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What Is the Complexity of Elliptic Systems?

Werschulz, Arthur G.

This paper deals with the optimal solution of the Petrovsky-elliptic system lu = f, where l is of homogeneous order t and f (x) ∈ H (Ω).Of particular interest is the strength of finite element information (FEI) of degree k, as well as the quality of the finite element method (FEM) using this information. We show that the FEM is quasi-optimal iff k ≥ r+t - 1. Suppose this inequality is violated; is the lack of optimality in the FEM due to the information that it uses, or is it because the FEM makes inefficient use of its information! We show that the latter is the ease. The FEI is always quasi-optimal information. That is, the spline algorithm using FEI is always a quasi-optimal algorithm. In addition, we show that the asymptotic penalty for using the FEM when k is too small (rather than the spline algorithm which uses the same finite element information as the FEI) 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-132-84
Published Here
February 22, 2012
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