Reference documentation for deal.II version 9.3.0
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)

R. Agelek, M. Anderson, W. Bangerth, and W. L. Barth. On orienting edges of unstructured two- and three-dimensional meshes. ACM Transactions on Mathematical Software, 44:5/1–22, 2017.


Mark Ainsworth and Bill Senior. An adaptive refinement strategy for hp-finite element computations. Applied Numerical Mathematics, 26(1–2):165–178, 1998.


Mark Ainsworth. A posteriori error estimation for discontinuous galerkin finite element approximation. SIAM Journal on Numerical Analysis, 45(4):1777–1798, 2007.


Philippe Angot, Charles-Henri Bruneau, and Pierre Fabrie. A penalization method to take into account obstacles in incompressible viscous flows. Numerische Mathematik, 81(4):497–520, February 1999.


I. Babuska and J. M. Melenk. The partition of unity method. International Journal for Numerical Methods in Engineering, 40(4):727–758, 1997.


Wolfgang Bangerth and Rolf Rannacher. Finite element approximation of the acoustic wave equation: Error control and mesh adaptation. East–West J. Numer. Math., 7(4):263–282, 1999.


Wolfgang Bangerth and Rolf Rannacher. Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoustics, 9(2):575–591, 2001.


Wolfgang Bangerth and Rolf Rannacher. Adaptive Finite Element Methods for Differential Equations. Birkhäuser Verlag, 2003.


Wolfgang Bangerth. Mesh adaptivity and error control for a finite element approximation of the elastic wave equation. In Alfredo Bermúdez, Dolores Gómez, Christophe Hazard, Patrick Joly, and Jean E. Roberts, editors, Proceedings of the Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation (Waves2000), Santiago de Compostela, Spain, 2000, pages 725–729. SIAM, 2000.


Wolfgang Bangerth. Adaptive Finite Element Methods for the Identification of Distributed Parameters in Partial Differential Equations. PhD thesis, University of Heidelberg, 2002.


S. Bartels, C. Carstensen, and G. Dolzmann. Inhomogeneous dirichlet conditions in a priori and a posteriori finite element error analysis. Numerische Mathematik, 99(1):1–24, September 2004.


J. Bebernes and D. Eberly. Mathematical Problems from Combustion Theory, volume 83 of Applied Mathematical Sciences. Springer-Verlag, New York, NY, 1989.


Roland Becker and Rolf Rannacher. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math, 4:237–264, 1996.


Roland Becker and Rolf Rannacher. Weighted a posteriori error control in FE methods. In H. G. Bock et al., editor, ENUMATH 97, pages 621–637. World Scientific Publ., Singapore, 1998.


Roland Becker and Rolf Rannacher. An optimal control approach to error estimation and mesh adaptation in finite element methods. Acta Numerica, 10:1–102, 2001.


Roland Becker. An Adaptive Finite Element Method for the Incompressible Navier-Stokes Equations on Time-dependent Domains. Dissertation, Universität Heidelberg, 1995.


Roland Becker. Weighted error estimators for the incompressible Navier-Stokes equations. Preprint 98-20, Universität Heidelberg, 1998.


Martin P. Bendsøe and Ole Sigmund. Topology Optimization. Springer Berlin Heidelberg, 2004.


Hande Y. Benson, Robert J. Vanderbei, and David F. Shanno. Interior-point methods for nonconvex nonlinear programs: Filter methods and merit functions. Computational Optimization and Applications, 23(2):257–272, 2002.


Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654, 1973.


Bruno Blais, Jean-Philippe Braeunig, Daniel Chauveheid, Jean-Michel Ghidaglia, and Raphaël Loubère. Dealing with more than two materials in the fvcf–enip method. European Journal of Mechanics-B/Fluids, 42:1–9, 2013.


Bruno Blais, David Vidal, Francois Bertrand, Gregory S Patience, and Jamal Chaouki. Experimental methods in chemical engineering: Discrete element method—dem. The Canadian Journal of Chemical Engineering, 97(7):1964–1973, 2019.


