Romberg Type Cubature over Arbitrary Triangles

We develop an extrapolation algorithm for numerical integration over arbitary non-standard triangles in IR², which parallels the well-known univariate Romberg method. This is done by a suitable generalization of the trapezoidal rule over triangles, for which we can prove the existence of an asymptotic expansion. Our approach relies mainly on two ideas: The use of barycentric coordinates and the interpretation of the trapezoidal rule as the integral over an interpolating linear spline function. Since our method works for arbitrary triangles, it yields - via triangulation - a method for cubature over arbitrary, possibly non-convex, polygon regions in IR². Moreover, also numerical integration over convex polyhedra in IR d, d > 2 , can be accomplished without difficulties. Numerical examples show the stability and efficiency of the algorithm.