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Analysis of Optimal Finite-Element Meshes in $R^1$

I. Babuska and W. C. Rheinboldt
Mathematics of Computation
Vol. 33, No. 146 (Apr., 1979), pp. 435-463
DOI: 10.2307/2006290
Stable URL: http://www.jstor.org/stable/2006290
Page Count: 29
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Analysis of Optimal Finite-Element Meshes in $R^1$
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Abstract

A theory of a posteriori estimates for the finite-element method was developed earlier by the authors. Based on this theory, for a two-point boundary value problem the existence of a unique optimal mesh distribution is proved and its properties analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10% accuracy. The approaches are not restricted to the model problem considered here only for clarity; in fact, they allow for rather straightforward extensions to more general problems in one dimension, as well as to higher-order elements.

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