Jean-Bernard Lasserre

Homogeneous Numerical Integration (HNI). We consider the (computational) problem of integrating a continuous function on a polytope in Rn. and more generally on domains whose boundary is defined by homogeneous functions. We have shown that integrating a homogeneous function reduces to integrating the same function on the boundary (e.g. faces of a polytope). Iterating simply reduces the integration problem to evaluation of the function and some of its derivatives at the vertices of the polytope.

J.B. Lasserre (1983). An analytical expression and an algorithm for the volume of a convex polytope. J. Optim. Theory. Appl. 39, pp. 363--377.
J.B. Lasserre (1998). Integration on a convex polytope. Proc. Amer. Math. Soc. 126, pp. 2433--2441
J.B. Lasserre (1999). Integration and homogeneous functions, Proc. Amer. Math. Soc. 127, pp. 813--818.

Therefore by iterating, integrating a polynomial on a polytope reduces to evaluations of the functions and its derivatives at the vertices only. It has been also extended to non convex polytopes. The HNI method (has been successfully implemented in numerical schemes for partial differential equations with sophisticated meshes and also in the well-known XFEM and NEM methods. For instance see the papers:

Therefore by iterating, integrating a polynomial on a polytope reduces to evaluations of the functions and its derivatives at the vertices only. It has been also extended to non convex polytopes in

``Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra", by E. Chin, J.B. Lasserre, N. Sukumar, Comput. Mech. 56 (2015), pp. 967--981.

.The HNI method (has been successfully implemented in numerical schemes for partial differential equations with sophisticated meshes and also in the well-known XFEM and NEM methods. For instance see the papers:

``Modeling crack discontinuities without element partitioning in the extended finite element method", by E. Chin, J.B. Lasserre, N. Sukumar, Int. J. Num. Methods Eng. 110 (2017), pp. 1201--1048.
"Attraction Controls the Inversion of Order by Disorder in Buckled Colloidal Monolayers", by Fabio Leoni and Yair Shokef, Phys. Rev. Lett. 118, 218002, 2017
"Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods", by P.F. Antonietti, P. Houston, and G. Pennesi, J. Scientific Computing 77 (2018), pp. 1339--1370.
"Modeling curved interfaces without element‐partitioning in the extended finite element method" by E.B. Chin and N. Sukumar, Int. J. Numer. Methods in Eng. (2019)
"Extended virtual element method for the Laplace problem with singularities and discontinuities" by E. Benvenuti, A. Chiosi, G. Manzini, and N. Sukumar, in Computer Methods in Appl. Mech. Eng. 354, pp. 571--597 (2019)
"A line integral approach for the computation of the potential harmonic coefficients of a constant density polyhedron" by O. Jamet and D. Tsoulis, in Journal of Geodesy (2020)
``Extended virtual element method for the torsion problem of cracked prismatic beams", by Andrea Chiozzi & Elena Benvenuti, Meccanica 55, pp. 637--648, 2020
`An efficient method to integrate polynomials over polytopes and curved solides", by E. B. Chin & N. Sukumar, Computer Aided Geometry Design 82, 2020
Spectral extended finite element method for band structure calculations in phononic crystals", by E. B. Chin, A.A. Mokhtari, A. Srivastava, N. Sukumar, Journal of Computational Physics 427 (2021), 110066
High–order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations, by Antonietti P.F., Facciolà C., Houston P., Mazzieri I., Pennesi G., Verani M. (2021) . In: Di Pietro D.A., Formaggia L., Masson R. (eds) Polyhedral Methods in Geosciences. SEMA SIMAI Springer Series, vol 27. Springer, Cham.
Minimizing numerical error in computing integrals on polynomial domains, by Fabian Gabor (2021). Annales Univ. Sci. Budapest, Section Comp. 52, pp. 109--129.
Robust high-order unfitted finite elements by interpolation-based discrete extension by S Badia, E Neiva and F Verdugo (2022), Computer & Mathematics with Applications 127, pp. 105--126.
NURBS enhanced virtual element methods for the spatial discretization of the multigroup neutron diffusion equation on curvilinear polygonal meshes by J.A. Ferguson, J. Kophasi, and M.D. Eaton (2022), J. Comput. Theor. Transport, 51(4), pp. 145--204.
Adaptive quadrature/cubature rule: Application to polytopes by B. Boroomand, N. Niknejadi (2023), Comput. Methods Appl. Mech. Engrg, 403, 115726 (2023).
Evaluation of inner products of implicitly defined finite element functions on multiply connected planar mesh cells, by J.S. Ovall, S.E Reynolds (2024), SIAM J. Sci. Computing 46(1), 10.1137/23M1569332
Quadrature-free polytopic discontinuous Galerkin methods for transport problems†, by Thomas J. Radley, Paul Houston* and Matthew E. Hubbard (2024), Mathematics in Engineering, 6(1): 192–220.
An issue on the surface integrals with face decomposition in the virtual element method and its improvement without the decomposition, by Min Ru, Guangtao Xu, Chuanqi Liu (2024), Comput. Methods Appl. Mech. Engrg. 428, 117107
Quadrature-free polytopic discontinuous Galerkin methods for transport problems, by T.J. Radley, P. Houston, M.E. Hubbard (2024), Mathematics in Engineering 6(1), pp. 192--220

We had already used a similar property to compute the volume of a polytope, with good performances. We have extended some of the previous results to arbitrary compact domains. We have also proposed another approach. We consider the result as a function of the right-hand-side and obtain a simple closed form expression of its Laplace transform. We then invert the latter by Cauchy residues techniques. For papers citing our work see the section ``Softwares".