Geodesic
A plane wave virtual element method for the Helmholtz problem
Perugia, Ilaria1, 2  ; Pietra, Paola3 ; Russo, Alessandro4
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

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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 :
DOI : 10.1051/m2an/2015066
Classification : 65N30, 65N12, 65N15, 35J05
Keywords: Helmholtz equation, virtual element method, plane wave basis functions, error analysis, duality estimates

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 
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     title = {A plane wave virtual element method for the {Helmholtz} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {783--808},
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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

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