  In this paper, we introduce the locally conservative enriched immersed finite
element method (EIFEM) to tackle the elliptic problem with interface. The
immersed finite element is useful for handling interface with mesh unfit with
the interface. However, all the currently available method under IFEM framework
may not be designed to consider the flux conservation. We provide an efficient
and effective remedy for this issue by introducing a local piecewise constant
enrichment, which provides the locally conservative flux. We have also
constructed and analyzed an auxiliary space preconditioner for the resulting
system based on the application of algebraic multigrid method. The new
observation in this work is that by imposing strong Dirichlet boundary
condition for the standard IFEM part of EIFEM, we are able to remove the zero
eigen-mode of the EIFEM system while still imposing the Dirichlet boundary
condition weakly assigned to the piecewise constant enrichment part of EIFEM. A
couple of issues relevant to the piecewise constant enrichment given for the
mesh unfit to the interface has been discussed and clarified as well. Numerical
tests are provided to confirm the theoretical development.

Numerical Analysis;Numerical Analysis