Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates
Geno Nikolov, Petar Nikolov
TL;DR
The paper develops two modified trapezoidal product cubature rules $S_n^{-}$ and $S_n^{+}$ for computing $I[f]$ over $[a,b]^2$ for functions in $\mathcal{C}^{2,2}[a,b]$, achieving one-sided definite bounds of order $(2,2)$ through blending and Peano-kernel analysis. It proves monotonicity of the remainders under refinement and provides explicit a-posteriori error estimates, with sharp constants: $|R[S_{2n}^{-};f]|\leq \tfrac{1}{2}|R[S_n^{-};f]|$ and $|R[S_{2n}^{-};f]|\leq |S_{2n}^{-}[f]-S_n^{-}[f]|$ for $f\in\mathcal{C}^{2,2}[a,b]$, and $|R[S_{2n}^{+};f]|\leq (\tfrac{1}{2}+\tfrac{1}{4(2n-1)})|R[S_n^{+};f]|$ and $|R[S_{2n}^{+};f]|\leq \tfrac{4n-1}{4n-3}|S_{2n}^{+}[f]-S_n^{+}[f]|$. The analysis relies on Peano kernel representations and blending interpolation, and is complemented by a numerical example that confirms the theoretical bounds and showcases practical stopping rules.
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.