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 (
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
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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.
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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.
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Z. Li, T. Lin, and X. Wu, New Cartesian Grid Methods for Interface
Problems Using Finite Element Formulation, Numerische
Mathematik, 96(2003), 61-93.
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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.
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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.
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Camp, T. Lin, Y. Lin, W-W. Sun, Quadratic Immersed Finite
Element Spaces and Their Approximation Capabilities, Advances in Computational, (2006)24: 81-112.
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Adjerid, S. and Lin, T., Higher-order immersed discontinuous
Galerkin methods, International Journal of
Information & Systems Sciences, 3(2007), 555-569.
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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.
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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.
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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).
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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
(b). Matlab code for the p-th (p <= 5) degree
Lagrange type IFE
2D linear IFE
(a). Plots for comparing the 2D linear IFE
The plots on the left column are for the
and the plots on the right column are for the standard FE basis functions.
(b). A linear IFE
A linear IFE
(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
2D bilinear IFE
(a). Plots for comparing the 2D bilinear IFE
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
A bilinear IFE
(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