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Variational-difference approximation

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Abstract

The article begins with a presentation of Courant's approach to the variational-difference method and of the definition of basis functions, as well as the method of construction of coordinate functions. Then the completeness of a system of coordinate functions is studied, and also the degree of approximation of functions of Sobolev classes and of some other (similar) classes, as well as basis functions of more complex structure. Some new problems related to approximate calculation of eigenvalues are also examined, as well as methods of construction of variational-difference equations and, in particular, difference schemes with a boundary layer; the stability of the variational-difference process is studied, as well as the condition number of the corresponding matrix. In the concluding part of the article it is shown that the obtained approximation formulas yield in the one-dimensional case the classical Euler-MacLaurin quadrature formula, and in the multidimensional case the obvious counterparts of this formula.

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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 48, pp. 32–188, 1974.

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Mikhlin, S.G. Variational-difference approximation. J Math Sci 10, 661–787 (1978). https://doi.org/10.1007/BF01083968

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