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 C 2 , 2 [ a , b ] = { f ( x , y ) : ∂ 4 f ∂ x 2 ∂ y 2 continuous and does not change sign in ( a , b ) 2 } . For integrands from C 2 , 2 [ a , b ] we prove monotonicity of the remainders and derive a posteriori error estimates.