Accepted

Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions

Shuhao Cao, Long Chen, Ruchi Guo

Mathematical Models and Methods in Applied Sciences

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ABSTRACT:

 Finite element methods for electromagnetic problems modeled
by Maxwell-type equations are highly sensitive to the conformity of
approximation spaces, and non-conforming methods may cause loss of
convergence. This fact leads to an essential obstacle for almost all
the interface-unfitted mesh methods in the literature regarding the
application to electromagnetic interface problems, as they are based
on non-conforming spaces. In this work, a novel immersed virtual
element method for solving a 3D H(curl) interface problem is
developed, and the motivation is to combine the conformity of virtual
element spaces and robust approximation capabilities of immersed
finite element spaces. The proposed method is able to achieve optimal
convergence. To develop a systematic framework, the H1,
H(curl) and H(div) interface problems and their
corresponding problem-orientated immersed virtual element spaces are
considered all together. In addition, the de Rham complex will be
established based on which the Hiptmair-Xu (HX) preconditioner can be
used to develop a fast solver for the H(curl) interface problem.