A multigrid algorithm for the p-version of the virtual element method

Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco

doi:10.1051/m2an/2018007

Antonietti, Paola F.¹ ; Mascotto, Lorenzo ¹ ; Verani, Marco ¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 337-364.

Résumé

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.

MR Zbl

DOI : 10.1051/m2an/2018007

Classification : 65N30, 65N55
Mots clés : Polygonal meshes, virtual element methods, p Galerkin methods, p multigrid

Affiliations des auteurs :

Antonietti, Paola F. ¹ ; Mascotto, Lorenzo ¹ ; Verani, Marco ¹

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     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://www.numdam.org/articles/10.1051/m2an/2018007/}
}

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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://www.numdam.org/articles/10.1051/m2an/2018007/

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