Daniele Boffi, Lucia Gastaldi, Luca Heltai, and Charles S. Peskin. On the hyper-elastic formulation of the immersed boundary method. Computer Methods in Applied Mechanics and Engineering, 197(25-28):2210–2231, April 2008.


Susanne Brenner and Ridgway Scott. The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics. Springer-Verlag, 3 edition, 2008.


Susanne C. Brenner and Li-Yeng Sung. c0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. Journal of Scientific Computing, 22-23(1-3):83–118, June 2005.


S. C. Brenner, T. Gudi, and L.-Y. Sung. An a posteriori error estimator for a quadratic c0-interior penalty method for the biharmonic problem. IMA Journal of Numerical Analysis, 30(3):777–798, March 2009.


Susanne C. Brenner. c0 interior penalty methods. In Lecture Notes in Computational Science and Engineering, pages 79–147. Springer Berlin Heidelberg, 2011.


Alexander N. Brooks and Thomas J. R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 32(1-3):199–259, 1982. FENOMECH ''81, Part I (Stuttgart, 1981).


A. Cangiani, J. Chapman, E. H. Georgoulis, and M. Jensen. Implementation of the continuous-discontinuous galerkin finite element method. In Numerical Mathematics and Advanced Applications 2011, pages 315–322. Springer Berlin Heidelberg, November 2012.


G. F. Castelli. Numerical Investigation of Cahn–Hilliard-Type Phase-Field Models for Battery Active Particles. PhD thesis, Karlsruhe Institute of Technology (KIT), 2021. (To be published).


Thomas C. Clevenger, Timo Heister, Guido Kanschat, and Martin Kronbichler. A flexible, parallel, adaptive geometric multigrid method for fem. submitted, 2019.


Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47(2):1319–1365, 2009.


B. D. Coleman and M. E. Gurtin. Thermodynamics with internal state variables. The Journal of Chemical Physics, 47(2):597–613, 1967.


B. D. Coleman and W. Noll. The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13(1):167–178, December


Stéphane Commend, Andrzej Truty, and Thomas Zimmermann. Stabilized finite elements applied to elastoplasticity: I. mixed displacement–pressure formulation. Computer Methods in Applied Mechanics and Engineering, 193(33):3559–3586, 2004.


Denis Davydov, Tymofiy Gerasimov, Jean-Paul Pelteret, and Paul Steinmann. Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics. Advanced Modeling and Simulation in Engineering Sciences, 4(1):7,


Daniele Antonio Di Pietro and Alexandre Ern. Mathematical aspects of discontinuous Galerkin methods, volume 69. Springer Science & Business Media, 2011.


Huo-Yuan Duan and Qun Lin. Mixed finite elements of least-squares type for elasticity. Computer Methods in Applied Mechanics and Engineering, 194(9):1093–1112, 2005.


Tino Eibner and Jens Markus Melenk. An adaptive strategy for hp-FEM based on testing for analyticity. Computational Mechanics, 39(5):575–595, 2007.


Stanley C. Eisenstat and Homer F. Walker. Choosing the forcing terms in an inexact Newton method. SIAM Journal on Scientific Computing, 17(1):16–32, 1996.


H. C. Elman, D. J. Silvester, and A. J. Wathen. Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, New York, 2005.


G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, and R.L. Taylor. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Computer Methods in Applied Mechanics and Engineering, 191(34):3669–3750, July 2002.


Alexandre Ern and Jean-Luc Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.


Marc Fehling. Algorithms for massively parallel generic hp-adaptive finite element methods. PhD thesis, Bergische Universität Wuppertal, 2020.


Niklas Fehn, Wolfgang A. Wall, and Martin Kronbichler. A matrix-free high-order discontinuous Galerkin compressible Navier–Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 89(3):71–102,


B. X. Fraeijs de Veubeke. Displacement and equilibrium models in the finite element method. In O. C. Zienkiewicz and G. S. Holister, editors, Stress Analysis, pages 275–284. Wiley, New York, 1965.


