Abstract. A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one-dimensional root finding and one-dimensional Gaussian quadrature. The computed quadrature scheme yields strictly positive quadrature weights and inherits the high-order accuracy of Gaussian quadrature: a range of different convergence tests demonstrate orders of accuracy up to 20th order. Also presented is an application of the quadrature algorithm to a high-order embedded boundary discontinuous Galerkin method for solving partial differential equations on curved domains.
Recently in REF ), it is proposed to convert the implicit geometry into the graph of an implicitly given height function, leading to a recursive algorithm on the number of dimensions and thus requiring only one-dimensional rootfinding and one-dimensional Gaussian quadratures.
HIGH-ORDER QUADRATURE METHODS FOR IMPLICITLY DEFINED SURFACES AND VOLUMES IN HYPERRECTANGLES ∗
SIAM J. Scientific Computing
SIAM J. Scientific Computing
Mathematics Computer Science