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]×[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. A 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 C2,2[a,b]={f(x,y):∂4f∂x2∂y2continuousanddoesnotchangesignin(a,b)2}. For integrands from C2,2[a,b] we prove monotonicity of the remainders and derive a posteriori error estimates.