Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T02:06:36.945Z Has data issue: false hasContentIssue false
  • English
  • Français

Topics in structure-preserving discretization*

Published online by Cambridge University Press:  28 April 2011

Snorre H. Christiansen ,
Hans Z. Munthe-Kaas  and
Brynjulf Owren
Snorre H. Christiansen
Affiliation:
Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway E-mail: [email protected]
Hans Z. Munthe-Kaas
Affiliation:
Department of Mathematics, University of Bergen, N-5008 Bergen, Norway E-mail: [email protected]
Brynjulf Owren
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives.

Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes.

The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

Type
Research Article
Information
Acta Numerica , Volume 20 , May 2011 , pp. 1 - 119
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

ÅAhlander, K. and Munthe-Kaas, H. (2005), ‘Applications of the generalized Fourier transform in numerical linear algebra’, BIT 45, 819–850.Google Scholar
Allgower, E. L., Böhmer, K., Georg, K. and Miranda, R. (1992), ‘Exploiting symmetry in boundary element methods’, SIAM J. Numer. Anal. 29, 534–552.CrossRefGoogle Scholar
Allgower, E. L., Georg, K. and Miranda, R. (1993), Exploiting permutation symmetry with fixed points in linear equations. In Lectures in Applied Mathematics (Allgower, E. L., Georg, K. and Miranda, R., eds), Vol. 29, AMS, pp. 23–36.Google Scholar
Allgower, E. L., Georg, K., Miranda, R. and Tausch, J. (1998), ‘Numerical exploitation of equivariance’, Z. Angew. Math. Mech. 78, 185–201.Google Scholar
Andreianov, B., Bendahmane, M. and Karlsen, K. H. (2010), ‘Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic–parabolic equations’, J. Hyperbolic Diff. Equations 7, 1–67.Google Scholar
Arnold, D. N., Bochev, P. B., Lehoucq, R. B., Nicolaides, R. A. and Shashkov, M., eds (2006 a), Compatible Spatial Discretizations, Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer.Google Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006 b), Finite element exterior calculus, homological techniques, and applications. In Acta Numerica, Vol. 15, Cambridge University Press, pp. 1–155.Google Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2010), ‘Finite element exterior calculus: From Hodge theory to numerical stability’, Bull. Amer. Math. Soc. (NS) 47, 281–354.CrossRefGoogle Scholar
Baker, H. F. (1905), ‘Alternants and continuous groups’, Proc. London Math. Soc. 3, 24–47.Google Scholar
Beerends, R. J. (1991), ‘Chebyshev polynomials in several variables and the radial part of the Laplace–Beltrami operator’, Trans. Amer. Math. Soc. 328, 779–814.CrossRefGoogle Scholar
Benjamin, T. B. (1972), ‘The stability of solitary waves’, Proc. Roy. Soc. London Ser. A 328, 153–183.Google Scholar
Benjamin, T. B., Bona, J. L. and Mahony, J. J. (1972), ‘Model equations for long waves in nonlinear dispersive systems’, Philos. Trans. Roy. Soc. London Ser. A 272, 47–78.Google Scholar
Blanes, S. and Moan, P. (2006), ‘Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems’, Appl. Numer. Math. 56, 1519–1537.CrossRefGoogle Scholar
Blanes, S., Casas, F., Oteo, J. A. and Ros, J. (2009), ‘The Magnus expansion and some of its applications’, Phys. Rep. 470, 151–238.Google Scholar
Bochev, P. B. and Hyman, J. M. (2006), Principles of mimetic discretizations of differential operators. In Compatible Spatial Discretizations, Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 89–119.CrossRefGoogle Scholar
Boffi, D. (2010), Finite element approximation of eigenvalue problems. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 1–120.Google Scholar
Bossavit, A. (1986), ‘Symmetry, groups, and boundary value problems: A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry’, Comput. Methods Appl. Mech. Engrg 56, 167–215.Google Scholar
