Immersed Virtual Element Methods for Elliptic Interface Problems

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 is the key to project immersed virtual functions to classic immersed finite element (IFE) functions for computing numerical solutions. 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. It also brings a connection between various methods such as body-fitted methods, IFE methods, and virtual element methods etc.

Immersed Virtual Element Methods for Maxwell Interface Problems in Three Dimensions

“…Nevertheless, the proposed IVE method follows the framework of VEM: the local PDEs need not be solved exactly, certain projections are computed instead with sufficient approximation capability to capture the jump conditions. It can successfully yield optimal convergence rates for the H(curl) interface problem, which has been rigorously proved in the 2D case [28]. In this work, we focus on the development of the IVE spaces, the scheme and the implementation in the 3D case.…”

“…This consideration motivates us to combine the conformity of virtual element spaces and the approximation capabilities of IFE spaces. In our previous work [28], we have successfully realized this idea for the 2D case, which is referred to as immersed virtual element (IVE) methods.…”

“…However, the treatment for H(curl) problems is quite different. The meta-framework for VEM developed in [22,25] cannot be directly used to obtain even optimal error estimates, and some more delicate techniques are needed, e.g., [15,28,12]. For interface problems, an extra layer of difficulty is to make error bounds robust with respect to potential anisotropic element shapes.…”

An immersed Crouzeix-Raviart finite element method in 2D and 3D based on discrete level set functions

“…We note that the local problems are also used in the virtual element method (VEM) with variable coefficients. As pointed out in [9], for 1D problems with a piecewise constant coefficient β, the IFE space in [34], the finite element space in [2], and the virtual element space are exactly identical due to the trivial 1D geometry, but they are distinguished in higher dimensions because of the more complicated geometry. For the existing 2D IFE methods (see, e.g., [35,36,30]), the interface inside an interface element is approximated by a straight line connecting the intersection points of the interface and the edges of the element, and a piecewise linear function is used as the IFE basis function so that the interface conditions can be satisfied on the straight line.…”

“…So, FEMs based on unfitted meshes, which are completely independent of the interface, have become highly attractive for interface problems. There are many FEMs using unfitted meshes (called unfitted mesh methods) in the literature, for example, the unfitted Nitsche's method [26,39,6], the extended FEM [17], the multiscale FEM [11], the FEM for high-contrast problems [24], the immersed virtual element method (IVEM) [9], and the immersed finite element (IFE) method [34,35,36,1,32], to name only a few.…”