Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods

Antonietti, Paola; Houston, Paul; Pennesi, Giorgio

Abstract

In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tetrahedral elements. The method is based on successive application of Stokes' theorem; thereby, the underlying integral may be evaluated using only the values of the integrand at the vertices of the polytopic domain, and hence leads to an exact cubature rule whose quadrature points are the vertices of the polytope. We demonstrate the capabilities of the proposed approach by efficiently computing the stiffness and mass matrices arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations.

• Journal Article Type: Article
• Acceptance Date: Jul 31, 2018
• Online Publication Date: Aug 29, 2018
• Publication Date: 2018-12
• Deposit Date: Aug 14, 2018
• Publicly Available Date: Sep 11, 2018
• Journal: Journal of Scientific Computing
• Print ISSN: 0885-7474
• Electronic ISSN: 1573-7691
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 77
• Issue: 3
• Pages: 1339-1370
• DOI: https://doi.org/10.1007/s10915-018-0802-y
• Public URL: https://nottingham-repository.worktribe.com/output/1031787
• Publisher URL: https://www.springerprofessional.de/en/fast-numerical-integration-on-polytopic-meshes-with-applications/16084184?fulltextView=true