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Coarse spaces for virtual element methods on irregular 3D subdomain decompositions

Ana Aguilar-Pineda, Luis F. Amey, Adrian Angulo-Paniagua, Juan G. Calvo

TL;DR

The paper introduces a two-level overlapping Schwarz preconditioner for 3D problems discretized with the Virtual Element Method on irregular polyhedral subdomain decompositions. It builds a reduced coarse space using vertex-based functions extended via discrete harmonic extensions on subdomain faces and interiors, integrated into a two-level preconditioner with local solvers. Numerical experiments demonstrate robustness to the number of subdomains and mesh parameters, with coarse-space dimensions remaining moderate and performance comparable to FEM-based coarse spaces. The work offers a practical, simple solver for VEM on challenging geometries, with potential extensions to high-contrast or multiscale settings.

Abstract

We present a two-level overlapping Schwarz preconditioner for three-dimensional problems discretized with the Virtual Element Method. Our approach handles general polyhedral meshes and irregular subdomains, extending the applicability of previous methods. Numerical experiments show robust performance with respect to the number of subdomains and mesh parameters, with condition-number bound comparable to classical finite element results. While alternative methods such as FETI-DP and BDDC are available, the simplicity and competitiveness of overlapping additive Schwarz methods underscore the practical significance of our contribution.

Coarse spaces for virtual element methods on irregular 3D subdomain decompositions

TL;DR

The paper introduces a two-level overlapping Schwarz preconditioner for 3D problems discretized with the Virtual Element Method on irregular polyhedral subdomain decompositions. It builds a reduced coarse space using vertex-based functions extended via discrete harmonic extensions on subdomain faces and interiors, integrated into a two-level preconditioner with local solvers. Numerical experiments demonstrate robustness to the number of subdomains and mesh parameters, with coarse-space dimensions remaining moderate and performance comparable to FEM-based coarse spaces. The work offers a practical, simple solver for VEM on challenging geometries, with potential extensions to high-contrast or multiscale settings.

Abstract

We present a two-level overlapping Schwarz preconditioner for three-dimensional problems discretized with the Virtual Element Method. Our approach handles general polyhedral meshes and irregular subdomains, extending the applicability of previous methods. Numerical experiments show robust performance with respect to the number of subdomains and mesh parameters, with condition-number bound comparable to classical finite element results. While alternative methods such as FETI-DP and BDDC are available, the simplicity and competitiveness of overlapping additive Schwarz methods underscore the practical significance of our contribution.

Paper Structure

This paper contains 5 sections, 7 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The virtual element method
  3. Coarse space
  4. Numerical results
  5. Conclusions

Figures (3)

  • Figure 1: (left) A polyhedral mesh, (middle) a subdomain $\Omega_i$ and its overlapping subdomain $\Omega_i'$, and (right) a piecewise coefficient $\rho$ with $\rho|_{\Omega_i} = \rho_i \in [1,10^3]$.
  • Figure 2: Subdomain faces for (left) square, (middle) Voronoi and (right) hexagonal meshes with subdomains based on the incenter of the elements. $R_0^T \varphi_{\bm{x}_0}$ is shown for each subdomain face.
  • Figure 3: Subdomain faces for (left) square, (middle) Voronoi and (right) hexagonal meshes with METIS subdomains. $R_0^T \varphi_{\bm{x}_0}$ is shown for each subdomain face.