We present a multigrid algorithm for the solution of the linear systems of equations stemming from the p-version of the virtual element discretization of a two-dimensional Poisson problem. The sequence of coarse spaces are constructed decreasing progressively the polynomial approximation degree of the virtual element space, as in standard p-multigrid schemes. The construction of the interspace operators relies on auxiliary virtual element spaces, where it is possible to compute higher order polynomial projectors. We prove that the multigrid scheme is uniformly convergent, provided the number of smoothing steps is chosen sufficiently large. We also demonstrate that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom that can be employed to accelerate the convergence of classical Krylov-based iterative schemes. Numerical experiments validate the theoretical results.
Mots clés : Polygonal meshes, virtual element methods, p Galerkin methods, p multigrid
@article{M2AN_2018__52_1_337_0,
author = {Antonietti, Paola F. and Mascotto, Lorenzo and Verani, Marco},
title = {A multigrid algorithm for the p-version of the virtual element method},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {337--364},
publisher = {EDP-Sciences},
volume = {52},
number = {1},
year = {2018},
doi = {10.1051/m2an/2018007},
mrnumber = {3808163},
zbl = {1397.65249},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018007/}
}
TY - JOUR AU - Antonietti, Paola F. AU - Mascotto, Lorenzo AU - Verani, Marco TI - A multigrid algorithm for the p-version of the virtual element method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 337 EP - 364 VL - 52 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018007/ DO - 10.1051/m2an/2018007 LA - en ID - M2AN_2018__52_1_337_0 ER -
%0 Journal Article %A Antonietti, Paola F. %A Mascotto, Lorenzo %A Verani, Marco %T A multigrid algorithm for the p-version of the virtual element method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 337-364 %V 52 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018007/ %R 10.1051/m2an/2018007 %G en %F M2AN_2018__52_1_337_0
Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco. A multigrid algorithm for the p-version of the virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 337-364. doi : 10.1051/m2an/2018007. http://archive.numdam.org/articles/10.1051/m2an/2018007/
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