Published

A Family of Immersed Finite Element Spaces and Applications to Three Dimensional H(Curl) Interface Problems

Long Chen, Ruchi Guo, and Jun Zou

SIAM Journal on Scientific Computing, 45( 6), 2023. A3121-A3149.

PDF   Bibtex   doi: https://doi.org/10.1137/22M1505360

ABSTRACT:

 Efficient and accurate computation of H(curl) interface
problems is of great importance in many electromagnetic
applications. Unfitted mesh methods are especially attractive in
three-dimensional (3D) computation as they can circumvent generating
complex 3D interface-fitted meshes. However, many unfitted mesh
methods rely on nonconforming approximation spaces, which may cause a
loss of accuracy for solving Maxwell-type equations, and the widely
used penalty techniques in the literature may not help in recovering
the optimal convergence. In this article, we provide a remedy by
developing Nédélec-type immersed finite element (IFE) spaces with a
Petrov–Galerkin scheme that is able to produce optimal-convergent
solutions. To establish a systematic framework, we analyze all the H1,
H(curl), and H(div) IFE spaces and form a discrete de Rham
complex. Based on these fundamental results, we further develop a fast
solver using a modified Hiptmair–Xu preconditioner which works for
both the generalized minimal residual (GMRES) and conjugate gradient
(CG) methods for solving the nonsymmetric linear algebraic system. The
approximation capabilities of the proposed IFE spaces will be also
established.