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## Preconditioning Legendre Spectral Collocation Approximations to Elliptic Problems

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**Author**

Parter, Seymour V.; Rothman, Ernest E.

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**Abstract**

This work deals with the H(exp 1) condition numbers and the distribution of the Beta~(sub N,M)-singular values of the preconditioned operators{Beta~(exp -1) (sub N,M) W(sub N,M) A^(sub N,M)}. A^(sub N,M) is the matrix representation of the Legendre Spectral Collocation discretization of the elliptic operator "A" defined by A(sub u) := -delta(u) + alpha(sub 1)u(sub x) + alpha(sub 2)u(sub y) + alpha(sub 0)u in omega (the unit square) with boundary conditions: u = 0 on Gamma(sub 0), delta(sub u) divided by delta(mu sub A) = alpha(u) on Gamma(sub 1). Beta~(sub N,M) is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator "B" defined by B(v) := -Delta(v) + b(sub 0i)v in omega with boundary conditions v = 0 on Gamma(sub 0), delta(v) divided by delta(mu sub B) = B(v) on Gamma(sub 1). The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the Legendre-Gauss-Lobatto (LGL) points or the space of continuous functions which are linear on a triangulation of omega determined by the LGL points. W(sub N,M) is the matrix of quadrature weights. When A = B we obtain results on the eigenvalues of Beta~(exp -1)(sub N,M) W(sub N,M) B^(sub N,M). We show that there is an integer N(sub 0) and constants alpha,beta with 0 less than alpha less than beta, such that: if min(N,M) gretaer than or equal to N(sub 0),i then all the Beta~(sub N,M)-singular values of Beta~(exp -1)(sub N,M) W(sub N,M) A^(sub N,M) lie in the interval [alpha,beta].Moreover, there is a smaller interval, [alpha(sub 0), beta(sub 0)], independent of the operator "A", such that: if min(N,M) greatre than or equal to N(sub 0), then all but a fixed finite number of the Beta~(sub N,M)-singular value lie in [alpha(sub 0),beta(sub 0)]. These results are related to results of Manteuffel and Parter [MP]Parter and Wong [PW] and Wong [W1], [W2] for finite element discretizations.

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**Date Issued**

1993-02#####
**Publisher**

Cornell University

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**Subject**

theory center

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**Previously Published As**

http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.tc/93-121

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**Type**

technical report