Hostname: page-component-76c49bb84f-6sxsx Total loading time: 0 Render date: 2025-07-13T14:27:53.427Z Has data issue: false hasContentIssue false
  • English
  • Français

The discontinuous Petrov–Galerkin method

Part of: General first-order equations and systems Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 July 2025

Leszek Demkowicz  and
Jay Gopalakrishnan Open the ORCID record for Jay Gopalakrishnan [Opens in a new window]
Leszek Demkowicz
Affiliation:
Oden Institute, The University of Texas at Austin, Austin, TX 78712-1229, USA E-mail: leszek@oden.utexas.edu
Jay Gopalakrishnan
Affiliation:
PO Box 751 (MTH), Portland State University, Portland, OR 97207-0751, USA E-mail: gjay@pdx.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research on theory and applications of the DPG methodology. The scope of the presented theory is restricted to linear problems on Hilbert spaces, but pointers to extensions are provided. Multiple viewpoints to the basic theory are provided. They show that the DPG method is equivalent to a method which minimizes a residual in a dual norm, as well as to a mixed method where one solution component is an approximate error representation function. Being a residual minimization method, the DPG method yields Hermitian positive definite stiffness matrix systems even for non-self-adjoint boundary value problems. Having a built-in error representation, the method has the out-of-the-box feature that it can immediately be used in automatic adaptive algorithms. Contrary to standard Galerkin methods, which are uninformed about test and trial norms, the DPG method must be equipped with a concrete test norm which enters the computations. Of particular interest are variational formulations in which one can tailor the norm to obtain robust stability. Key techniques to rigorously prove convergence of DPG schemes, including construction of Fortin operators, which in the DPG case can be done element by element, are discussed in detail. Pointers to open frontiers are presented.

MSC classification

Primary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N12: Stability and convergence of numerical methods
Secondary: 35F45: Boundary value problems for linear first-order systems

Information

Type
Research Article
Information
Acta Numerica , Volume 34 , July 2025 , pp. 293 - 384
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

References

Al-Mohy, A. H. and Higham, N. J. (2011), Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput. 33, 488–511.10.1137/100788860CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006), Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155.10.1017/S0962492906210018CrossRefGoogle Scholar
Babuška, I. (1971), Error-bounds for finite element method, Numer . Math. 16, 322–333.Google Scholar
Babuška, I., Aziz, A. K., Fix, G. and Kellogg, R. B. (1972), Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Aziz, A. K., ed.), Academic Press, pp. 1–359.Google Scholar
Bachmayr, M., Dahmen, W. and Oster, M. (2024), Variationally correct neural residual regression for parametric PDEs: On the viability of controlled accuracy. Available at arXiv:2405.20065.Google Scholar
Bacuta, C., Demkowicz, L., Mora, J. and Xenophontos, C. (2021a), Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms, Comput. Math. Appl. 95, 215–241.10.1016/j.camwa.2020.09.018CrossRefGoogle Scholar
Bacuta, C., Hayes, D. and Jacavage, J. (2021b), Notes on a saddle point reformulation of mixed variational problems, Comput. Math. Appl. 95, 4–18.10.1016/j.camwa.2020.07.016CrossRefGoogle Scholar
