The Reduced Basis Multigrid scheme for the Virtual Element Method
Paola F. Antonietti, Silvia Bertoluzza, Fabio Credali
TL;DR
The paper addresses the challenge of applying multigrid to the lowest-order Virtual Element Method on polygonal meshes, where VEM spaces are defined implicitly and spaces are non-nested across levels. It introduces a computable rbVEM auxiliary space to construct intergrid operators within a non-nested W-cycle multigrid, using an $L^2$- projection prolongation and Richardson smoothing, while solving the problem on the original VEM space at the finest level. The key contributions are the first use of rbVEM to design a multigrid for the $h$-version of VEM, and numerical evidence that the convergence factor is independent of mesh refinement and reduced-basis size, with improved performance as smoothing increases. This yields scalable solvers for VEM on general polygonal meshes and motivates further convergence analysis and V-cycle extensions.
Abstract
We present a non-nested W-cycle multigrid scheme for the lowest order Virtual Element Method on polygonal meshes. To avoid the implicit definition of the Virtual Element space, which poses several issues in the computation of intergrid operators that underpin multigrid methods, the proposed scheme uses a fully-conforming auxiliary space constructed by cheaply computing the virtual basis functions via the reduced basis method.