Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Calcolo
Published in: Calcolo 4/2017

04-05-2017

Multigrid algorithms for \(\varvec{hp}\)-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

Authors: P. F. Antonietti, P. Houston, X. Hu, M. Sarti, M. Verani

Published in: Calcolo | Issue 4/2017

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied.
previous article The representations and computations of generalized inverses , and the group inverse
next article On the convergence of Schröder’s method for the simultaneous computation of polynomial zeros of unknown multiplicity

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Antonietti, P.F., Beirão Da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386–404 (2014)MathSciNetMATHCrossRef
2.
Antonietti, P.F., Beirão Da Veiga, L., Scacchi, S., Verani, M.: A C1 virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34–56 (2016)MathSciNetMATHCrossRef
3.
Antonietti, P.F., Brezzi, F., Marini, L.: Stabilizations of the Baumann-Oden DG formulation: the 3D case. Boll. Unione Mat. Ital. (9) 1(3), 629–643 (2008)MathSciNetMATH
4.
Antonietti, P.F., Brezzi, F., Marini, L.D.: Bubble stabilization of discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 198(21–26), 1651–1659 (2009)MathSciNetMATHCrossRef
5.
Antonietti, P.F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E.H., Giani, S., Houston, P.: Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains. In: Barrenechea G. R. et al. (eds.) Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Lecture Notes in Computational Science and Engineering, vol. 114, pp. 279–307 (2016)
6.
Antonietti, P.F., Facciola, C., Russo, A., Verani, M.: Discontinuous Galerkin approximation of flows in fractured porous media on polygonal and polyhedral meshes. MOX Report 55/2016 (2016)
7.
Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Nicola, V.: Mimetic finite difference approximation of flows in fractured porous media. Math. Model. Numer. Anal. 50(3), 809–832 (2016)MathSciNetMATHCrossRef
8.
Antonietti, P.F., Giani, S., Houston, P.: \(hp\)-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35(3), A1417–A1439 (2013)MathSciNetMATHCrossRef
9.
Antonietti, P.F., Giani, S., Houston, P.: Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comput. 60(1), 203–227 (2014)MathSciNetMATHCrossRef
10.
Antonietti, P.F., Houston, P.: A class of domain decomposition preconditioners for \(hp\)-discontinuous Galerkin finite element methods. J. Sci. Comput. 46(1), 124–149 (2011)MathSciNetMATHCrossRef
11.
Antonietti, P.F., Houston, P.: Preconditioning high-order discontinuous Galerkin discretizations of elliptic problems. Lect. Notes Comput. Sci. Eng. 91, 231–238 (2013)MathSciNetCrossRef
12.
Antonietti, P.F., Houston, P., Smears, I.: A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods. Int. J. Numer. Anal. Model. 13(4), 513–524 (2016)MathSciNetMATH
13.
Antonietti, P.F., Sarti, M., Verani, M.: Multigrid algorithms for \(hp\)-discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53(1), 598–618 (2015)MathSciNetMATHCrossRef
14.
Antonietti, P.F., Sarti, M., Verani, M.: Multigrid algorithms for high order discontinuous Galerkin methods. Lect. Notes Comput. Sci. Eng. 104, 3–13 (2016)MathSciNetMATHCrossRef
15.
Antonietti, P.F., Sarti, M., Verani, M., Zikatanov, L.T.: A uniform additive Schwarz preconditioner for high-order discontinuous Galerkin approximations of elliptic problems. J. Sci. Comput. 70(2), 608–630 (2017)MathSciNetMATHCrossRef
16.
17.
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001)MathSciNetMATHCrossRef
18.
Aubin, J.: Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires. J. Math. Anal. Appl. 30, 510–521 (1970)MathSciNetMATHCrossRef
19.
20.
21.
Bassi, F., Botti, L., Colombo, A., Brezzi, F., Manzini, G.: Agglomeration-based physical frame dg discretizations: an attempt to be mesh free. Math. Models Methods Appl. Sci. 24(8), 1495–1539 (2014)MathSciNetMATHCrossRef
22.
Bassi, F., Botti, L., Colombo, A., Di Pietro, D.A., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231(1), 45–65 (2012)MathSciNetMATHCrossRef
23.
Bassi, F., Botti, L., Colombo, A., Rebay, S.: Agglomeration based discontinuous Galerkin discretization of the Euler and Navier–Stokes equations. Comput. Fluids 61, 77–85 (2012)MathSciNetMATHCrossRef
24.
Bastian, P., Blatt, M., Scheichl, R.: Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. Numer. Linear Algebra Appl. 19(2), 367–388 (2012)MathSciNetMATHCrossRef
25.
Beirão Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(01), 199–214 (2013)MathSciNetMATHCrossRef
26.
Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Mixed virtual element methods for general second order elliptic problems on polygonal meshes. M2AN. Math. Model. Numer. Anal. 50(3), 727–747 (2016)MathSciNetMATHCrossRef
27.
Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)MathSciNetMATHCrossRef
28.
Beirão da Veiga, L., Lipnikov, K., Manzini, G.: The mimetic finite difference method for elliptic problems, MS&A. Modeling, Simulation and Applications, vol. 11, Springer, Cham (2014)
29.
Bramble, J.: Multigrid Methods. Number 294 in Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1993)
