A plane wave virtual element method for the Helmholtz problem

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 ingred...

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Published inESAIM. Mathematical modelling and numerical analysis Vol. 50; no. 3; pp. 783 - 808
Main Authors Perugia, Ilaria, Pietra, Paola, Russo, Alessandro
Format Journal Article
LanguageEnglish
Published Les Ulis EDP Sciences 01.05.2016
Subjects
35J05
65N12
65N15
65N30
Apexes
Basis functions
Boundary conditions
duality estimates
error analysis
Helmholtz equation
Helmholtz equations
Mathematical analysis
Nonlinear programming
plane wave basis functions
Plane waves
virtual element method
Online AccessGet full text
ISSN0764-583X
1290-3841
DOI10.1051/m2an/2015066

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Abstract 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.
AbstractList 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.
Author Pietra, Paola
Perugia, Ilaria
Russo, Alessandro
Author_xml – sequence: 1
  givenname: Ilaria
  surname: Perugia
  fullname: Perugia, Ilaria
  organization: Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
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  givenname: Paola
  surname: Pietra
  fullname: Pietra, Paola
  organization: Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR, 27100 Pavia, Italy
– sequence: 3
  givenname: Alessandro
  surname: Russo
  fullname: Russo, Alessandro
  organization: University of Milano Bicocca, 20126 Milano, Italy
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Snippet 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...
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SubjectTerms 35J05
65N12
65N15
65N30
Apexes
Basis functions
Boundary conditions
duality estimates
error analysis
Helmholtz equation
Helmholtz equations
Mathematical analysis
Nonlinear programming
plane wave basis functions
Plane waves
virtual element method
Title A plane wave virtual element method for the Helmholtz problem
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