Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms


Authors: James H. Bramble, Joseph E. Pasciak and Jinchao Xu
Journal: Math. Comp. 56 (1991), 1-34
MSC: Primary 65N22; Secondary 65N55
MathSciNet review: 1052086
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the $ \mathcal{V}$ cycle, $ \mathcal{W}$ cycle and a $ \mathcal{V}$ cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way.

We apply this theory to various numerical approximations of second-order elliptic boundary value problems. In our first example, we consider certain finite difference multigrid algorithms. In the second example, we consider a finite element multigrid algorithm with nested spaces, which however uses a prolongation operator that does not coincide with the natural subspace imbedding. The third example gives a multigrid algorithm derived from a loosely coupled sequence of approximation grids. Such a loosely coupled grid structure results from the most natural standard finite element application on a domain with curved boundary. The fourth example develops and analyzes a multigrid algorithm for a mixed finite element method using the so-called Raviart-Thomas elements.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N22, 65N55

Retrieve articles in all journals with MSC: 65N22, 65N55


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1991-1052086-4
PII: S 0025-5718(1991)1052086-4
Article copyright: © Copyright 1991 American Mathematical Society