Bossavit, A. (1988), Mixed finite elements and the complex of Whitney forms. In The Mathematics of Finite Elements and Applications VI (Uxbridge 1987), Academic Press, pp. 137–144.Google Scholar
Brezzi, F. (1974), ‘On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers’, Rev. Fraçaise Automat. In format. 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.CrossRefGoogle Scholar
Brezzi, F., Douglas, J. Jr, and Marini, L. D. (1985), ‘Two families of mixed finite elements for second order elliptic problems’, Numer. Math. 47, 217–235.Google Scholar
Brezzi, F., Lipnikov, K. and Shashkov, M. (2005), ‘Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes’, SIAM J. Numer. Anal. 43, 1872–1896.CrossRefGoogle Scholar
Bryant, R. L. (1995), An introduction to Lie groups and symplectic geometry. In Geometry and Quantum Field Theory (Freed, D. S. and Uhlenbeck, K. K., eds), Vol. 1 of IAS/Park City Mathematics Series, AMS.CrossRefGoogle Scholar
Buffa, A. and Christiansen, S. H. (2007), ‘A dual finite element complex on the barycentric refinement’, Math. Comp. 76, 1743–1769.Google Scholar
Bump, D. (2004), Lie Groups, Springer.Google Scholar
Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T. (2006), Spectral Methods: Fundamentals in Single Domains, Scientific Computation series, Springer.CrossRefGoogle Scholar
Celledoni, E. (2005), Eulerian and semi-Lagrangian schemes based on commutator-free exponential integrators. In Group Theory and Numerical Analysis, Vol. 39 of CRM Proc. Lecture Notes, AMS, pp. 77–90.Google Scholar
Celledoni, E. and Iserles, A. (2000), ‘Approximating the exponential from a Lie algebra to a Lie group’, Math. Comp. 69, 1457–1480.Google Scholar
Celledoni, E. and Iserles, A. (2001), ‘Methods for the approximation of the matrix exponential in a Lie-algebraic setting’, IMA J. Numer. Anal. 21, 463–488.Google Scholar
Celledoni, E. and Kometa, B. K. (2009), ‘Semi-Lagrangian Runge–Kutta exponential integrators for convection dominated problems’, J. Sci. Comput. 41, 139–164.Google Scholar
Celledoni, E., Cohen, D. and Owren, B. (2008), ‘Symmetric exponential integrators with an application to the cubic Schrödinger equation’, Found. Comput. Math. 8, 303–317.CrossRefGoogle Scholar
Celledoni, E., Marthinsen, A. and Owren, B. (2003), ‘Commutator-free Lie group methods’, Future Generation Computer Systems 19, 341–352.CrossRefGoogle Scholar
Certaine, J. (1960), The solution of ordinary differential equations with large time constants. In Mathematical Methods for Digital Computers, Wiley, pp. 128–132.Google Scholar
Christiansen, S. H. (2007), ‘Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension’, Numer. Math. 107, 87–106.CrossRefGoogle Scholar
Christiansen, S. H. (2008 a), ‘A construction of spaces of compatible differential forms on cellular complexes’, Math. Models Methods Appl. Sci. 18, 739–757.Google Scholar
Christiansen, S. H. (2008 b), On the linearization of Regge calculus. E-print, Department of Mathematics, University of Oslo.Google Scholar
Christiansen, S. H. (2009), Foundations of finite element methods for wave equations of Maxwell type. In Applied Wave Mathematics, Springer, pp. 335–393.Google Scholar
Christiansen, S. H. (2010), ‘Éléments finis mixtes minimaux sur les polyèdres’, CR Math. Acad. Sci. Paris 348, 217–221.CrossRefGoogle Scholar
Christiansen, S. H. and Scheid, C. (2011), ‘Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation’, ESAIM: Math. Model. Numer. Anal. 45, 739–760.Google Scholar
Christiansen, S. H. and Winther, R. (2006), ‘On constraint preservation in numerical simulations of Yang–Mills equations’, SIAM J. Sci. Comput. 28, 75–101.CrossRefGoogle Scholar
Christiansen, S. H. and Winther, R. (2008), ‘Smoothed projections in finite element exterior calculus’, Math. Comp. 77, 813–829.Google Scholar
Christiansen, S. H. and Winther, R. (2010), On variational eigenvalue approximation of semidefinite operators. Preprint: arXiv.org/abs/1005.2059.Google Scholar