Badger, J., Henneking, S., Petrides, S. and Demkowicz, L. (2023), Scalable DPG multigrid solver for Helmholtz problems: A study on convergence, Comput. Math. Appl. 148, 81–92.10.1016/j.camwa.2023.07.006CrossRefGoogle Scholar
Barbone, P. E. and Harari, I. (2001), Nearly H 1-optimal finite element methods, Comput. Methods Appl. Mech. Engrg 190, 5679–5690.10.1016/S0045-7825(01)00191-8CrossRefGoogle Scholar
Barker, A. T., Dobrev, V., Gopalakrishnan, J. and Kolev, T. (2018), A scalable preconditioner for a DPG method, SIAM J. Sci. Comput. 40, A1187–A1203.10.1137/16M1104780CrossRefGoogle Scholar
Barrett, J. W. and Morton, K. W. (1984), Approximate symmetrization and Petrov–Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Engrg 45, 97–122.10.1016/0045-7825(84)90152-XCrossRefGoogle Scholar
Bergh, J. and Löfström, J. (1976), Interpolation Spaces: An Introduction, Springer.10.1007/978-3-642-66451-9CrossRefGoogle Scholar
Bertrand, F., Boffi, D. and Schneider, H. (2023), Discontinuous Petrov–Galerkin approximation of eigenvalue problems, Comput. Methods Appl. Math. 23, 1–17.10.1515/cmam-2022-0069CrossRefGoogle Scholar
Bochev, P. B. and Gunzburger, M. D. (2009), Least-Squares Finite Element Methods, Vol. 166 of Applied Mathematical Sciences, Springer.Google Scholar
Bouma, T., Gopalakrishnan, J. and Harb, A. (2014), Convergence rates of the DPG method with reduced test space degree, Comput. Math. Appl. 68, 1550–1561.10.1016/j.camwa.2014.08.004CrossRefGoogle Scholar
Bramble, J. H. and Pasciak, J. E. (2004), A new approximation technique for div-curl systems, Math. Comp. 73, 1739–1762.10.1090/S0025-5718-03-01616-8CrossRefGoogle Scholar
Bramble, J. H., Lazarov, R. D. and Pasciak, J. E. (1997), A least-squares approach based on a discrete minus one inner product for first order systems, Math. Comp. 66, 935–955.10.1090/S0025-5718-97-00848-XCrossRefGoogle Scholar
Brevis, I., Muga, I. and van der Zee, K. G. (2022), Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights, Comput. Methods Appl. Mech. Engrg 402, art. 115716.10.1016/j.cma.2022.115716CrossRefGoogle Scholar
Brevis, I., Muga, I., Pardo, D., Rodriguez, O. and van der Zee, K. G. (2024), Learning quantities of interest from parametric PDEs: An efficient neural-weighted minimal residual approach, Comput. Math. Appl. 164, 139–149.10.1016/j.camwa.2024.04.006CrossRefGoogle Scholar
Brezis, H. (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer.10.1007/978-0-387-70914-7CrossRefGoogle Scholar
Brezzi, F. (1974), On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151.Google Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Vol. 15 of Springer Series in Computational Mathematics, Springer.10.1007/978-1-4612-3172-1CrossRefGoogle Scholar
Bristeau, M. O., Pironneau, O., Glowinski, R., Périaux, J., Perrier, J. P. and Poirier, G. (1979), On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods, I: Least square formulations and conjugate gradient solution of the continuous problems, Comput. Methods Appl. Mech. Engrg 17-18, 619–657.10.1016/0045-7825(79)90048-3CrossRefGoogle Scholar
Bristeau, M. O., Pironneau, O., Glowinski, R., Périaux, J., Perrier, J. P. and Poirier, G. (1985), On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods, II: Application to transonic flow simulations, Comput. Methods Appl. Mech. Engrg 51, 363–394.10.1016/0045-7825(85)90039-8CrossRefGoogle Scholar
