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Least squares methods for $2m$th order elliptic boundary-value problems


Authors: J. H. Bramble and A. H. Schatz
Journal: Math. Comp. 25 (1971), 1-32
MSC: Primary 65N99
DOI: https://doi.org/10.1090/S0025-5718-1971-0295591-8
MathSciNet review: 0295591
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Abstract: In this paper we consider a general class of boundary-value problems for 2mth order elliptic equations including nonhomogeneous essential boundary conditions and nonselfadjoint problems. Approximation methods involving least squares approximation of the data are presented and corresponding error estimates are proved. These methods can be considered in the category of Rayleigh-Ritz-Galerkin methods and have the special feature that the trial functions need not satisfy the boundary conditions. A special case of the trial functions which is studied are spline functions defined on a uniform mesh of width h (or more generally piecewise polynomial functions). For a given "well set" boundary-value problem for a 2mth order operator the theory presented will provide a method with any prescribed order of accuracy r which is optimal in the sense that the best approximation in the underlying subspace is of order of accuracy r.


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Keywords: Rayleigh-Ritz-Galerkin methods, least squares approximation, 2<I>m</I>th order elliptic boundary-value problems, numerical solution of higher order elliptic problems
Article copyright: © Copyright 1971 American Mathematical Society