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, Vol. 4 of Studies in Mathematics and its Applications, North-Holland.Google Scholar
Clément, P. (1975), ‘Approximation by finite element functions using local regularization’, RAIRO Analyse Numérique 9, 77–84.Google Scholar
Courant, R., Friedrichs, K. and Lewy, H. (1928), ‘Über die partiellen Differenzengleichungen der mathematischen Physik’, Math. Ann. 100, 32–74.Google Scholar
Cox, S. M. and Matthews, P. C. (2002), ‘Exponential time differencing for stiff systems’, J. Comput. Phys. 176, 430–455.Google Scholar
Crouch, P. E. and Grossman, R. (1993), ‘Numerical integration of ordinary differential equations on manifolds’, J. Nonlinear Sci. 3, 1–33.Google Scholar
Dahlby, M. and Owren, B. (2010), A general framework for deriving integral preserving numerical methods for PDEs. Technical report 8/2010, Norwegian University of Science and Technology. arXiv.org/abs/1009.3151.Google Scholar
Dahlby, M., Owren, B. and Yaguchi, T. (2010), Preserving multiple first integrals by discrete gradients. Technical report 11/2010, Norwegian University of Science and Technology. arXiv.org/abs/1011.0478.Google Scholar
Demkowicz, L. and Babuška, I. (2003), ‘p interpolation error estimates for edge finite elements of variable order in two dimensions’, SIAM J. Numer. Anal. 41, 1195–1208.Google Scholar
Demkowicz, L. and Buffa, A. (2005), ‘H1, H(curl) and H(div)-conforming projection-based interpolation in three dimensions: Quasi-optimal p-interpolation estimates’, Comput. Methods Appl. Mech. Engrg 194, 267–296.Google Scholar
Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W. and Zdunek, A. (2008), Computing with hp-adaptive Finite Elements, Vol. 2, Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC.Google Scholar
Diele, F., Lopez, L. and Peluso, R. (1998), ‘The Cayley transform in the numerical solution of unitary differential systems’, Adv. Comput. Math. 8, 317–334.Google Scholar
Dodziuk, J. and Patodi, V. K. (1976), ‘Riemannian structures and triangulations of manifolds’, J. Indian Math. Soc. (NS) 40, 1–52.Google Scholar
Douglas, C. C. and Mandel, J. (1992), ‘Abstract theory for the domain reduction method’, Computing 48, 73–96.CrossRefGoogle Scholar
Droniou, J., Eymard, R., Gallouët, T. and Herbin, R. (2010), ‘A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods’, Math. Models Methods Appl. Sci. 20, 265–295.Google Scholar
Dubiner, M. (1991), ‘Spectral methods on triangles and other domains’, J. Sci. Comput. 6, 345–390.CrossRefGoogle Scholar
Eier, R. and Lidl, R. (1982), ‘A class of orthogonal polynomials In K variables’, Math. Ann. 260, 93–99.CrossRefGoogle Scholar
Fässler, A. F. and Stiefel, E. (1992), Group Theoretical Methods and their Applications, Birkhäuser.Google Scholar
Furihata, D. (1999), ‘Finite difference schemes for ∂u/∂t = (∂/∂x)α∂G/γu that inherit energy conservation or dissipation property’, J. Comput. Phys. 156, 181–205.Google Scholar
Furihata, D. (2001 a), ‘Finite-difference schemes for nonlinear wave equation that inherit energy conservation property’, J. Comput. Appl. Math. 134, 37–57.Google Scholar
Furihata, D. (2001 b), ‘A stable and conservative finite difference scheme for the Cahn-Hilliard equation’, Numer. Math. 87, 675–699.Google Scholar
Furihata, D. and Matsuo, T. (2003), ‘A stable, convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation’, Japan J. Indust. Appl. Math. 20, 65–85.CrossRefGoogle Scholar
Georg, K. and Miranda, R. (1992), Exploiting symmetry in solving linear equations. In Bifurcation and Symmetry (Allgower, E. L., Böhmer, K. and Golubisky, M., eds), Vol. 104 of International Series of Numerical Mathematics, Birkhäuser, pp. 157–168.Google Scholar
Giraldo, F. X. and Warburton, T. (2005), ‘A nodal triangle-based spectral element method for the shallow water equations on the sphere’, J. Comput. Phys. 207, 129–150.Google Scholar
Gonzalez, O. (1996), ‘Time integration and discrete Hamiltonian systems’, J. Nonlinear Sci. 6, 449–467.Google Scholar
