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What is the Complexity of the Fredholm Problem of the Second Kind?

Werschulz, Arthur G.

This paper deals with the approximate solution of the Fredholm problem Lu = f of the second kind, with f ∈Wr,p(I). Of particular interest is the quality of the finite element method (FEM) of degree k using n inner products of f. The error of the approximation is measured in the Lp (I)-norm. We find that the FEM has minimal error iff k ≥ r - 1. However in the Hilbert case p = 2, there always exists a linear combination (called the spline algorithm) of the inner products used by the FEM which does have minimal error; this holds regardless of whether k ≥ r - 1. We also investigate the case where the inner products used by the FEM are not available. Suppose, however, that we can evaluate f (x) for any x ∈ I. In this case, it is reasonable to consider a finite element method with quadrature (FEMQ), in which the inner products required by the FEM are approximated via numerical quadrature. We prove that the FEMQ has minimal error iff k ≥ r - 1. Moreover, we show that the asymptotic penalty for using the FEM or FEMQ with a value of k that is too small is unbounded.

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Academic Units
Computer Science
Publisher
Department of Computer Science, Columbia University
Series
Columbia University Computer Science Technical Reports, CUCS-131-84
Published Here
February 22, 2012
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