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 -version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.
DOI : 10.1051/m2an/2015066
Mots clés : Helmholtz equation, virtual element method, plane wave basis functions, error analysis, duality estimates
@article{M2AN_2016__50_3_783_0, author = {Perugia, Ilaria and Pietra, Paola and Russo, Alessandro}, title = {A plane wave virtual element method for the {Helmholtz} problem}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {783--808}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/m2an/2015066}, zbl = {1343.65137}, mrnumber = {3507273}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015066/} }
TY - JOUR AU - Perugia, Ilaria AU - Pietra, Paola AU - Russo, Alessandro TI - A plane wave virtual element method for the Helmholtz problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 783 EP - 808 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015066/ DO - 10.1051/m2an/2015066 LA - en ID - M2AN_2016__50_3_783_0 ER -
%0 Journal Article %A Perugia, Ilaria %A Pietra, Paola %A Russo, Alessandro %T A plane wave virtual element method for the Helmholtz problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 783-808 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015066/ %R 10.1051/m2an/2015066 %G en %F M2AN_2016__50_3_783_0
Perugia, Ilaria; Pietra, Paola; Russo, Alessandro. A plane wave virtual element method for the Helmholtz problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 783-808. doi : 10.1051/m2an/2015066. http://www.numdam.org/articles/10.1051/m2an/2015066/
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