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High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. (English) Zbl 1328.65070

Summary: 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.

MSC:

65D30 Numerical integration
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

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References:

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