On the identification of symmetric quadrature rules for finite element methods. In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

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  1. Arndt, Daniel; Bangerth, Wolfgang; Blais, Bruno; Fehling, Marc; Gassmöller, Rene; Heister, Timo; Heltai, Luca; Köcher, Uwe; Kronbichler, Martin; Maier, Matthias; Munch, Peter; Pelteret, Jean-Paul; Proell, Sebastian; Simon, Konrad; Turcksin, Bruno; Wells, David; Zhang, Jiaqi: The deal. II library, version 9.3 (2021)
  2. Frontin, Cory V.; Walters, Gage S.; Witherden, Freddie D.; Lee, Carl W.; Williams, David M.; Darmofal, David L.: Foundations of space-time finite element methods: polytopes, interpolation, and integration (2021)
  3. You, Hojun; Kim, Chongam: Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes III. Code optimization via tensor contraction (2021)
  4. Marchildon, André L.; Zingg, David W.: Optimization of multidimensional diagonal-norm summation-by-parts operators on simplices (2020)
  5. Modarreszadeh, Amir; Timofeev, Evgeny: High-order numerical simulation of axisymmetric wave phase conjugation (2020)
  6. Williams, David M.; Frontin, Cory V.; Miller, Edward A.; Darmofal, David L.: A family of symmetric, optimized quadrature rules for pentatopes (2020)
  7. You, Hojun; Kim, Chongam: Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes. II: Surface integration (2020)
  8. Chan, Jesse; Wilcox, Lucas C.: On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes (2019)
  9. Codony, D.; Marco, O.; Fernández-Méndez, S.; Arias, I.: An immersed boundary hierarchical B-spline method for flexoelectricity (2019)
  10. Geevers, S.; Mulder, W. A.; van der Vegt, J. J. W.: Efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for linear wave problems (2019)
  11. Vondřejc, Jaroslav: Double-grid quadrature with interpolation-projection (DoGIP) as a novel discretisation approach: an application to FEM on simplexes (2019)
  12. You, Hojun; Kim, Chongam: Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes. I: Volume integration (2019)
  13. Dawson, Mark; Sevilla, Ruben; Morgan, Kenneth: The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes (2018)
  14. Poya, Roman; Gil, Antonio J.; Ortigosa, Rogelio; Sevilla, Ruben; Bonet, Javier; Wall, Wolfgang A.: A curvilinear high order finite element framework for electromechanics: from linearised electro-elasticity to massively deformable dielectric elastomers (2018)
  15. Chan, Jesse; Hewett, Russell J.; Warburton, T.: Weight-adjusted discontinuous Galerkin methods: curvilinear meshes (2017)
  16. Xie, Bin; Xiao, Feng: Toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids: the THINC method with quadratic surface representation and Gaussian quadrature (2017)
  17. Chan, Jesse; Warburton, T.: Orthogonal bases for vertex-mapped pyramids (2016)
  18. Gillette, Andrew: Serendipity and tensor product affine pyramid finite elements (2016)
  19. Hicken, Jason E.; Del Rey Fernández, David C.; Zingg, David W.: Multidimensional summation-by-parts operators: general theory and application to simplex elements (2016)
  20. Papanicolopulos, Stefanos-Aldo: New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases (2016)

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