# Auxiliary space preconditioners for virtual element methods on polytopal meshes

## TL;DR

This paper introduces auxiliary space preconditioners for virtual element methods on polytopal meshes, improving the efficiency of solving linear systems from second order elliptic equations with robust performance regardless of mesh size or coefficient jumps.

## Contribution

The paper develops novel auxiliary space preconditioners specifically designed for virtual element methods on polytopal meshes, ensuring uniform boundedness of condition numbers.

## Key findings

- Condition numbers are uniformly bounded, independent of problem size.
- Preconditioners perform well across various numerical experiments.
- Effective for second order elliptic equations with coefficient jumps.

## Abstract

In this paper, we develop the auxiliary space preconditioners for solving the linear system arising from the virtual element methods discretization on polytopal meshes for the second order elliptic equations. The preconditioners are constructed based on an auxiliary simplicial mesh. The condition numbers of the preconditioned systems are uniformly bounded, independent of the problem size and the jump in coefficients. Several numerical experiments are presented to demonstrate the performance of the preconditioners.