Griffiths, P. A. and Morgan, J. W. (1981), Rational Homotopy Theory and Differential Forms, Vol. 16 of Progress in Mathematics, Birkhäuser.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, second edition, Vol. 31 of Springer Series in Computational Mathematics, Springer.Google Scholar
Hausdorff, F. (1906), ‘Die symbolische Exponential Formel in der Gruppentheorie’, Leipziger Ber. 58, 19–48.Google Scholar
Hesthaven, J. S. (1998), ‘From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex’, SIAM J. Numer. Anal. 35, 655–676.Google Scholar
Hilbert, S. (1973), ‘A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations’, Math. Comp. 27, 81–89.CrossRefGoogle Scholar
Hiptmair, R. (1999), ‘Canonical construction of finite elements’, Math. Comp. 68, 1325–1346.Google Scholar
Hiptmair, R. (2002), Finite elements in computational electromagnetism. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 237–339.Google Scholar
Hochbruck, M. and Ostermann, A. (2005), ‘Explicit exponential Runge–Kutta methods for semilinear parabolic problems’, SIAM J. Numer. Anal. 43, 1069–1090.Google Scholar
Hochbruck, M. and Ostermann, A. (2010), Exponential integrators. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 209–286.Google Scholar
Hoffman, M. E. and Withers, W. D. (1988), ‘Generalized Chebyshev polynomials associated with affine Weyl groups’, Trans. Amer. Math. Soc. 308, 91–104.Google Scholar
Huybrechs, D. (2010), ‘On the Fourier extension of non-periodic functions’, SIAM J. Numer. Anal. 47, 4326–4355.CrossRefGoogle Scholar
Huybrechs, D., Iserles, A. and Nørsett, S. (2010), ‘From high oscillation to rapid approximation V: The equilateral triangle’, IMA J. Numer. Anal. doi:10.1093/imanum/drq010.Google Scholar
Iserles, A. and Nørsett, S. P. (1999), ‘On the solution of linear differential equations in Lie groups’, Philos. Trans. Roy. Soc. London Ser. A 357, 983–1019.Google Scholar
Iserles, A. and Zanna, A. (2005), ‘Efficient computation of the matrix exponential by generalized polar decompositions’, SIAM J. Numer. Anal. 42, 2218–2256.Google Scholar
Iserles, A., Munthe-Kaas, H., Nørsett, S. P. and Zanna, A. (2000), Lie-group methods. In Acta Numerica, Vol. 9, Cambridge University Press, pp. 215–365.Google Scholar
James, G. and Liebeck, M. (2001), Representations and Characters of Groups, second edition, Cambridge University Press.Google Scholar
Kang, F. and Shang, Z.-J. (1995), ‘Volume-preserving algorithms for source-free dynamical systems’, Numer. Math. 71, 451–463.CrossRefGoogle Scholar
Karlsen, K. H. and Karper, T. K. (2010), ‘Convergence of a mixed method for a semi-stationary compressible Stokes system’, Math. Comp. doi:10.1090/S0025–5718–2010–02446–9.Google Scholar
Kennedy, C. A. and Carpenter, M. H. (2003), ‘Additive Runge–Kutta schemes for convection–diffusion–reaction equations’, Appl. Numer. Math. 44, 139–181.Google Scholar
Koornwinder, T. (1974), ‘Orthogonal polynomials in two variables which are eigen-functions of two algebraically independent partial differential operators I–IV’, Indag. Math. 36, 48–66 and 357–381.Google Scholar
Krogstad, S. (2005), ‘Generalized integrating factor methods for stiff PDEs’, J. Comput. Phys. 203, 72–88.Google Scholar
Krogstad, S., Munthe-Kaas, H. and Zanna, A. (2009), ‘Generalized polar coordinates on Lie groups and numerical integrators’, Numer. Math. 114, 161–187.Google Scholar
Kuznetsov, Y. and Repin, S. (2005), ‘Convergence analysis and error estimates for mixed finite element method on distorted meshes’, J. Numer. Math. 13, 33–51.Google Scholar
LaBudde, R. A. and Greenspan, D. (1974), ‘Discrete mechanics: A general treatment’, J. Comput. Phys. 15, 134–167.Google Scholar
Lawson, J. D. (1967), ‘Generalized Runge–Kutta processes for stable systems with large Lipschitz constants’, SIAM J. Numer. Anal. 4, 372–380.CrossRefGoogle Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Lewis, D. and Simo, J. C. (1994), ‘Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups’, J. Nonlinear Sci. 4, 253–299.Google Scholar
