Joseph E. Pasciak

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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)
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)
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)
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)
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)