Adaptive quadrature/cubature rule: Application to polytopes

Abstract

A new quadrature rule for 2D/3D domains is proposed. We first embed the domain in a circumscribing square/cube and then distribute a set of Gauss points in the square/cube. A generic function is approximated through a set of Legendre polynomials whose coefficients are determined by numerical integration. The information needed for the points falling outside the physical domain is re-generated through the polynomials and the unknown coefficients. The reproducibility of the polynomials and the sufficiency of the number of Gauss points may be analyzed through a simple factorization process during the solution of the resulting system of equations. Two simple adaptive algorithms are proposed for when the integration points are to be increased so that the targeted polynomials are fully reproduced. The adaptive processes are designed so that the coefficient matrix, which always is a square matrix, becomes regular. This contributes to the robustness of the whole procedure and also allows the use of conventional inverse processes The integration of the generic function is then summarized into a set of integrals (of the Legendre polynomials) multiplied by the function’s discrete values. The Legendre polynomials are integrated through the divergence theorem (or Stokes theorem). Although the formulation is suitable for arbitrary domains, the focus of this paper is on polytopes since the divergence theorem can easily be applied. To demonstrate the performance of the method, several 2D/3D integral problems are solved and discussed.