Cubature from rational approximation

We present a numerical construction of cubature rules for area integrals of analytic functions over planar domains with rectifiable Jordan boundary.

The starting point is the Cauchy--Green identity.

Given a weight $w$, we choose a $\bar\partial$-antiderivative $W$ and reduce the area integral to a contour integral involving the boundary values of $W$.

These values are then approximated by a rational function with free poles, computed by the AAA algorithm.

The poles inside the domain become cubature nodes, the corresponding residues become weights, and the boundary residual controls the error through an a posteriori estimate, rigorous once the continuous boundary residual is bounded.

The same rule admits a dual reading, as the exact integral of a rational interpolant to the integrand, the area analogue of the one-dimensional interpolatory viewpoint.

The numerical examples recover the disk mean-value rule and the focal-segment rule of the ellipse to machine precision, reproduce the exact finite quadrature identities of quadrature domains with both separated and confluent nodes, and evaluate logarithmic and Cauchy volume potentials from boundary data alone.

The interior poles trace analytic skeletons that we identify tentatively with the mother bodies of potential theory, along with image points that appear without being imposed; for the square the observed convergence is root-exponential.