Abstract
Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver.
The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology.
Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron.
The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions.
Consistent with existing results, the presented algorithms for the integration of
Funding source: Air Force Office of Scientific Research
Award Identifier / Grant number: FA9550-12-1-0484
Funding statement: Jaime Mora has been sponsored by a 2015 Colciencias-Fulbright scholarship, granted by the Government of Colombia and the Fulbright Commission-Colombia. Leszek Demkowicz has been supported by a grant from AFOSR (FA9550-12-1-0484).
Acknowledgements
The authors thank the anonymous reviewers for their valuable contributions to the quality of the paper.
References
[1] M. Ainsworth, G. Andriamaro and O. Davydov, Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures, SIAM J. Sci. Comput. 33 (2011), no. 6, 3087–3109. 10.1137/11082539XSearch in Google Scholar
[2] P. Antolin, A. Buffa, F. Calabrò, M. Martinelli and G. Sangalli, Efficient matrix computation for tensor-product isogeometric analysis: The use of sum factorization, Comput. Methods Appl. Mech. Engrg. 285 (2015), 817–828. 10.1016/j.cma.2014.12.013Search in Google Scholar
[3] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, A posteriori error control for DPG methods, SIAM J. Numer. Anal. 52 (2014), no. 3, 1335–1353. 10.1137/130924913Search in Google Scholar
[4] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. 72 (2016), no. 3, 494–522. 10.1016/j.camwa.2016.05.004Search in Google Scholar
[5] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations 27 (2011), no. 1, 70–105. 10.1002/num.20640Search in Google Scholar
[6] L. Demkowicz and J. Gopalakrishnan, An overview of the discontinuous Petrov Galerkin method, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, IMA Vol. Math. Appl. 157, Springer, Cham (2014), 149–180. 10.1007/978-3-319-01818-8_6Search in Google Scholar
[7] L. Demkowicz and J. Gopalakrishnan, Discontinuous Petrov–Galerkin (DPG) method, ICES Report 15-20, The University of Texas at Austin, 2015. Search in Google Scholar
[8] L. Demkowicz, J. Gopalakrishnan, S. Nagaraj and P. Sepúlveda, A spacetime DPG method for the Schrödinger equation, SIAM J. Numer. Anal. 55 (2017), no. 4, 1740–1759. 10.1137/16M1099765Search in Google Scholar
[9] L. Demkowicz, J. Gopalakrishnan and A. H. Niemi, A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity, Appl. Numer. Math. 62 (2012), no. 4, 396–427. 10.1016/j.apnum.2011.09.002Search in Google Scholar
[10] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszyński, W. Rachowicz and A. Zdunek, Computing with hp Finite Elements. II. Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, Chapman & Hall/CRC, New York, 2007. 10.1201/9781420011692Search in Google Scholar
[11] F. Fuentes, L. Demkowicz and A. Wilder, Using a DPG method to validate DMA experimental calibration of viscoelastic materials, Comput. Methods Appl. Mech. Engrg. 325 (2017), 748–765. 10.1016/j.cma.2017.07.012Search in Google Scholar
[12] F. Fuentes, B. Keith, L. Demkowicz and S. Nagaraj, Orientation embedded high order shape functions for the exact sequence elements of all shapes, Comput. Math. Appl. 70 (2015), no. 4, 353–458. 10.1016/j.camwa.2015.04.027Search in Google Scholar
[13] F. Hellwig, Three low-order dPG methods for linear elasticity, Master’s thesis, Humboldt-Universität zu Berlin, Berlin, 2014. Search in Google Scholar
[14] G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd ed., Numer. Math. Sci. Comput., Oxford University, New York, 2005. 10.1093/acprof:oso/9780198528692.001.0001Search in Google Scholar
[15] B. Keith, F. Fuentes and L. Demkowicz, The DPG methodology applied to different variational formulations of linear elasticity, Comput. Methods Appl. Mech. Engrg. 309 (2016), 579–609. 10.1016/j.cma.2016.05.034Search in Google Scholar
[16] B. Keith, P. Knechtges, N. V. Roberts, S. Elgeti, M. Behr and L. Demkowicz, An ultraweak DPG method for viscoelastic fluids, J. Non-Newton. Fluid Mech. 247 (2017), 107–122. 10.1016/j.jnnfm.2017.06.006Search in Google Scholar
[17] B. Keith, S. Petrides, F. Fuentes and L. Demkowicz, Discrete least-squares finite element methods, Comput. Methods Appl. Mech. Engrg. 327 (2017), 226–255. 10.1016/j.cma.2017.08.043Search in Google Scholar
[18] J. P. Kurtz, Fully Automatic hp-Adaptivity for Acoustic and Electromagnetic Scattering in Three Dimensions, ProQuest LLC, Ann Arbor, 2007. Search in Google Scholar
[19] J. M. Melenk, K. Gerdes and C. Schwab, Fully discrete hp-finite elements: Fast quadrature, Comput. Methods Appl. Math. 190 (2001), no. 32, 4339–4364. 10.1016/S0045-7825(00)00322-4Search in Google Scholar
[20] S. Nagaraj, J. Grosek, S. Petrides, L. F. Demkowicz and J. Mora, A 3D DPG Maxwell approach to nonlinear Raman gain in fiber laser amplifiers, J. Comput. Phys. X 2 (2019), Article ID 100002. 10.1016/j.jcpx.2019.100002Search in Google Scholar
[21] S. Nagaraj, S. Petrides and L. F. Demkowicz, Construction of DPG Fortin operators for second order problems, Comput. Math. Appl. 74 (2017), no. 8, 1964–1980. 10.1016/j.camwa.2017.05.030Search in Google Scholar
[22]
J.-C. Nédélec,
Mixed finite elements in
[23] S. A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980), no. 1, 70–92. 10.1016/0021-9991(80)90005-4Search in Google Scholar
[24] S. Petrides and L. F. Demkowicz, An adaptive DPG method for high frequency time-harmonic wave propagation problems, Comput. Math. Appl. 74 (2017), no. 8, 1999–2017. 10.1016/j.camwa.2017.06.044Search in Google Scholar
[25] N. V. Roberts, A discontinuous Petrov–Galerkin methodology for incompressible flow problems, PhD thesis, The University of Texas at Austin, Austin, 2013. Search in Google Scholar
[26] A. Vaziri Astaneh, F. Fuentes, J. Mora and L. Demkowicz, High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations, Comput. Methods Appl. Mech. Engrg. 332 (2018), 686–711. 10.1016/j.cma.2017.12.011Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