Broersen, D., Dahmen, W. and Stevenson, R. P. (2018), On the stability of DPG formulations of transport equations, Math. Comp. 87, 1051–1082.10.1090/mcom/3242CrossRefGoogle Scholar
Buffa, A., Costabel, M. and Sheen, D. (2002), On traces for in Lipschitz domains, J. Math. Anal. Appl. 276, 845–867.10.1016/S0022-247X(02)00455-9CrossRefGoogle Scholar
Bui-Thanh, T. and Ghattas, O. (2014), A PDE-constrained optimization approach to the discontinuous Petrov–Galerkin method with a trust region inexact Newton-CG solver, Comput . Methods Appl. Mech. Engrg 278, 20–40.10.1016/j.cma.2014.04.018CrossRefGoogle Scholar
Cai, Z., Lazarov, R., Manteuffel, T. A. and McCormick, S. F. (1994), First-order system least squares for second-order partial differential equations I, SIAM J. Numer. Anal. 31, 1785–1799.10.1137/0731091CrossRefGoogle Scholar
Cai, Z., Manteuffel, T. A., McCormick, S. F. and Ruge, J. (2001), First-order system (FOSLL*): Scalar elliptic partial differential equations, SIAM J. Numer. Anal. 39, 1418–1445.10.1137/S0036142900388049CrossRefGoogle Scholar
Calo, V. M., Collier, N. O. and Niemi, A. H. (2014), Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model, Comput. Math. Appl. 66, 2570–2586.10.1016/j.camwa.2013.07.012CrossRefGoogle Scholar
Cantin, P. and Heuer, N. (2018), A DPG framework for strongly monotone operators, SIAM J. Numer. Anal. 56, 2731–2750.10.1137/18M1166663CrossRefGoogle Scholar
Carstensen, C., Bringmann, P., Hellwig, F. and Wriggers, P. (2018), Nonlinear discontinuous Petrov–Galerkin methods, Numer. Math. 139, 529–561.10.1007/s00211-018-0947-5CrossRefGoogle Scholar
Carstensen, C., Demkowicz, L. and Gopalakrishnan, J. (2014a), A posteriori error control for DPG methods, SIAM J. Numer. Anal. 52, 1335–1353.10.1137/130924913CrossRefGoogle Scholar
Carstensen, C., Demkowicz, L. and Gopalakrishnan, J. (2016), Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. 72, 494–522.10.1016/j.camwa.2016.05.004CrossRefGoogle Scholar
Carstensen, C., Gallistl, D., Hellwig, F. and Weggler, L. (2014b), Low-order dPG-FEM for an elliptic PDE, Comput. Math. Appl. 68, 1503–1512.10.1016/j.camwa.2014.09.013CrossRefGoogle Scholar
Celia, A., Russell, T. F., Ismael, H. and Ewing, R. E. (1990), An Eulerian–Lagrangian localized adjoint method for the advection–diffusion equation, Adv . Water Resources 13, 187–206.10.1016/0309-1708(90)90041-2CrossRefGoogle Scholar
Chan, J., Demkowicz, L. and Moser, R. (2014a), A DPG method for steady viscous compressible flow, Comput. Fluids 98, 69–90.10.1016/j.compfluid.2014.02.024CrossRefGoogle Scholar
Chan, J., Heuer, N., Bui-Thanh, T. and Demkowicz, L. (2014b), Robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms, Comput. Math. Appl. 67, 771–795.10.1016/j.camwa.2013.06.010CrossRefGoogle Scholar
Cockburn, B. and Gopalakrishnan, J. (2004), A characterization of hybridized mixed methods for the Dirichlet problem, SIAM J. Numer. Anal. 42, 283–301.10.1137/S0036142902417893CrossRefGoogle Scholar
Cohen, A., Dahmen, W. and Welper, G. (2012), Adaptivity and variational stabilization for convection–diffusion equations, ESAIM Math. Model. Numer. Anal. 46, 1247–1273.10.1051/m2an/2012003CrossRefGoogle Scholar
Courant, R. and Friedrichs, K. O. (1948), Supersonic Flow and Shock Waves, Interscience.Google Scholar
Demkowicz, L. (2018), Lecture notes on energy spaces. Report 18-13, Oden Institute, The University of Texas at Austin.Google Scholar
Demkowicz, L. (2024), Mathematical Theory of Finite Elements, Computational Science and Engineering, SIAM.Google Scholar