J. Freund and R. Stenberg. On weakly imposed boundary conditions for second order problems. In Proceedings of the Ninth International Conference on Finite Elements in Fluids, pages 327–336, 1995.


Isaac Fried and David S Malkus. Finite element mass matrix lumping by numerical integration with no convergence rate loss. International Journal of Solids and Structures, 11(4):461–466, 1975.


Christian Führer and Guido Kanschat. A posteriori error control in radiative transfer. Computing, 58(4):317–334, 1997.


Rene Gassmöller, Harsha Lokavarapu, Eric Heien, Elbridge Gerry Puckett, and Wolfgang Bangerth. Flexible and scalable particle-in-cell methods with adaptive mesh refinement for geodynamic computations. Geochemistry, Geophysics, Geosystems, 19(9):3596–3604, 2018.


Rene Gassmöller, Harsha Lokavarapu, Wolfgang Bangerth, and Elbridge Gerry Puckett. Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible stokes flow. Geophysical Journal International, 219(3):1915–1938, 2019.


Gregor J. Gassner. A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM Journal on Scientific Computing, 35(3):A1233–A1253, 2013.


S. Geevers, W. A. Mulder, and J. J. W. van der Vegt. New higher-order mass-lumped tetrahedral elements for wave propagation modelling. SIAM Journal on Scientific Computing, 40(5):A2830–A2857, Jan 2018.


R. Glowinski, T.-W. Pan, T.I. Hesla, and D.D. Joseph. A distributed lagrange multiplier/fictitious domain method for particulate flows. International Journal of Multiphase Flow, 25(5):755–794, August 1999.


Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stability-preserving high-order time discretization methods. SIAM review, 43(1):89–112, 2001.


I. Griva, S. G. Nash, and A. Sofer. Linear and nonlinear optimization. SIAM, 2nd edition, 2008.


Jean-Luc Guermond and Bojan Popov. Fast estimation of the maximum wave speed in the riemann problem for the euler equations. J. Comput. Phys., 321:908–926, 2016.


Jean-Luc Guermond and Bojan Popov. Invariant domains and first-order continuous finite element approximation for hyperbolic systems. SIAM J. Numer. Anal., 54(4):2466–2489, 2016.


Jean-Luc Guermond, Murtazo Nazarov, Bojan Popov, and Ignacio Tomas. Second-order invariant domain preserving approximation of the Euler equations using convex limiting. SIAM J. Sci. Comput., 40(5):A3211–A3239, 2018.


R. J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3(2):380–380, feb 1965.


Ralf Hartmann and Paul Houston. Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws.


Ralf Hartmann and Paul Houston. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. Journal of Computational Physics, 183(2):508–532, 2002.


Ralf Hartmann. Adaptive Finite Element Methods for the Compressible Euler Equations. PhD thesis, Universität Heidelberg, 2002.


Luca Heltai and Francesco Costanzo. Variational implementation of immersed finite element methods. Computer Methods in Applied Mechanics and Engineering, 229-232:110–127, July 2012.


Torsten Hoefler, Christian Siebert, and Andrew Lumsdaine. Scalable communication protocols for dynamic sparse data exchange. ACM Sigplan Notices, 45(5):159–168, 2010.


G. A. Holzapfel and J. C. Simo. A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. International Journal of Solids and Structures, 33(20–22):3019–3034, August 1996.


G. A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons Ltd., West Sussex, England, 2007.


Paul Houston and Endre Süli. A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 194(2):229–243, 2005.


Volker John and Petr Knobloch. On discontinuity—capturing methods for convection—diffusion equations. In Numerical Mathematics and Advanced Applications, pages 336–344. Springer, 2006.


