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

TL;DR

This paper introduces modified trapezoidal product cubature rules for double integrals that are definite, monotonic, and provide a posteriori error estimates for specific smooth functions, enhancing numerical integration accuracy.

Contribution

It develops new modified cubature formulas using mixed data that are definite of order (2,2), with proven monotonicity and error estimation methods for certain smooth functions.

Findings

Cubature formulas are definite of order (2,2).

Monotonicity of remainders is established.

A-posteriori error estimates are derived.

Abstract

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.