Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates
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arXiv
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| Format: | Preprint |
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2024
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| author | Nikolov, Geno Nikolov, Petar |
| author_facet | Nikolov, Geno Nikolov, Petar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates Nikolov, Geno Nikolov, Petar Numerical Analysis Classical Analysis and ODEs 41A55, 65D30, 65D32 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. |
| title | Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates |
| topic | Numerical Analysis Classical Analysis and ODEs 41A55, 65D30, 65D32 |
| url | https://arxiv.org/abs/2404.17796 |