A Virtual Element Method for the wave equation on curved edges in two dimensions

TL;DR

This paper introduces an extension of the Virtual Element Method with curved edges for accurately solving the wave equation in two dimensions, precisely representing domain boundaries and interfaces to improve solution accuracy.

Contribution

The work develops a Virtual Element Method that exactly models curved geometries, surpassing traditional finite element approaches in accuracy for wave equation simulations.

Findings

Exact geometry representation improves solution accuracy.

The method achieves the expected order of convergence.

Numerical results confirm theoretical predictions.

Abstract

In this work we present an extension of the Virtual Element Method with curved edges for the numerical approximation of the second order wave equation in a bidimensional setting. Curved elements are used to describe the domain boundary, as well as internal interfaces corresponding to the change of some mechanical parameters. As opposite to the classic and isoparametric Finite Element approaches, where the geometry of the domain is approximated respectively by piecewise straight lines and by higher order polynomial maps, in the proposed method the geometry is exactly represented, thus ensuring a highly accurate numerical solution. Indeed, if in the former approach the geometrical error might deteriorate the quality of the numerical solution, in the latter approach the curved interfaces/boundaries are approximated exactly guaranteeing the expected order of convergence for the numerical scheme. Theoretical results and numerical findings confirm the validity of the proposed approach.