High-order quadrature on multi-component domains implicitly defined by multivariate polynomials

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry implicitly defined by the level sets of (one or more) multivariate polynomials. The recasts as graph an defined, multi-valued height function, applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore use Gauss-Legendre tanh-sinh methods demonstrate that inherits their convergence rates. Under action h-refinement with q fixed, schemes yield order accuracy 2q, where node count; numerical experiments up to 22nd order. q-refinement approximately exponential, i.e., doubling doubles number accurate digits computed integral. Complex automatically handled algorithm, including, e.g., multi-component domains, tunnels, junctions arising from multiple polynomial sets, well self-intersections, cusps, other kinds singularities. variety demonstrates on two- three-dimensional problems, including: randomly generated involving high-curvature pieces; challenging examples high degree singularities such cusps; adaptation simplex constraint cells in addition hyperrectangular cells; boolean operations compute overlapping domains.