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 and its derivatives 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.
Highlights
In recent years the exploitation of computational meshes composed of polygonal and polyhedral elements has become very popular in the field of numerical methods for partial differential equations
Several conforming numerical discretization methods which admit polygonal/polyhedral meshes have been proposed within the current literature; here, we mention, for example, the Composite Finite Element Method [5,41,42], the Mimetic Finite Difference (MFD) method [4,18,19,20,21,44], the Polygonal Finite Element Method [63], the Extended Finite Element Method [37,64], the Virtual Element Method (VEM) [10,11,15,16,17] and the Hybrid High-Order (HHO) method [33,34,35]
In the non-conforming setting, we mention Discontinuous Galerkin (DG) methods [1,2,3,6,9,14,22,23,24,25,25], Hybridizable DG methods [29,30,31,32], non-conforming VEM [8,13,26], and the Gradient Schemes [36]; here the possibility of defining local polynomial discrete spaces follows naturally with the flexibility provided by polytopic meshes
Summary
In recent years the exploitation of computational meshes composed of polygonal and polyhedral elements has become very popular in the field of numerical methods for partial differential equations. The essential idea here is to exploit the generalized Stokes’ theorem together with Euler’s homogeneous function theorem, cf [58], in order to reduce the integration over a polytope only to boundary evaluations. The main difference with respect to the work presented in [59] is the possibility to apply the same idea recursively, leading to a quadrature formula which exactly evaluates integrals over a polygon/polyhedron by employing only point-evaluations of the integrand and its derivatives at the vertices of the polytope. 2 we recall the work introduced in [27], and outline how this approach can be utilized to efficiently compute the integral of d-variate polynomial functions over general polytopes. Remark 3 We point out that in (12), cf. (13), the shape of the underlying polytope can be general: nonconvex -connected domains E are admissable
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