A study on unfitted 1D finite element methods.

In the present paper we consider a 1D Poisson model characterized by the presence of an
interface, where a transmission condition arises due to jumps of the coefficients. We aim
at studying finite element methods with meshes not fitting such an interface. It is well
known that when the mesh does not fit the material discontinuities the resulting scheme
provides in general lower order accurate solutions. We focus on so-called embedded
approaches, frequently adopted to treat fluid–structure interaction problems, with the aim
of recovering higher order of approximation also in presence of non fitting meshes; we
implement several methods inspired by: the Immersed Boundary method, the Fictitious
Domain method, and the Extended Finite Element method. In particular, we present four
formulations in a comprehensive and unified format, proposing several numerical tests and
discussing their performance. Moreover, we point out issues that may be encountered in
the generalization to higher dimensions and we comment on possible solutions.