Demkowicz, L. and Gopalakrishnan, J. (2010), A class of discontinuous Petrov–Galerkin methods, Part I: The transport equation, Comput. Methods Appl. Mech. Engrg 199, 1558–1572.10.1016/j.cma.2010.01.003CrossRefGoogle Scholar
Demkowicz, L. and Gopalakrishnan, J. (2011a), Analysis of the DPG method for the Poisson equation, SIAM J. Numer. Anal. 49, 1788–1809.10.1137/100809799CrossRefGoogle Scholar
Demkowicz, L. and Gopalakrishnan, J. (2011b), A class of discontinuous Petrov–Galerkin methods, Part II: Optimal test functions, Numer. Methods Partial Differential Equations 27, 70–105.10.1002/num.20640CrossRefGoogle Scholar
Demkowicz, L. and Gopalakrishnan, J. (2013), A primal DPG method without a first-order reformulation, Comput. Math. Appl. 66, 1058–1064.10.1016/j.camwa.2013.06.029CrossRefGoogle Scholar
Demkowicz, L. and Gopalakrishnan, J. (2017), Discontinuous Petrov Galerkin (DPG) method, in Encyclopedia of Computational Mechanics, second edition, Wiley Computational Mechanics Online.Google Scholar
Demkowicz, L. and Heuer, N. (2013), Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal. 51, 2514–2537.10.1137/120862065CrossRefGoogle Scholar
Demkowicz, L. and Oden, J. T. (1986a), An adaptive characteristic Petrov–Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in one space variable, J. Comput. Phys. 67, 188–213.10.1016/0021-9991(86)90121-XCrossRefGoogle Scholar
Demkowicz, L. and Oden, J. T. (1986b), An adaptive characteristic Petrov–Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables, Comput. Methods Appl. Mech. Engrg 55, 63–87.10.1016/0045-7825(86)90086-1CrossRefGoogle Scholar
Demkowicz, L. and Zanotti, P. (2020), Construction of DPG Fortin operators revisited, Comput . Math. Appl. 80, 2261–2271.Google Scholar
Demkowicz, L., Gopalakrishnan, J. and Keith, B. (2020), The DPG-star method, Comput . Math. Appl. 79, 3092–3116.Google Scholar
Demkowicz, L., Gopalakrishnan, J. and Niemi, A. (2012a), A class of discontinuous Petrov–Galerkin methods, Part III: Adaptivity, Appl. Numer. Math. 62, 396–427.10.1016/j.apnum.2011.09.002CrossRefGoogle Scholar
Demkowicz, L., Gopalakrishnan, J., Muga, I. and Zitelli, J. (2012b), Wavenumber explicit analysis for a DPG method for the multidimensional Helmholtz equation, Comput. Methods Appl. Mech. Engrg 213/216, 126–138.10.1016/j.cma.2011.11.024CrossRefGoogle Scholar
Demkowicz, L., Gopalakrishnan, J., Nagaraj, S. and Sepúlveda, P. (2017), A spacetime DPG method for the Schrödinger equation, SIAM J. Numer. Anal. 55, 1740–1759.10.1137/16M1099765CrossRefGoogle Scholar
Diening, L. and Storn, J. (2022), A space-time DPG method for the heat equation, Comput. Math. Appl. 105, 41–53.10.1016/j.camwa.2021.11.013CrossRefGoogle Scholar
Ekeland, I. and Témam, R. (1999), Convex Analysis and Variational Problems , Vol. 28 of Classics in Applied Mathematics, SIAM.10.1137/1.9781611971088CrossRefGoogle Scholar
Ellis, T., Chan, J. and Demkowicz, L. (2016), Robust DPG methods for transient convection–diffusion, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (Barrenechea, G. R. et al., eds), Vol. 114 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Ellis, T., Demkowicz, L. and Chan, J. (2014), Locally conservative discontinuous Petrov–Galerkin finite elements for fluid problems, Comput. Math. Appl. 68, 1530–1549.10.1016/j.camwa.2014.07.005CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2021), Finite Elements II, Springer.10.1007/978-3-030-56923-5CrossRefGoogle Scholar
Ern, A., Guermond, J.-L. and Caplain, G. (2007), An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs’ systems, Commun . Partial Differential Equations 32, 317–341.10.1080/03605300600718545CrossRefGoogle Scholar
