Tao Lin's Research Page

 

My research interests involve numerical methods for solving partial differential equations and integro-differential equations. My recent focus is the immersed finite element methods for interface problems and their applications.


A very brief introduction to the immersed finite elements (IFE): 

 

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 IFE methods: (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 IFE spaces and sample Matlab codes for 2nd order elliptic interface problems:

 

1D IFE p-th degree IFE space:

 

(a). Plots for comparing the quadratic IFE basis and the standard quadratic FE basis:

 

The plots on the left column are for the IFE basis functions,

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

The interface is located at -3/4 for these IFE basis functions

 

(b). Matlab code for the p-th (p <= 5) degree Lagrange type IFE basis: (coming soon)

2D linear IFE space based on a triangular mesh:

(a). Plots for comparing the 2D linear IFE basis functions and the linear FE basis functions on a triangular element: 

The plots on the left column are for the IFE basis functions,

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

(b). A linear IFE global basis function and a linear FE global basis function:

A linear IFE global basis function and FE global basis function on a structured mesh. The interface is a circle.

(c). Matlab can be used to see more details of the global IFE basis above and the corresponding global FE function as follows:

n       Download the fig file for the linear IFE basis

n       Download the fig file for the linear FE basis

n       Load these fig files into Matlab as follows:

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.

(d). Matlab code for the 2D linear IFE basis functions on a triangular element (coming soon)

2D bilinear IFE space based on a rectangular mesh:

(a). Plots for comparing the 2D bilinear IFE basis functions and bilinear FE basis functions on a rectangular element: 

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

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

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

 

 

(b). A bilinear IFE global basis function and a bilinear FE global basis function:

A bilinear IFE global basis function and FE global basis function on a rectangular mesh. The interface is a circle.

(c). Matlab can be used to see more details of the global IFE basis above and the corresponding global FE function as follows:

n       Download the fig file for the bilinear IFE basis

n       Download the fig file for the bilinear FE basis

n       Load these fig files into Matlab as follows:

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.

(d). Matlab code for the 2D bilinear IFE basis functions (coming soon)

 


Some FE links:


Links for inverse problems:


Research links from our department


Miscellaneous links