The authors study two modifications of the trapezoidal product cubature formulae, S-n and S+n, which approximate double integrals over the square domain [a,b]2. These modified cubature formulae use mixed type data, involving both evaluations of the integrand on a uniform grid and univariate integrals.
The key highlights and insights are:
S-n and S+n are definite cubature formulae of order (2,2), meaning they provide one-sided approximations to the double integral for integrands from the class C2,2[a,b] (functions with continuous fourth mixed partial derivative that does not change sign).
The authors prove monotonicity properties for the remainders of S-n and S+n. Doubling the number of grid points reduces the error magnitude by at least a factor of two.
A posteriori error estimates are derived, which can serve as stopping rules in automated numerical integration routines.
The error bounds in the monotonicity and a posteriori results are shown to be the best possible.
Numerical examples illustrating the theoretical results are provided.
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