A plane wave virtual element method for the Helmholtz problem

Perugia, Ilaria; Pietra, Paola; Russo, Alessandro

doi:10.1051/m2an/2015066

Perugia, Ilaria ^1,² ; Pietra, Paola ³ ; Russo, Alessandro ⁴

¹ Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
² Department of Mathematics, University of Pavia, 27100 Pavia, Italy
³ Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR, 27100 Pavia, Italy
⁴ University of Milano Bicocca, 20126 Milano, Italy

ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 783-808.

Résumé

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: (i) a low order VEM space whose basis functions, which are associated to the mesh vertices, are not explicitly computed in the element interiors; (ii) a proper local projection operator onto the plane wave space; (iii) an approximate stabilization term. A convergence result for the $h$ -version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.

Reçu le : 2015-04-14

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/m2an/2015066

Classification : 65N30, 65N12, 65N15, 35J05
Mots-clés : Helmholtz equation, virtual element method, plane wave basis functions, error analysis, duality estimates

Affiliations des auteurs :

Perugia, Ilaria ^{1, 2} ; Pietra, Paola ³ ; Russo, Alessandro ⁴

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Perugia, Ilaria; Pietra, Paola; Russo, Alessandro. A plane wave virtual element method for the Helmholtz problem. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 783-808. doi : 10.1051/m2an/2015066. http://www.numdam.org/articles/10.1051/m2an/2015066/

Bibliographie
Cité par

P.F. Antonietti, L. Beirão Da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52 (2014) 386–404. | DOI | MR | Zbl

B. Ayuso De Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. To appear in Special issue – Polyhedral discretization for PDE. ESAIM: M2AN 50 (2016). DOI: | DOI | Numdam | MR

I.M. Babuška and S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation? SIAM Rev. 42 (2000) 451–484. | MR | Zbl

L. Beirão Da Veiga, and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl

L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, $H$ (div) and $H$ (curl)-conforming virtual element method. To appear in Numer. Math. (2015). DOI: | DOI | MR

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems. To appear in Special issue – Polyhedral discretization for PDE. ESAIM M2AN 50 (2016). DOI: | DOI

L Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016) 729–750. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, The Hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24 (2014) 1541–1573. | DOI | MR | Zbl

L. Beirão Da Veiga, C. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes. Comput. Methods Appl. Mech. Eng. 295 (2015) 327–346. | DOI | MR | Zbl

M.F. Benedetto, S. Berrone, S. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280 (2014) 135–156. | DOI | MR | Zbl

J.H. Bramble and L.E. Payne, Bounds in the Neumann problem for second order uniformly elliptic operators. Pacific J. Math 12 (1962) 823–833. | DOI | MR | Zbl

F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253 (2013) 455–462. | DOI | MR | Zbl

F. Brezzi and L.D. Marini, Virtual Element and Discontinuous Galerkin Methods. In Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. Springer (2014) 209–221. | MR | Zbl

F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl

A. Buffa and P. Monk, Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation. ESAIM: M2AN 42 (2008) 925–940. | DOI | Numdam | MR | Zbl

O. Cessenat, Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques, Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine (1996).

O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35 (1998) 255–299. | DOI | MR | Zbl

E. Deckers, O. Atak, L. Coox, R. D’Amico, H. Devriendt, S. Jonckheere, K. Koo, B. Pluymers, D. Vandepitte and W. Desmet, The wave based method: An overview of 15 years of research. Innovations in Wave Modelling. Wave Motion 51 (2014) 550–565. | DOI | MR | Zbl

W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis. Ph.D. thesis, KU Leuven, Belgium, 1998.

C. Farhat, I. Harari and L. Franca, The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6455–6479. | DOI | MR | Zbl

C. Farhat, I. Harari and U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192 (2003) 1389–1419. | DOI | MR | Zbl

G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225 (2007) 1961–1984. | DOI | MR | Zbl

G. Gabard, Exact integration of polynomial-exponential products with application to wave-based numerical methods. Comm. Numer. Methods Eng. 25 (2009) 237–246. | DOI | MR | Zbl

A.L. Gain, C. Talischi and G.H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282 (2014) 132–160. | DOI | MR | Zbl

C.J. Gittelson, Plane wave discontinuous Galerkin methods. Master’s thesis, SAM-ETH Zürich, Switzerland (2008).

C.J. Gittelson, R. Hiptmair and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the $h$ -version. ESAIM: M2AN 43 (2009) 297–332. | DOI | Numdam | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Approximation by plane waves. Technical report 2009-27, SAM-ETH Zürich, Switzerland (2009). Available at http://www.sam.math.ethz.ch/reports/2009/27.

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$ -version. SIAM J. Numer. Anal. 49 (2011) 264–284. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79 (2014) 79–91. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation. “Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations”. Edited by G.R. Barrenechea, A. Cangiani, E.H. Geogoulis. In Lect. Notes Comput. Sci. Eng. Springer. Preprint [math.NA] (2015). | arXiv | MR

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin Methods: Exponential convergence of the $h p$ -version. To appear in Found. Comput. Math. (2015). DOI: | DOI | MR

F. Ihlenburg and I. Babuska, Solution of Helmholtz problems by knowledge-based fem. Comp. Ass. Mech. Eng. Sci. 4 (1997) 397–416. | Zbl

J. Ladevèze and P. Ladevèze, Bounds of the Poincaré constant with respect to the problem of star-shaped membrane regions. Z. Angew. Math. Phys. 29 (1978) 670–683. | MR | Zbl

P. Ladevèze and H. Riou, On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 278 (2014) 729–743. | DOI | MR | Zbl

G. Manzini, A. Russo and N. Sukumar. New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24 (2014) 1665–1699. | DOI | MR | Zbl

J.M. Melenk, On Generalized Finite Element Methods. Ph.D. thesis, University of Maryland, 1995. | MR

J.M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1996) 289–314. | DOI | MR | Zbl

J.M. Melenk and I Babuska, Approximation with harmonic and generalized harmonic polynomials in the partition of unity method. Comp. Ass. Mech. Eng. Sci. 4 (1997) 607–632. | Zbl

J.M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49 (2011) 1210–1243. | DOI | MR | Zbl

A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems. Ph.D. thesis, Seminar for applied mathematics, ETH Zürich (2011). Available at http://e-collection.library.ethz.ch/view/eth:4515.

A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62 (2011) 809–837. | DOI | MR | Zbl

P. Monk and D.Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 175 (1999) 121–136. | DOI | MR | Zbl

L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl

H. Riou, P. Ladevéze and B. Sourcis, The multiscale VTCR approach applied to acoustics problems. J. Comput. Acoust. 16 (2008) 487–505. | DOI | Zbl

M. Stojek, Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Methods Eng. 41 (1998) 831–849. | DOI | MR | Zbl

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61 (2004) 2045–2066. | DOI | MR | Zbl

R. Tezaur and C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Methods Eng. 66 (2006) 796–815. | DOI | MR | Zbl

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