# 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:

n  Guo, R., Lin, T., and Lin, Y., Error Estimates For A Partially Penalized Immersed Finite Element, ESAIM Mathematical Modelling and Numerical Analysis, DOI: 10.1051/m2an/2019051.

n  Guo, R., Lin, T., A higher degree immersed finite element method based on a Cauchy extension for elliptic interface problems, SIAM Journal on Numerical Analysis, (2019) 57, No. 4, 1545-1573.

n  Adjerid, S., Chaabane, N., Lin, T., and Yue, P., An immersed discontinuous finite element method for the Stokes problem with a moving interface, Journal of Computational and Applied Mathematics, (2019) 362, 540-559.

n  Lin, T., Lin, Y., and Zhuang, Q., Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods, Communications on Applied Mathematics and Computation (2019). https://doi.org/10.1007/s42967-019-0002-2

n  Guo, R., Lin, T., A group of immersed finite-element spaces for elliptic interface problems, IMA Journal of Numerical Analysis (2019) 39, 482?11.

n  Guo, R., Lin, T., Lin, Y., Approximation capabilities of immersed finite element spaces for elasticity Interface problems, Numerical Methods for Partial Differential Equations, 35(2019), 1243-1268.

n  Guo, R., Lin, T., Lin, Y., A Fixed Mesh Method with Immersed Finite Elements for Solving Interface Inverse Problems, Journal of Scientific Computing, 79(2019), 148?75.

n  Adjerid, S., Ben-Romdhane, M., Lin, T., Higher degree immersed finite element spaces constructed according to the actual interface, Computers and Mathematics with Applications, 75 (2018) 1868?881.

n  Guo, R., Lin, T., Zhang, X., Nonconforming immersed finite element spaces for elliptic interface problems, Computers and Mathematics with Applications, 75(2018), 2002-2016.

n  Adjerid, S., Guo, R., Lin, T., High Degree Immersed Finite Element Spaces By A Least Squares Method, International Journal Of Numerical Analysis And Modeling, 14(2017), 604-626.

n  Lin, M., Lin, T., Zhang, H., Error Analysis Of An Immersed Finite Element Method For Euler-Bernoulli Beam Interface Problems, International Journal Of Numerical Analysis And Modeling, 14(2017), 822-841.

n  Han, D., Wang, P., He, X., Lin, T., Wang, J., A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma–lunar surface interactions, Journal of Computational Physics, 321(2016), 965-980.

n  Adjerid, S., Chaabane, N., Lin, T., An immersed discontinuous finite element method for Stokes interface problems, Computer Methods in Applied Mechanics and Engineering, 293(2015) 170?90.

n  Lin, T., Lin, Y., Zhang, X., Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems, SIAM Journal on Numerical Analysis, 53(2015), 1121-1144.

n  Adjerid, S., Ben-Romdhane, M., Lin, T., Higher Degree Immersed Finite Element Methods For Second-Order Elliptic Interface Problems, International Journal Of Numerical Analysis And Modeling, 11(2014), 541-566.

n  He, X., Lin, T., Lin, Y., A selective immersed discontinuous Galerkin method for elliptic interface problems, Mathematical Methods in the Applied Sciences, 37(2014), 983?002.

n  He, X., Lin, T., Lin, Y., Zhang, X., Immersed Finite Element Methods for Parabolic Equations with Moving Interface, Numerical Methods for Partial Differential Equations, 29(2013), 619-646.

n  Lin, T., Lin, Y., Zhang, X., Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump, Contemporary Mathematics, 586(2013), 257-265.

n  He, X., Lin, T., Lin, Y., Immersed Finite Element Methods For Elliptic Interface Problems With Non-Homogeneous Jump Conditions, Journal Of Numerical Analysis And Modeling, 8(2011), 284-301.

n  Adjerid, S. and Lin, T., A p-th degree immersed finite element for boundary value problems with discontinuous coefficients, Journal of Applied Numerical Mathematics, 59(2009), 1303-1321.

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, 6(2009), 185-202.

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

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

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

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

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

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.

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

# (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.

# 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.

# (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.  # 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)

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