C. Johnson and J. Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46(173):1–26, 1986.


Guido Kanschat. Parallel and Adaptive Galerkin Methods for Radiative Transfer Problems. Dissertation, Universität Heidelberg, 1996.


Guido Kanschat. Notes on applied mathematics: Iterative methods, schwarz preconditioners and multigrid. 2015.


Ohannes A Karakashian and Frederic Pascal. A posteriori error estimates for a discontinuous galerkin approximation of second-order elliptic problems. SIAM Journal on Numerical Analysis, 41(6):2374–2399, 2003.


R. Bruce Kellogg. On the Poisson equation with intersecting interfaces. Applicable Analysis, 4(2):101–129, 1974.


D. W. Kelly, J. P. De S. R. Gago, O. C. Zienkiewicz, and I. Babuska. A posteriori error analysis and adaptive processes in the finite element method: Part I–error analysis. Int. J. Num. Meth. Engrg., 19:1593–1619, 1983.


Christopher A. Kennedy, Mark H. Carpenter, and R. Micheal Lewis. Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Applied Numerical Mathematics, 35:177–219, 2000.


N. Koprowski-Theiss, M. Johlitz, and S. Diebels. Characterizing the time dependence of filled EPDM. Rubber Chemistry and Technology, 84(2):147–165, June 2011.


J. Korelc and P. Wriggers. Automation of Finite Element Methods. Springer Nature, 2016.


LIG Kovasznay. Laminar flow behind a two-dimensional grid. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 44, pages 58–62. Cambridge University Press, 1948.


Martin Kronbichler and Katharina Kormann. Fast matrix-free evaluation of discontinuous Galerkin finite element operators. ACM Transactions on Mathematical Software, 45(3):29:1–29:40, 2019.


Martin Kronbichler, Svenja Schoeder, Christopher Müller, and Wolfgang A. Wall. Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. International Journal for Numerical Methods in Engineering, 106(9):712–739, 2016.


Hao Li and Xiangxiong Zhang. Superconvergence of c0-qk finite element method for elliptic equations with approximated coefficients. Journal of Scientific Computing, 82(1), December 2019.


C. Linder, M. Tkachuk, and C. Miehe. A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids, 59(10):2134–2156, October 2011.


A. Logg, K.-A. Mardal, and G. Wells, editors. Automated Solution of Differential Equations by the Finite Element Method. Springer Berlin Heidelberg, 2012.


Rainald Lohner. Edge-Based Compressible Flow Solvers. John Wiley & Sons, Ltd, 2008.


Catherine Mavriplis. Adaptive mesh strategies for the spectral element method. Computer Methods in Applied Mechanics and Engineering, 116(1):77–86, 1994.


Steve McConnell. Code Complete. Microsoft Press, second edition, 2004.


J.M. Melenk and I. Babuska. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1–4):289 – 314, 1996.


Jens Markus Melenk and Barbara I. Wohlmuth. On residual-based a posteriori error estimation in hp-FEM. Advances in Computational Mathematics, 15(1):311–331, 2001.


William F. Mitchell and Marjorie A. McClain. A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations. ACM Trans. Math. Softw., 41(1):2/1–39, Oct 2014.


William F. Mitchell. The hp‐multigrid method applied to hp‐adaptive refinement of triangular grids. Numerical Linear Algebra with Applications, 17(2-3):211–228, Apr


Mo Mu. PDE.Mart: A network-based problem-solving environment for PDEs. ACM Trans. Math. Software., 31(4):508–531, 2005.


Peter Munch, Katharina Kormann, and Martin Kronbichler. An efficient, matrix-free finite-element library for high-dimensional partial differential equations, 2020.


N. C. Nguyen and J. Peraire. Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J. Comput. Phys., 231(18):5955–5988, 2012.


J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer, New York, 1999.


Jorge Nocedal and Stephen Wright. Numerical Optimization. Springer New York, 2006.


