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- Joseph E. Pasciak
- SIAM Review
- 1995

Invariant manifolds form one of the most important subjects in the theory ofdifferential equations and dynamical systems and have found many applications. In the first four pages of the book, the author gives a list of papers where invariant manifolds have been applied with a short description of each paper. He also gives a short historical survey ofthe… (More)

The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm discussed recently together with J. Xu and the standard V-cycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a uniform reduction per iteration… (More)

- James H. Bramble, Raytcho D. Lazarov, Joseph E. Pasciak
- Math. Comput.
- 1997

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner… (More)

In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a… (More)

- James H. Bramble, Joseph E. Pasciak, Andrew V. Knyazev
- Adv. Comput. Math.
- 1996

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V . In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall… (More)

- Chisup Kim, Raytcho D. Lazarov, Joseph E. Pasciak, Panayot S. Vassilevski
- SIAM J. Numerical Analysis
- 2001

We consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions. Abstract conditions are given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation. Three examples of multipliers satisfying these conditions are given. The first one is a dual basis… (More)

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… (More)

In this paper, new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems. Our theory relates the convergence of multigrid algorithms to a "regularity and approximation" parameter a 6 (0,1] and the number of relaxations m. We show that for the symmetric and nonsymmetric V… (More)

This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by… (More)

In this paper, we present an analysis of a multigrid method for nonsym-metric and/or indeenite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is deened in terms of an associated symmetric problem and includes point and line, Jacobi and Gauss-Seidel iterations. We also study… (More)