Agglomeration-based geometric multigrid schemes for the Virtual Element Method

In this paper we analyse the convergence properties of two-level, W-cycle and V-cycle agglomeration-based geometric multigrid schemes for the numerical solution of the linear system of equations stemming from the lowest order C-conforming Virtual Element discretization of two-dimensional second-order elliptic partial differential equations. The sequence of agglomerated tessellations are nested, but the corresponding multilevel virtual discrete spaces are generally non-nested thus resulting into non-nested multigrid algorithms. We prove the uniform convergence of the two-level method with respect to the mesh size and the uniform convergence of the W-cycle and the V-cycle multigrid algorithms with respect to the mesh size and the number of levels. Numerical experiments confirm the theoretical findings.

[1]  Franco Dassi,et al.  Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems , 2020, Computer Methods in Applied Mechanics and Engineering.

[2]  Stefano Berrone,et al.  Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method , 2017 .

[3]  Paola F. Antonietti,et al.  V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes , 2017, Journal of Scientific Computing.

[4]  Silvia Bertoluzza,et al.  BDDC and FETI-DP for the virtual element method , 2017, 1708.03599.

[5]  Shangyou Zhang,et al.  A generalized BPX multigrid framework covering nonnested V-cycle methods , 2007, Math. Comput..

[6]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[7]  Juan G. Calvo,et al.  An overlapping Schwarz method for virtual element discretizations in two dimensions , 2019, Comput. Math. Appl..

[8]  Xiaozhe Hu,et al.  Multigrid algorithms for $hp$-version Interior Penalty Discontinuous Galerkin methods on polygonal and polyhedral meshes , 2014 .

[9]  Alessandro Russo,et al.  Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes , 2014, 1506.07328.

[10]  Lorenzo Mascotto,et al.  Ill‐conditioning in the virtual element method: Stabilizations and bases , 2017, 1705.10581.

[11]  Jianguo Huang,et al.  Some error analysis on virtual element methods , 2017, 1708.08558.

[12]  P. F. Antonietti,et al.  A multigrid algorithm for the $p$-version of the Virtual Element Method , 2017, 1703.02285.

[13]  P. Alam.  ‘L’ , 2021, Composites Engineering: An A–Z Guide.

[14]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[15]  Micol Pennacchio,et al.  Algebraic Multigrid Methods For Virtual Element Discretizations: A Numerical Study , 2018, 1812.02161.

[16]  S. Bertoluzza,et al.  FETI-DP Preconditioners for the Virtual Element Method on General 2D Meshes , 2017, Lecture Notes in Computational Science and Engineering.

[17]  Ahmed Alsaedi,et al.  Equivalent projectors for virtual element methods , 2013, Comput. Math. Appl..

[18]  Joseph E. Pasciak,et al.  THE ANALYSIS OF SMOOTHERS FOR MULTIGRID ALGORITHMS , 1992 .

[19]  J. Pasciak,et al.  New convergence estimates for multigrid algorithms , 1987 .

[20]  J. Pasciak,et al.  The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms , 1991 .

[21]  Juan G. Calvo,et al.  On the approximation of a virtual coarse space for domain decomposition methods in two dimensions , 2018, Mathematical Models and Methods in Applied Sciences.