The Mixed Virtual Element Method on curved edges in two dimensions

TL;DR

This paper extends the mixed Virtual Element Method to handle curved edges in 2D domains, improving geometric accuracy and convergence rates for complex geometries.

Contribution

It introduces a new VEM approximation space that accurately models curvilinear features, ensuring optimal convergence in 2D computational grids.

Findings

Achieves optimal convergence rates with curved edges

Validates the method through numerical tests

Provides theoretical analysis of the scheme

Abstract

In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical feature, such as a portion of domain boundary or an internal interface, may introduce a geometrical error that degrades the expected order of convergence of the scheme. In the present work a suitable VEM approximation space is proposed to consistently handle curvilinear geometrical objects, thus recovering optimal convergence rates. The resulting numerical scheme is presented along with its theoretical analysis and several numerical test cases to validate the proposed approach.