Romberg extrapolation for Euler summation-based cubature on regular regions

Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube [ 0 , 1 ] q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q -dimensional regular regions G ⊂ R q and derive an explicit representation for the remainder term.