We introduce and analyze families of Galerkin–collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin–collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2020033
Mots-clés : Wave equation, variational time discretization, collocation methods, space-time finite element methods, error estimates, post-processing
@article{M2AN_2020__54_6_2099_0,
author = {Anselmann, Mathias and Bause, Markus and Becher, Simon and Matthies, Gunar},
title = {Galerkin{\textendash}collocation approximation in time for the wave equation and its post-processing},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2099--2123},
publisher = {EDP-Sciences},
volume = {54},
number = {6},
year = {2020},
doi = {10.1051/m2an/2020033},
mrnumber = {4160327},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2020033/}
}
TY - JOUR AU - Anselmann, Mathias AU - Bause, Markus AU - Becher, Simon AU - Matthies, Gunar TI - Galerkin–collocation approximation in time for the wave equation and its post-processing JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 2099 EP - 2123 VL - 54 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2020033/ DO - 10.1051/m2an/2020033 LA - en ID - M2AN_2020__54_6_2099_0 ER -
%0 Journal Article %A Anselmann, Mathias %A Bause, Markus %A Becher, Simon %A Matthies, Gunar %T Galerkin–collocation approximation in time for the wave equation and its post-processing %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 2099-2123 %V 54 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2020033/ %R 10.1051/m2an/2020033 %G en %F M2AN_2020__54_6_2099_0
Anselmann, Mathias; Bause, Markus; Becher, Simon; Matthies, Gunar. Galerkin–collocation approximation in time for the wave equation and its post-processing. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 2099-2123. doi : 10.1051/m2an/2020033. http://www.numdam.org/articles/10.1051/m2an/2020033/
, , and , Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 200 (2011) 1747–1756. | DOI | MR | Zbl
, and , Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem. Comput. Methods Appl. Mech. Eng. 313 (2017) 28–52. | DOI | MR
, and , Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27 (2006) 5–40. | DOI | MR | Zbl
, and , Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods. Numer. Math. 114 (2009) 133–160. | DOI | MR | Zbl
, and , Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118 (2011) 429–456. | DOI | MR | Zbl
and , Numerical study of Galerkin–collocation approximation in time for the wave equation. Preprint arXiv:1905.00606 (2019). To appear in Mathematics of Wave Phenomena, edited by et al. Trends in Mathematics. Birkhäuser (2019).
, , and , Galerkin–collocation approximation in time for the wave equation and its post-processing. Preprint arXiv:1908.08238v1 (2019). | MR
and , Continuous finite elements in space and time for the heat equation. Math. Comput. 52 (1989) 255–274. | DOI | MR | Zbl
and , Continuous finite elements in space and time for the nonhomogeneous wave equation. Comput. Math. Appl. 27 (1994) 91–102. | DOI | MR | Zbl
, and , Adaptive Galerkin finite element methods for the wave equation. Comput. Methods Appl. Math. 10 (2010) 3–48. | DOI | MR | Zbl
, and , deal.II, Differential equations analysis library. Technical reference. Technical reference. http://www.dealii.org (2013).
and , Comparative study of continuously differentiable Galerkin time discretizations for the wave equation. PAMM 19 (2019) e201900144. | DOI
and , Variational time discretization for mixed finite element approximations of nonstationary diffusion problems. J. Comput. Appl. Math. 289 (2015) 208–224. | DOI | MR
, and , Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320 (2017) 745–768. | DOI | MR
, and , Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space. Numer. Math. 137 (2017) 773–818. | DOI | MR
, , and , Post-processed Galerkin approximation of improved order for wave equations. Math. Comput. 89 (2020) 595–627. | DOI | MR
and , Variational time discretizations of higher order and higher regularity. Preprint arXiv:2003.04056 (2020). | MR
, and , Variational methods for stable time discretization of first-order differential equations. In: Advanced Computing in Industrial Mathematics, edited by , and . BGSIAM. Springer, Cham (2018) 63–75.
and , Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 50 (2012) 1181–1206. | DOI | MR | Zbl
, and , The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175 (2008) 83–95. | DOI
and , An overview of the discontinuous Petrov-Galerkin method In: Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, edited by , and . Springer, Cham (2014) 149–180. | DOI | MR | Zbl
, and , Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl. Math. 16 (2016) 409–428. | DOI | MR
and , Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs. Math. Comput. 85 (2016) 2099–2129. | DOI | MR
, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island (2010). | Zbl
, A parallel and adaptive space-time method for Maxwell’s equations. Ph.D. thesis, KIT, Karlruhe (2016).
and , A continuous space-time finite element method for the wave equation. Math. Comput. 65 (1996) 491–506. | DOI | MR | Zbl
and , Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38 (2016) A2173–A2208. | DOI | MR
, One-step piecewise polynomial Galerkin methods for initial value problems. Math. Comput. 26 (1972) 416–426. | DOI | MR | Zbl
, Discrete Galerkin and related one-step methods for ordinary differential equations. Math. Comput. 26 (1972) 881–891. | DOI | MR | Zbl
, and , Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer. Math. 19 (2011) 41–61. | DOI | MR | Zbl
, and , An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow. Int. J. Numer. Methods Fluids 73 (2013) 927–952. | DOI | MR
, Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Eng. 107 (1993) 117–129. | DOI | MR | Zbl
and , On Gautschi’s conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae. J. Comput. Appl. Math. 233 (2009) 768–774. | DOI | MR | Zbl
and , Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations. Math. Comput. 74 (2004) 85–102. | DOI | MR | Zbl
, Variational space-time methods for the elastic wave equation and the diffusion equation. Ph.D. thesis, Helmut Schmidt University, Hamburg (2015).
and , Variational space-time methods for the wave equation. J. Sci. Comput. 61 (2014) 424–453. | DOI | MR | Zbl
, and , Space-time isogeometric analysis of parabolic evoluation equations. Comput. Methods Appl. Mech. Eng. 306 (2016) 342–363. | DOI | MR
, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971). | DOI | MR | Zbl
and , Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972). | MR | Zbl
and , A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104 (2006) 489–514. | DOI | MR | Zbl
and , Higher order variational time discretizations for nonlinear systems of ordinary differential equations. Preprint 23/2011, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg (2011).
and , Theory of the dynamic Biot–Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53 (2012) 123702. | DOI | MR | Zbl
and , Numerical Approximation of Partial Differential Equations. Springer, Berlin (2008). | Zbl
, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15 (2015) 551–566. | DOI | MR
and , Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D. Numer. Linear Algebra App. 25 (2018) e2143. | DOI | MR
, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin (2006). | MR | Zbl
and , On Runge-Kutta, collocation and discontinuous Galerkin methods: mutual connections and resulting consequences to the analysis. Programs Algorithms Numer. Math. 17 (2015) 231–236. | MR
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