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Quadrature error expansions. II: The full corner singularity. (English) Zbl 0793.41023

Summary: We continue the work of Part I, treating in detail the theory of numerical quadrature over a square \([0,1)^ 2\) using an \(m^ 2\) copy, \(Q^{(m)}\), of a one-point quadrature rule. As before, we determine the nature of an asymptotic expansion for the quadrature error functional \(Q^{(m)}F-IF\) in inverse powers of \(m\) and related functions, valid for specified classes of the integrand function \(F\). The extreme case treated here is one in which the integrand function has a full-corner algebraic singularity. This has the form \(x^ \lambda y^ \mu r_ \rho (x,y)\). Here \(\lambda\), \(\mu\), and \(\rho\) need not be integer, and \(r_ \rho\) is \((x^ 2+y^ 2)^{\rho/2}\) or some other similar homogeneous function. The error expansion forms the theoretic basis for the use of extrapolation for this kind of integrand.

MSC:

41A55 Approximate quadratures
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References:

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