Jorge Nocedal, Andreas Wächter, and Richard A. Waltz. Adaptive barrier update strategies for nonlinear interior methods. SIAM Journal on Optimization, 19(4):1674–1693, January 2009.


Y. H. Pao. Electromagnetic forces in deformable continua. In S. Nemat-Nasser, editor, Mechanics Today, volume 4 of Pergamon Mechanics Today Series, chapter IV, pages 209–305. Elsevier, New York, 1978.


Chunjae Park and Dongwoo Sheen. P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM Journal on Numerical Analysis, 41(2):624–640, 2003.


J-P. V. Pelteret and P. Steinmann. Magneto-active polymers: Fabrication, characterisation, modelling and simulation at the micro- and macro-scale. De Gruyter Mouton, 1 edition, 2019.


J.-P. V. Pelteret, B. Walter, and P. Steinmann. Application of metaheuristic algorithms to the identification of nonlinear magneto-viscoelastic constitutive parameters. Journal of Magnetism and Magnetic Materials, 464:116–131, October


Rolf Rannacher and Franz-Theo Suttmeier. A feed-back approach to error control in finite element methods: Application to linear elasticity. Computational Mechanics, 19(5):434–446, 1997.


Rolf Rannacher and Franz-Theo Suttmeier. A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity. Computational Mechanics, 21(2):123–133, 1998.


Rolf Rannacher and Franz-Theo Suttmeier. A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Computer Methods in Applied Mechanics and Engineering, 176(1):333–361, 1999.


Béatrice Rivière, Mary F. Wheeler, and Vivette Girault. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Computational Geosciences, 3(3/4):337–360, 1999.


Y. Saad. A Flexible Inner-Outer Preconditioned GMRES Algorithm. Technical Report 91-279, Minnesota Supercomputer Institute, University of Minnesota, 1991.


Nico Schlömer. quadpy: Your one-stop shop for numerical integration in python, 2021.


Svenja Schoeder, Katharina Kormann, Wolfgang A. Wall, and Martin Kronbichler. Efficient explicit time stepping of high order discontinuous Galerkin schemes for waves. SIAM J. Sci. Comput., 40(6):C803–C826, 2018.


D. Silvester and A. Wathen. Fast iterative solution of stabilised Stokes systems. Part II: Using general block preconditioners. SIAM J. Numer. Anal., 31:1352–1367, 1994.


Barry Smith, Petter Bjorstad, and William Gropp. Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, 2004.


Hans R Stoll. The relationship between put and call option prices. The Journal of Finance, 24(5):801–824, 1969.


Franz-Theo Suttmeier. Adaptive Finite Element Approximation of Problems in Elasto-Plasticity Theory. Dissertation, Universität Heidelberg, 1996.


Eleuterio F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg, 2009.


Andrea Toselli and Olof Widlund. Domain decomposition methods-algorithms and theory, volume 34. Springer Science & Business Media, 2006.


C. Truesdell and R. Toupin. Encyclopedia of Physics: Principles of Thermodynamics and Statics, volume 1, chapter 2: The classical field theories, pages 226–794. Springer-Verlag Berlin Heidelberg, 1960.


Kostas Tselios and Theodore E. Simos. Optimized Runge–Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. Physics Letters A, 363:38–48, 2007.


Andreas Wächter and Lorenz T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25–57, April 2005.


Zhuoran Wang, Graham Harper, Patrick O'Leary, Jiangguo Liu, and Simon Tavener. deal.II implementation of a weak galerkin finite element solver for darcy flow. In Lecture Notes in Computer Science, pages 495–509. Springer International Publishing, 2019.


Garth N. Wells and Nguyen Tien Dung. A c0 discontinuous galerkin formulation for kirchhoff plates. Computer Methods in Applied Mechanics and Engineering, 196(35-36):3370–3380, July 2007.


Freddie D Witherden and Peter E Vincent. On the identification of symmetric quadrature rules for finite element methods. Computers & Mathematics with Applications, 69(10):1232–1241, 2015.