Ernesti, J. and Wieners, C. (2019), A space-time discontinuous Petrov–Galerkin method for acoustic waves, in Space-Time Methods: Applications to Partial Differential Equations (Langer, U. and Steinbach, O., eds), De Gruyter, pp. 89–116.10.1515/9783110548488-003CrossRefGoogle Scholar
Friedrichs, K. O. (1958), Symmetric positive linear differential equations, Commun. Pure Appl. Math. 11, 333–418.10.1002/cpa.3160110306CrossRefGoogle Scholar
Führer, T. (2018), Superconvergence in a DPG method for an ultra-weak formulation, Comput. Math. Appl. 75, 1705–1718.10.1016/j.camwa.2017.11.029CrossRefGoogle Scholar
Führer, T. and Heuer, N. (2017), Robust coupling of DPG and BEM for a singularly perturbed transmission problem, Comput. Math. Appl. 74, 1940–1954.10.1016/j.camwa.2016.09.016CrossRefGoogle Scholar
Führer, T. and Heuer, N. (2019), Fully discrete DPG methods for the Kirchhoff–Love plate bending model, Comput. Methods Appl. Mech. Engrg 343, 550–571.10.1016/j.cma.2018.08.041CrossRefGoogle Scholar
Führer, T. and Heuer, N. (2024), Robust DPG test spaces and Fortin operators: The H 1 and H(div) cases, SIAM J. Numer. Anal. 62, 718–748.10.1137/23M1550360CrossRefGoogle Scholar
Führer, T., González, R. and Karkulik, M. (2024), Well-posedness of first-order acoustic wave equations and space-time finite element approximation. Available at arXiv:2311.10536.Google Scholar
Führer, T., Heuer, N. and Karkulik, M. (2017), On the coupling of DPG and BEM, Math. Comp. 86, 2261–2284.10.1090/mcom/3170CrossRefGoogle Scholar
Führer, T., Heuer, N. and Niemi, A. H. (2019), An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation, Math. Comp. 88, 1587–1619.10.1090/mcom/3381CrossRefGoogle Scholar
Führer, T., Heuer, N. and Niemi, A. H. (2022), A DPG method for shallow shells, Numer. Math. 152, 76–99.10.1007/s00211-022-01308-wCrossRefGoogle Scholar
Führer, T., Heuer, N. and Niemi, A. H. (2023), A DPG method for Reissner–Mindlin plates, SIAM J. Numer. Anal. 61, 995–1017.10.1137/22M1498838CrossRefGoogle Scholar
Führer, T., Heuer, N. and Sayas, F.-J. (2020), An ultraweak formulation of the Reissner–Mindlin plate bending model and DPG approximation, Numer. Math. 145, 313–344.10.1007/s00211-020-01116-0CrossRefGoogle Scholar
Führer, T., Heuer, N. and Stephan, E. P. (2018a), On the DPG method for Signorini problems, IMA J. Numer. Anal. 38, 1893–1926.10.1093/imanum/drx048CrossRefGoogle Scholar
Führer, T., Heuer, N., Karkulik, M. and Rodríguez, R. (2018b), Combining the DPG method with finite elements, Comput. Methods Appl. Math. 18, 639–652.10.1515/cmam-2017-0041CrossRefGoogle Scholar
Garg, V. V., Prudhomme, S., van der Zee, K. G. and Carey, G. F. (2014), Adjoint-consistent formulations of slip models for coupled electroosmotic flow systems, Adv. Model. Simul. Engrg 2, art. 15.10.1186/s40323-014-0015-3CrossRefGoogle Scholar
Gopalakrishnan, J. (2013), Five lectures on DPG methods. Available at arXiv:1306.0557.Google Scholar
Gopalakrishnan, J. and Qiu, W. (2014), An analysis of the practical DPG method, Math. Comput. 83, 537–552.10.1090/S0025-5718-2013-02721-4CrossRefGoogle Scholar
Gopalakrishnan, J. and Sepúlveda, P. (2019), A spacetime DPG method for acoustic waves, in Space-Time Methods: Applications to Partial Differential Equations (Langer, U. and Steinbach, O., eds), Radon Series on Computational and Applied Mathematics, De Gruyter, pp. 129–154.Google Scholar