Lidl, R. (1975), ‘Tchebyscheffpolynome in mehreren Variablen’, J. Reine Angew. Math. 273, 178–198.Google Scholar
Lomont, J. S. (1959), Applications of Finite Groups, Academic Press.Google Scholar
Lopez, L. and Politi, T. (2001), ‘Applications of the Cayley approach in the numerical solution of matrix differential systems on quadratic groups’, Appl. Numer. Math. 36, 35–55.Google Scholar
Marthinsen, A. and Owren, B. (2001), ‘Quadrature methods based on the Cayley transform’, Appl. Numer. Math. 39, 403–413.Google Scholar
Matsuo, T. (2007), ‘New conservative schemes with discrete variational derivatives for nonlinear wave equations’, J. Comput. Appl. Math. 203, 32–56.Google Scholar
Matsuo, T. (2008), ‘Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations’, J. Comput. Appl. Math. 218, 506–521.Google Scholar
Matsuo, T. and Furihata, D. (2001), ‘Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations’, J. Comput. Phys. 171, 425–447.Google Scholar
Matsuo, T., Sugihara, M., Furihata, D. and Mori, M. (2000), ‘Linearly implicit finite difference schemes derived by the discrete variational method’, Sūrikaise-kikenkyūsho Kōkyūroku 1145, 121–129.Google Scholar
Matsuo, T., Sugihara, M., Furihata, D. and Mori, M. (2002), ‘Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method’, Japan J. Indust. Appl. Math. 19, 311–330.CrossRefGoogle Scholar
McLachlan, R. I. (1995), ‘On the numerical integration of ordinary differential equations by symmetric composition methods’, SIAM J. Sci. Comput. 16, 151–168.Google Scholar
McLachlan, R. I., Quispel, G. R. W. and Robidoux, N. (1999), ‘Geometric integration using discrete gradients’, Philos. Trans. Roy. Soc. London Ser. A 357, 1021–1045.Google Scholar
McLachlan, R. I., Quispel, G. R. W. and Tse, P. S. P. (2009), ‘Linearization-preserving self-adjoint and symplectic integrators’, BIT 49, 177–197.Google Scholar
Minchev, B. V. (2004), Exponential Integrators for Semilinear Problems, University of Bergen. PhD thesis, University of Bergen, Norway.Google Scholar
Minesaki, Y. and Nakamura, Y. (2006), ‘New numerical integrator for the Stäckel system conserving the same number of constants of motion as the degree of freedom’, J. Phys. A 39, 9453–9476.Google Scholar
Moler, C. and Van Loan, C. (1978), ‘Nineteen dubious ways to compute the exponential of a matrix’, SIAM Review 20, 801–836.CrossRefGoogle Scholar
Moler, C. B. and van Loan, C. F. (2003), ‘Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later’, SIAM Review 45, 3–49.Google Scholar
Monk, P. (2003), Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press.Google Scholar
Morton, K. W. (2010), ‘The convection–diffusion Petrov–Galerkin story’, IMA J. Numer. Anal. 30, 231–240.Google Scholar
Munthe-Kaas, H. (1989), Symmetric FFTs: A general approach. In Topics in Linear Algebra for Vector and Parallel Computers, PhD thesis, NTNU, Trondheim, Norway. Available at: hans.munthe-kaas.no.Google Scholar
Munthe-Kaas, H. (1999), ‘High order Runge–Kutta methods on manifolds’, Appl. Numer. Math. 29, 115–127.Google Scholar
Munthe-Kaas, H. (2006), ‘On group Fourier analysis and symmetry preserving discretizations of PDEs’, J. Phys. A 39, 5563.Google Scholar
Munthe-Kaas, H. Z. (1995), ‘Lie–Butcher theory for Runge–Kutta methods’, BIT 35, 572–587.Google Scholar
Munthe-Kaas, H. Z. (1998), ‘Runge–Kutta methods on Lie groups’, BIT 38, 92–111.Google Scholar
Munthe-Kaas, H. Z. and Owren, B. (1999), ‘Computations in a free Lie algebra’, Philos. Trans. Roy. Soc. London Ser. A 357, 957–981.Google Scholar
Munthe-Kaas, H. Z. and Zanna, A. (1997), Numerical integration of differential equations on homogeneous manifolds. In Foundations of Computational Mathematics (Cucker, F. and Shub, M., eds), Springer, pp. 305–315.CrossRefGoogle Scholar
Nédélec, J.-C. (1980), ‘Mixed finite elements in R3’, Numer. Math. 35, 315–341.Google Scholar
