Error Analysis of Adaptively Weighted Numerical 1 Integration

With Adaptively Weighted (AW) numerical integration, for a given set of quadrature 4 nodes, order and domain of integration, the quadrature weights are obtained by solving a system of 5 suitable moment fitting equations in least square sense. The moments in the moment equations 6 are approximated over a simplified domain that is homeomorphic to the original domain, and then 7 are corrected for the deviation from the original domain using first-order shape sensitivity analysis. 8 The moment corrections enable the resulting quadrature weights to adapt to the original domain of 9 integration automatically and efficiently. 10 In this paper, we study the convergence of the AW method and a practical way to estimate 11 the error due to moment approximations in an a posteriori sense. We prove that the AW method 12 converges quadratically w.r.t. the maximum design velocity based on Taylor’s remainder theorem. 13 We also propose a simple a posteriori error estimate, based on higher-order Taylor series, that can 14 be used in adaptive numerical integration. We support the developed error estimates with suitable 15 computational examples in 2D. 16