30.
Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005). (electronic)MathSciNetMATHCrossRef
31.
Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16(2), 275–297 (2006)MathSciNetMATHCrossRef
32.
Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)MathSciNetMATHCrossRef
33.
Cangiani, A., Dong, Z., Georgoulis, E.: \(hp\)-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes. Submitted for publication (2016)
34.
Cangiani, A., Dong, Z., Georgoulis, E., Houston, P.: \(hp\)–Version discontinuous Galerkin methods on polygonal and polyhedral meshes. 2016, in preparation (2016)
35.
Cangiani, A., Dong, Z., Georgoulis, E., Houston, P.: \(hp\)-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. M2AN. Math. Model. Numer. Anal. 50(3), 699–725 (2016)MathSciNetMATHCrossRef
36.
Cangiani, A., Georgoulis, E.H., Houston, P.: \(hp\)-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24(10), 2009–2041 (2014)MathSciNetMATHCrossRef
37.
Chan, T.F., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids. In: Domain decomposition methods, 10 (Boulder, CO, 1997), volume 218 of Contemp. Math., pp. 67–81. American Mathematical Society, Providence, RI (1998)
38.
Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin methods, Springer, Berlin, 2000. Theory, computation and applications. Papers from the 1st International Symposium held in Newport, RI, May 24-26 (1999)
39.
Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications(Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012)
40.
Fries, T.-P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)MathSciNetMATH
41.
Georgoulis, E.H.: Inverse-type estimates on \(hp\)-finite element spaces and applications. Math. Comput. 77(261), 201–219 (2008). (electronic)MathSciNetMATHCrossRef
42.
Hackbusch, W.: Multi-grid methods and applications, volume 4of Springer series in computational mathematics. Springer, Berlin (1985)
43.
Hackbusch, W., Sauter, S.: Composite finite elements for problems containing small geometric details. Part II: implementation and numerical results. Comput. Vis. Sci. 1(4), 15–25 (1997)MATHCrossRef
44.
Hackbusch, W., Sauter, S.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75(4), 447–472 (1997)MathSciNetMATHCrossRef
45.
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, 1st edn. Springer, Berlin (2007)MATH
46.
Hyman, J., Shashkov, M., Steinberg, S.: The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132(1), 130–148 (1997)MathSciNetMATHCrossRef
47.
Lions, J.-L.: Problèmes aux limites non homogènes à donées irrégulières: Une méthode d’approximation. In: Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Rome, pp. 283–292 (1968)
48.
49.
Moulitsas, I., Karypis, G.: Multilevel algorithms for generating coarse grids for multigrid methods,. In: Supercomputing 2001 Conference Proceedings (2001)
50.
Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Uni. Hamburg 36, 9–15 (1971)MATHCrossRef
51.
Olson, L.N., Schroder, J.B.: Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems. J. Comput. Phys. 230(18), 6959–6976 (2011)MathSciNetMATHCrossRef
52.
Pavarino, L.F.: Additive Schwarz methods for the \(p\)-version finite element method. Numer. Math. 66(4), 493–515 (1994)MathSciNetMATH
53.
Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)
54.
Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, volume 35 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008)
55.
Schöberl, J., Melenk, J.M., Pechstein, C., Zaglmayr, S.: Additive Schwarz preconditioning for \(p\)-version triangular and tetrahedral finite elements. IMA J. Numer. Anal. 28(1), 1–24 (2008)MathSciNetMATHCrossRef
56.
Schwab, C.: \(p\)- and \(hp\)-Finite Element Methods. Numerical Mathematics and Scientific Computation: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, New York (1998)MATH
57.
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)MATH
58.
59.
Tabarraei, A., Sukumar, N.: Extended finite element method on polygonal and quadtree meshes. Comput. Methods Appl. Mech. Eng. 197(5), 425–438 (2008)MathSciNetMATHCrossRef
60.
Talischi, C., Paulino, G., Pereira, A., Menezes, I.: Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscipl. Optim. 45(3), 309–328 (2012)MathSciNetMATHCrossRef
61.
Metadata
Title
Multigrid algorithms for -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes
Authors
P. F. Antonietti
P. Houston
X. Hu
M. Sarti
M. Verani
Publication date
04-05-2017
Publisher
Springer Milan
Published in
Calcolo / Issue 4/2017
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-017-0223-6

Other articles of this Issue 4/2017

Calcolo 4/2017 Go to the issue

OriginalPaper

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

OriginalPaper

Critical lengths of cycloidal spaces are zeros of Bessel functions

OriginalPaper

The representations and computations of generalized inverses , and the group inverse

OriginalPaper

A real structure-preserving method for the quaternion LU decomposition, revisited

OriginalPaper

Partial condition number for the equality constrained linear least squares problem

OriginalPaper

BDDC and FETI-DP for the virtual element method

Premium Partner

    Image Credits