Nørsett, S. P. (1969), An A-stable modification of the Adams–Bashforth methods. In Conf. Numerical Solution of Differential Equations (Dundee 1969), Springer, pp. 214–219.Google Scholar
Olver, P. J. (1993), Applications of Lie Groups to Differential Equations, second edition, Vol. 107 of Graduate Texts in Mathematics, Springer.Google Scholar
Ostermann, A., Thalhammer, M. and Wright, W. M. (2006), ‘A class of explicit exponential general linear methods’, BIT 46, 409–431.Google Scholar
Owren, B. (2006), ‘Order conditions for commutator-free Lie group methods’, J. Phys. A 39, 5585–5599.CrossRefGoogle Scholar
Owren, B. and Marthinsen, A. (1999), ‘Runge–Kutta methods adapted to manifolds and based on rigid frames’, BIT 39, 116–142.Google Scholar
Owren, B. and Marthinsen, A. (2001), ‘Integration methods based on canonical coordinates of the second kind’, Numer. Math. 87, 763–790.Google Scholar
Pasciak, J. E. and Vassilevski, P. S. (2008), ‘Exact de Rham sequences of spaces defined on macro-elements in two and three spatial dimensions’, SIAM J. Sci. Comput. 30, 2427–2446.Google Scholar
Raviart, P.-A. and Thomas, J. M. (1977), A mixed finite element method for second order elliptic problems. In Mathematical Aspects of Finite Element Methods, Vol. 606 of Lecture Notes in Mathematics, Springer, pp. 292–315.Google Scholar
Roberts, J. E. and Thomas, J.-M. (1991), Mixed and hybrid methods. In Handbook of Numerical Analysis, Vol. II, North-Holland, pp. 523–639.Google Scholar
Ryland, B. N. and Munthe-Kaas, H. (2011), On multivariate Chebyshev polynomials and spectral approximations on triangles. In Spectral and High Order Methods for Partial Differential Equations, Vol. 76 of Lecture Notes in Computational Science and Engineering, Springer, pp. 19–41.Google Scholar
Sanz-Serna, J. M. and Calvo, M. P. (1994), Numerical Hamiltonian Problems, Vol. 7 of Applied Mathematics and Mathematical Computation, Chapman & Hall.Google Scholar
Schöberl, J. (2008), ‘A posteriori error estimates for Maxwell equations’, Math. Comp. 77, 633–649.Google Scholar
Schöberl, J. and Sinwel, A. (2007), Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. RICAM report.Google Scholar
Serre, J. P. (1977), Linear Representations of Finite Groups, Springer.Google Scholar
Stein, E. M. (1970), Singular Integrals and Differentiability Properties of Functions, Vol. 30 of Princeton Mathematical Series, Princeton University Press.Google Scholar
Strang, G. (1972), ‘Approximation in the finite element method’, Numer. Math. 19, 81–98.Google Scholar
Trønnes, A. (2005), Symmetries and generalized Fourier transforms applied to computing the matrix exponential. Master's thesis, University of Bergen, Norway.Google Scholar
Varadarajan, V. S. (1984), Lie Groups, Lie Algebras, and Their Representations, Vol. 102 of Graduate Texts in Mathematics, Springer.Google Scholar
Warburton, T. (2006), ‘An explicit construction of interpolation nodes on the simplex’, J. Engng Math. 56, 247–262.Google Scholar
Warner, F. W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Vol. 94 of Graduate Texts in Mathematics, Springer.Google Scholar
Weil, A. (1952), ‘Sur les thèorémes de de Rham’, Comment. Math. Helv. 26, 119–145.Google Scholar
Wensch, J., Knoth, O. and Galant, A. (2009), ‘Multirate infinitesimal step methods for atmospheric flow simulation’, BIT 49, 449–473.CrossRefGoogle Scholar
Whitney, H. (1957), Geometric Integration Theory, Princeton University Press.Google Scholar
Yaguchi, T., Matsuo, T. and Sugihara, M. (2010), ‘Conservative numerical schemes for the Ostrovsky equation’, J. Comput. Appl. Math. 234, 1036–1048.Google Scholar
Zanna, A. and Munthe-Kaas, H. Z. (2001/2002), ‘Generalized polar decompositions for the approximation of the matrix exponential’, SIAM J. Matrix Anal. Appl. 23, 840–862.Google Scholar
Zanna, A., Engø, K. and Munthe-Kaas, H. Z. (2001), ‘Adjoint and selfadjoint Lie-group methods’, BIT 41, 395–421.Google Scholar