried out by Babuska, Craig, Mandel, and Pitkaranta =-=[1]-=-, and Babuska, Griebel, and Pitkaranta [2]. For the p-version finite element method, the elements themselves, which are associated with many degrees of freedom, are treated as subdomains. Russian math... |

26 |
The Problem of Selecting the Shape Functions for a p-type Finite Element
- Babuˇska, Griebel, et al.
- 1989
(Show Context)
Citation Context ...orithms for elasticity [60],[61],[62],[63]. Other work for the p-version finite element method has been carried out by Babuska, Craig, Mandel, and Pitkaranta [1], and Babuska, Griebel, and Pitkaranta =-=[2]-=-. For the p-version finite element method, the elements themselves, which are associated with many degrees of freedom, are treated as subdomains. Russian mathematicians are also working on domain deco... |

23 | On the spectra of sums of orthogonal projections with applications to parallel computing
- Bjørstad, Mandel
- 1991
(Show Context)
Citation Context ...has been done by Lions [58]; see also Nepomnyaschikh [68]. For the finite element formulation the two subspace case has been discussed in Bjrstad [5], Mandel and McCormick [64] and Bjrstad and Mandel =-=[7]-=-. More general theories for the case of more then two subspaces have been introduced by Widlund [85] and Mathew [65]. The most successful results so far have been obtained very recently by Bramble, Pa... |

16 |
O.B.: Solving elliptic problems on regions partitioned into substructures
- Bjørstad, Widlund
- 1984
(Show Context)
Citation Context ...ternating algorithm which has some similarities to the algorithm presented in Chapter 4. Much work using domain decomposition has focused on the scalar elliptic problem, including Bjrstad and Widlund =-=[8]-=-,[9]; Bramble, Pasciak, and Schatz [12],[15]; Chan and Resasco [23],[24]; Dryja [33],[34]; Dryja and Widlund [35],[36]; and Widlund [84]. Experimental work on the relative effectiveness of various dom... |

15 |
Multiplicative and Additive Schwarz Methods: Convergence in the 2 domain case
- Bjrstad
- 1989
(Show Context)
Citation Context ...k on the two subspace case for the continuous problem has been done by Lions [58]; see also Nepomnyaschikh [68]. For the finite element formulation the two subspace case has been discussed in Bjrstad =-=[5]-=-, Mandel and McCormick [64] and Bjrstad and Mandel [7]. More general theories for the case of more then two subspaces have been introduced by Widlund [85] and Mathew [65]. The most successful results ... |

8 |
a programming system for the finite element method. User’s manual, Part 1, General description, The
- BELL, HATLESTAD, et al.
- 1973
(Show Context)
Citation Context ... elimination is referred to as static condensation. Hence it has eight degrees of freedom, two for each vertex node. This element has been used in some of the experiments reported in this thesis, see =-=[4]-=-. The flat, thin shell quadrilateral element is based on Kirchoff's plate bending theory, see [4],[74]. In general such an element has six degrees of freedom for each node, three displacements and thr... |

8 |
A.: Iterative Methods for Substructured Elasticity Problems in Structural Analysis
- Bjorstad, Hvidsten
- 1987
(Show Context)
Citation Context ...rz alternating method converges for the equations of linear elasticity. More recent work on elasticity has been done by several scientists. Using the Neumann-Dirichlet algorithm, Bjrstad and Hvidsten =-=[6]-=- have solved actual industrial problems with some success. De Roeck [28], De Roeck, Le Tallec, and Vidrascu [56] and De Roeck and Le Tallec [29] have implemented the algorithm proposed in Bourgat, Glo... |

1 | To Overlap or Not to Overlap: A - Bjrstad, Widlund |