**A very brief introduction to the immersed finite elements (**:

In many applications, we often
need to simulate a procedure in a domain consisting of several materials
separated from each other by curves or surfaces. This usually leads to the so
called interface problem which is a boundary value problem of a PDE whose
coefficients are discontinuous across the material interface.

The conventional finite
element methods can be used to solve interface problems, but they usually
require that the each element contains essentially one of the materials.
Geometrically, this means that each element has to be on one side of a material
interface.

The immersed finite
elements are developed such that each element can contain multiple materials so
that elements are allowed to sit on an interface. The basic features of the
immersed finite elements are:

A. Their meshes can be
independent of the interface location; hence, if preferred, structured meshes
can be used to solve a problem with non-trivial interface.

B. The basis functions on an
interface element are formed according the interface jump conditions.

**Some Publications on **:
(links to PDE files will be given soon)

n
R.E. Ewing, Z. Li, T. Lin, Y. Lin, The immersed finite
volume element method for the elliptic interface problems, *Mathematics and Computers in Simulation*, 50(1999), 63-76.

n
T. Lin, Y. Lin, R.C. Rogers, and L.M. Ryan, A rectangular
immersed finite element method for interface problems, *Advances in
Computation: Theory and Practice, *Vol. 7, 2001, pp 107-114.

n
Z. Li, T. Lin, and X. Wu, New Cartesian Grid Methods for Interface
Problems Using Finite Element Formulation, *Numerische
Mathematik*, 96(2003), 61-93.

n
Z. Li, T. Lin, Y. Lin, R. Rogers, An Immersed Finite Element
Space and Its Approximation Capability, *Numerical
Methods for Partial Differential Equations, *20(2004), pp. 338-367.

n
R. Kafafy, T. Lin, Y. Lin, and J. Wang, 3-D immersed finite
element methods for electric field simulation in composite materials, *International
Journal for Numerical Methods in Engineering, *2005, 64:940-972.

n
R. Kafay, T. Lin, and J. Wang, A hybrid-grid
immersed-finite-element particle-in-cell simulation model of ion optics plasma
dynamics, *Dynamics of Continuous,
Discrete and Impulsive Systems, Series B-Applications & Algorithms*, 12:
103-118, Suppl. S. 2005.

n
Camp, T. Lin, Y. Lin, W-W. Sun, Quadratic Immersed Finite
Element Spaces and Their Approximation Capabilities, *Advances in Computational, *(2006)24: 81-112.

n
Adjerid, S. and Lin, T., Higher-order immersed discontinuous
Galerkin methods,* International Journal of
Information & Systems Sciences, *3(2007), 555-569.

n
Lin, T., Lin, Y., and Sun, W.-W., Error Estimation of a
Class of Quadratic Immersed Finite Element Methods for Elliptic Interface
Problems, *Discrete and Continuous
Dynamical System*, Series B, 8 (2007),, 757-780.* *

n
He, X., Lin, T., and Lin, Y., Approximation capability of a
bilinear immersed finite element space, *Numerical
Methods for Partial Differential Equations, *24(2008), 1265-1300.

n
He, X., Lin, T., and Lin, Y., A bilinear immersed finite
volume element method for the diffusion equation with discontinuous
coefficient, *Communications in
Computational Physics (to appear).*

n
Adjerid, S. and Lin, T., p-th degree immersed finite element
for boundary value problems with discontinuous coefficients, *Journal of Applied Numerical Mathematics (to
appear).*

**Some **:

** 1D **:

(a). Plots for comparing
the quadratic

The plots on the left column are for the

and the plots on the right column are for the standard FE basis functions.

The interface is located at -3/4 for these

The plots on the left column are for the

and the plots on the right column are for the standard FE basis functions.

uiopen('ife_2D_linear_basis_global.fig',1)

uiopen('fe_2D_linear_basis_global.fig',2)

These Matlab commands
will open two figure windows to display the plots for the global basis
functions. Then the plots can be rotated for more details.

The plots in the 1st column are for the Type I IFE basis functions,

the plots in the 2^{nd} column are for the Type II IFE basis
functions,

and the plots in 3^{rd}
column are for the standard FE basis functions.

uiopen('bilinear_ife_global_basis.fig',1)

uiopen('bilinear_fe_global_basis.fig’,2)

These Matlab commands
will open two figure windows to display the plots for the global basis
functions. Then the plots can be rotated for more details.

**Miscellaneous links
**