Immersed Virtual Element Methods for Elliptic Interface Problems in Two Dimensions

TL;DR

This paper introduces an immersed virtual element method that effectively solves elliptic interface problems in two dimensions, combining advantages of body-fitted and unfitted mesh approaches with robust convergence and improved performance.

Contribution

The paper develops a novel immersed virtual element method with local interface problem solutions and establishes convergence analysis robust to interface positioning.

Findings

Method handles complex interface configurations.

Provides better performance than penalty-type IFE methods.

Establishes a connection between various numerical methods.

Abstract

This article presents an immersed virtual element method for solving a class of interface problems that combines the advantages of both body-fitted mesh methods and unfitted mesh methods. A background body-fitted mesh is generated initially. On those interface elements, virtual element spaces are constructed as solution spaces to local interface problems, and exact sequences can be established for these new spaces involving discontinuous coefficients. The discontinuous coefficients of interface problems are recast as Hodge star operators that are the key to project immersed virtual functions to classic immersed finite element (IFE) functions for computing numerical solutions. An a priori convergence analysis is established robust with respect to the interface location. The proposed method is capable of handling more complicated interface element configuration and provides better performance than the conventional penalty-type IFE method for the H(curl)-interface problem arising from Maxwell equations. It also brings a connection between various methods such as body-fitted methods, IFE methods, virtual element methods, etc.