Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain \([a,b]^2=[a,b]\times [a,b]\). Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on \([a,b]^2\), t...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Nikolov, Geno, Nikolov, Petar
Format: Article
Language:English
Subjects:
Approximation
Error analysis
Estimates
Integrals
Online Access:Get full text
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Summary:We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain \([a,b]^2=[a,b]\times [a,b]\). Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on \([a,b]^2\), they involve two or four univariate integrals. An useful property of these cubature formulae is that they are definite of order \((2,2)\), that is, they provide one-sided approximation to the double integral for real-valued integrands from the class $$ \mathcal{C}^{2,2}[a,b]=\{f(x,y)\,:\,\frac{\partial^4 f}{\partial x^2\partial y^2}\ \text{continuous and does not change sign in}\ (a,b)^2\}. $$ For integrands from \(\mathcal{C}^{2,2}[a,b]\) we prove monotonicity of the remainders and derive a-posteriori error estimates.
ISSN:2331-8422