The elliptic equations with discontinuous coefficients are often used to describe the
problems of the multiple materials or fluids with different densities or conductivities
or diffusivities. In this paper we develop a partially penalty immersed finite element
(PIFE) method on triangular grids for anisotropic flow models, in which the diffusion
coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart
type finite element space is used on non-interface elements and the piecewise linear
Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface
elements. The piecewise linear functions satisfying the interface jump conditions are
uniquely determined by the integral averages on the edges as degrees of freedom.
The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete
interior penalty discontinuous Galerkin formulation. The solvability of the method is
proved and the optimal error estimates in the energy norm are obtained. Numerical
experiments are presented to confirm our theoretical analysis and show that the
newly developed PIFE method has optimal-order convergence in the L2 norm as well.
In addition, numerical examples also indicate that this method is valid for both the
isotropic and the anisotropic elliptic interface problems.

• A partially penalty immersed Crouzeix-Raviart finite element method for interface problems