Gopalakrishnan, J., Grubišić, L., Ovall, J. and Parker, B. Q. (2020), Analysis of FEAST spectral approximations using the DPG discretization, Comput. Methods Appl. Math. 89, 203–228.Google Scholar
Gopalakrishnan, J., Muga, I. and Olivares, N. (2014), Dispersive and dissipative errors in the DPG method with scaled norms for the Helmholtz equation, SIAM J. Sci. Comput. 36, A20–A39.10.1137/130918186CrossRefGoogle Scholar
Grisvard, P. (1985), Elliptic Problems in Nonsmooth Domains, Vol. 24 of Monographs and Studies in Mathematics, Pitman Advanced Publishing Program.Google Scholar
Heuer, N. and Karkulik, M. (2015), DPG method with optimal test functions for a transmission problem, Comput. Math. Appl. 70, 1504–1518.10.1016/j.camwa.2015.06.032CrossRefGoogle Scholar
Heuer, N. and Karkulik, M. (2017a), Discontinuous Petrov–Galerkin boundary elements, Numer. Math. 135, 1011–1043.10.1007/s00211-016-0824-zCrossRefGoogle Scholar
Heuer, N. and Karkulik, M. (2017b), A robust DPG method for singularly perturbed reaction–diffusion problems, SIAM J. Numer. Anal. 55, 1218–1242.10.1137/15M1041304CrossRefGoogle Scholar
Jensen, M. (2004), Discontinuous Galerkin methods for Friedrichs systems with irregular solutions. PhD thesis, University of Oxford.Google Scholar
Kato, T. (1995), Perturbation Theory for Linear Operators, Classics in Mathematics, Springer.10.1007/978-3-642-66282-9CrossRefGoogle Scholar
Keith, B. (2018), New ideas in adjoint methods for PDEs: A saddle-point paradigm for finite element analysis and its role in the DPG methodology. PhD thesis, The University of Texas at Austin.Google Scholar
Keith, B., Astaneh, A. V. and Demkowicz, L. (2019), Goal-oriented adaptive mesh refinement for discontinuous Petrov–Galerkin methods, SIAM J. Numer. Anal. 57, 1649–1676.10.1137/18M1181754CrossRefGoogle Scholar
Kharazmi, E., Zhang, Z. and Karniadakis, G. E. M. (2021), hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Engrg 374, art. 113547.10.1016/j.cma.2020.113547CrossRefGoogle Scholar
Li, J. (2024), A nonlinear mixed problem framework for discontinuous Petrov Galerkin (DPG) methods. PhD thesis, The University of Texas at Austin.Google Scholar
Li, J. and Demkowicz, L. (2024), A DPG method for planar div-curl problems, Comput. Math. Appl. 159, 31–43.10.1016/j.camwa.2024.01.023CrossRefGoogle Scholar
Loula, A. F. D. and Fernandes, D. T. (2009), A quasi optimal Petrov–Galerkin method for Helmholtz problem, Internat. J. Numer. Methods Engrg 80, 1595–1622.10.1002/nme.2677CrossRefGoogle Scholar
Loula, A. F. D., Hughes, T. J. R. and Franca, L. P. (1987), Petrov–Galerkin formulations of the Timoshenko beam problem, Comput. Methods Appl. Mech. Engrg 63, 115–132.10.1016/0045-7825(87)90167-8CrossRefGoogle Scholar
Melenk, J. M. (1995), On generalized finite element methods. PhD thesis, University of Maryland.Google Scholar
Millar, F., Muga, I., Rojas, S. and Van der Zee, K. G. (2022), Projection in negative norms and the regularization of rough linear functionals, Numer. Math. 150, 1087–1121.10.1007/s00211-022-01278-zCrossRefGoogle Scholar
Monk, P. (2003), Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press.10.1093/acprof:oso/9780198508885.001.0001CrossRefGoogle Scholar
Muga, I. and van der Zee, K. G. (2020), Discretization of linear problems in Banach spaces: Residual minimization, nonlinear Petrov–Galerkin, and monotone mixed methods, SIAM J. Numer. Anal. 58, 3406–3426.10.1137/20M1324338CrossRefGoogle Scholar
Muñoz-Matute, J. and Demkowicz, L. (2024), Multistage discontinuous Petrov–Galerkin time-marching scheme for nonlinear problems, SIAM J. Numer. Anal. 62, 1956–1978.10.1137/23M1598088CrossRefGoogle Scholar
Muñoz-Matute, J., Demkowicz, L. and Pardo, D. (2022), Error representation of the time-marching DPG scheme, Comput. Methods Appl. Mech. Engrg 391, art. 114480.10.1016/j.cma.2021.114480CrossRefGoogle Scholar
Muñoz-Matute, J., Pardo, D. and Demkowicz, L. (2021), Equivalence between the DPG method and the exponential integrators for linear parabolic problems, J. Comput. Phys. 429, art. 110016.10.1016/j.jcp.2020.110016CrossRefGoogle Scholar
Nečas, J. (1962), Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Sc. Norm. Super. Pisa Cl. Sci. 16, 305–326.Google Scholar
Nédélec, J.-C. (1980), Mixed finite elements in , Numer. Math. 35, 315–341.10.1007/BF01396415CrossRefGoogle Scholar
Niesen, J. and Wright, W. M. (2012), Algorithm 919: A Krylov subspace algorithm for evaluating the ϕ-functions appearing in exponential integrators, ACM Trans. Math. Softw. 10.1145/2168773.2168781Google Scholar
Nocedal, J. and Wright, S. J. (2006), Numerical Optimization, second edition, Springer.Google Scholar
Petrides, S. and Demkowicz, L. (2017), An adaptive DPG method for high frequency time-harmonic wave propagation problems, Comput. Math. Appl. 74, 1999–2017.10.1016/j.camwa.2017.06.044CrossRefGoogle Scholar
Raviart, P.-A. and Thomas, J. M. (1977a), A mixed finite element method for 2-nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Vol. 606 of Lecture Notes in Mathematics, Springer, pp. 292–315.10.1007/BFb0064470CrossRefGoogle Scholar
Raviart, P.-A. and Thomas, J. M. (1977b), Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31, 391–413.10.1090/S0025-5718-1977-0431752-8CrossRefGoogle Scholar
Roberts, N. V., Bui-Thanh, T. and Demkowicz, L. (2014), The DPG method for the Stokes problem, Comput. Math. Appl. 67, 966–995.10.1016/j.camwa.2013.12.015CrossRefGoogle Scholar
Roberts, N. V., Demkowicz, L. and Moser, R. (2015), A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations, J. Comput. Phys. 301, 456–483.10.1016/j.jcp.2015.07.014CrossRefGoogle Scholar
Rojas, S., Maczuga, P., Muñoz-Matute, J., Pardo, D. and Paszyński, M. (2024), Robust variational physics-informed neural networks, Comput. Methods Appl. Mech. Engrg 425, art. 116904.10.1016/j.cma.2024.116904CrossRefGoogle Scholar
Sepúlveda, P. (2018), Spacetime numerical techniques for the wave and Schrödinger equations. PhD thesis, Portland State University.Google Scholar
Sheen, D. (1992), A generalized Green’s theorem, Appl. Math. Lett. 5, 95–98.10.1016/0893-9659(92)90096-RCrossRefGoogle Scholar
Valseth, E. and Dawson, C. (2022), A stable space-time FE method for the shallow water equations, Comput. Geosci. pp. 1–18.Google Scholar
Valseth, E., Romkes, A. and Kaul, A. R. (2021), A stable FE method for the space-time solution of the Cahn–Hilliard equation, J. Comput. Phys. 441, art. 110426.10.1016/j.jcp.2021.110426CrossRefGoogle Scholar
Astaneh, A. Vaziri, Fuentes, F., Mora, J. and Demkowicz, L. (2018), High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations, Comput. Methods Appl. Mech. Engrg 332, 686–711.10.1016/j.cma.2017.12.011CrossRefGoogle Scholar
Xu, J. and Zikatanov, L. (2003), Some observations on Babuška and Brezzi theories, Numer. Math. 94, 195–202.10.1007/s002110100308CrossRefGoogle Scholar
Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D. and Calo, V. (2011), A class of discontinuous Petrov–Galerkin methods, Part IV: Wave propagation, J. Comput. Phys. 230, 2406–2432.10.1016/j.jcp.2010.12.001CrossRefGoogle Scholar