Zhilin Li
CRSC & Department of Mathematics, North Carolina State University, USA; & School of Mathematical Sciences, XinJiang University, China
Keywords: Interface problem, augmented IIM, fast solvers, ADI method, discontinuous coefficient, multi-scale, jump conditions.

The immersed interface method (IIM) ?rst proposed in is an accurate numerical method for solving elliptic interface problems on Cartesian meshes. It is a sharp interface method that was intended to improve accuracy of the immersed boundary (IB) method. The IIM is second order accurate in the maximum norm (pointwise, strongest) while the IB method is ?rst order accurate. The ?rst IIM paper is one of the most downloaded one from the SIAM website and is one of the most cited papers. While IIM provided a way of accurate discretization of the partial differential equations (PDEs) with discontinuous coefficients, the augmented IIM ?rst proposed in made the IIM much more efficient and faster by utilizing existing fast Poisson solvers. More important is that the augmented IIM provides an efficient way for multi-physics models with different governing equations, problems on irregular domains, multi-scales and multi-connected domains. A brie?y introduction of the augmented strategy including some recently progress is presented in this article.

Article Information

Identifiers and Pagination:
First Page:1
Last Page:6
Publisher Id:Adv Calc Anal (2018 ). 3. 1-6
Article History:
Received:April 14, 2018
Accepted:July 19, 2018
Collection year:2018
First Published:August 17, 2018

1. Introduction

Interface problems arise in many applications. For example, when there are two different materials, such as a ?uid ?ow and a porous media, or the same material but at different states, such as water and ice, we are dealing with an interface problem. If ordinary or partial differential equations are used to model multi-phase ?ows, multi-physics problems, the parameters in the governing differential equations are typically discontinuous, or even the governing PDEs are different as in the Stokes and Darcy’s coupling, and the source terms are often singular to re?ect source/sink distributions along co-dimensional interfaces. Because of these irregularities, the solutions to the differential equations are typically non-smooth, or even discontinuous as in the example of the pressure inside and outside an in?ated balloon. As a result, many standard numerical methods based on the assumption of smoothness of solutions do not work or work poorly for interface problems.

With some initial and boundary conditions, where  is an interface, see Figure 1 for an illustration. We assume that the coefficient ß   has a ?nite jump across the interface  corresponding to two different materials or states and with a source term along the interface as in Peskin’s Immersed Boundary model (Peskin & McQueen,1995). It is known that the IB method is a smoothing method and has shown to be ?rst order accurate by this author and other scholars, see (Li, 2015; Mori, 2018).

Where the second jump is the flux jump and n is the normal direction with |n|=1 pointing to the ?+ side.

For a regular heat equation, the alternating direction implicit (ADI) method using a ?nite difference (FD) discretization is very efficient for time-dependent parabolic PDEs, see for example (Li, Qiao, & Tang, 2017). The classical ADI method is second-order accurate and unconditionally stable if the standard 3-point central ?nite difference discretization is used in each coordinate direction. An ADI method only requires to solve a sequence of tridiagonal systems of equations which can be done in O(N2) operations, and is considered as one of the most efficient algorithms.

Figure 2: Snap contour plots of the computed solution with N = 160 of a ?ow through a porous media.


In Figure 2, we show plots of the streamlines of the ?ow from left passes different objects with different permeability. For particles with large permeability, the ?ow gets saturated while the ?ow goes around if the permeability is very small.

2. A brief review of augmented IIM

There are at least two motivations for using augmented strategies. The ?rst one is to get a faster algorithm compared to a direct discretization, particularly to take advantage of existing fast solvers. The second one (more important one) is that, for some interface problems, an augmented approach may be the only way to derive an accurate algorithm. The augmented techniques enable us to decouple the jump conditions so that the idea of the immersed interface method can be applied, see for example, (Li, 2015; Li, Ito, & Lai, 2007; Li & Qin, 2017).


The application of the augmented approach for two dimensional interior irregular domains can be found in (Li, Zhao, & Gao,1999) for simulating electro-migration of voids in an integrated circuit; in (Hunter, Li, & Zhao, 2002 ) for simulating the motion of drops; in (Li & Wang, 2003) for solving Navier-Stokes equations on irregular domains using the stream- vorticity formulation; in (Jin & Wang, 2002) for micro-structure evolution in the chemical vapor in?ltration process, and (Chen, Li, & Lin, 2008)  for solving bi-harmonic equations on irregular domains; and recently work for bi-harmonic interface problems (Li & Qin, 2017). The augmented approach for generalized Helmholtz equations has also been developed in three dimensions and has been applied to an inverse problem of shape optimization in (Deng, Ito, & Li, 2003; Ito, Kunisch, & Li, 2001). While augmented methods have some similarities to boundary integral methods or integral equation approaches to ?nd a source strength, the augmented methods have a few special features: (1) no Green function is needed, and therefore there is no need to evaluate singular integrals; (2) there is no need to set up the system of equations for the augmented variable explicitly; (3) they are applicable to general PDEs with or without source terms; (4) the process does not depend on the boundary conditions. On the other hand, we may have less knowledge about the condition number of the Schur complement system and how to apply pre-conditioning techniques.

Figure 3: Simulations with the augmented IIM. (a): An interface with corners in a cavity ?ow, see (Li, Lai, Peng, & Zhang, 2018). (b): An irregular domain problem: a ?ow passes through a stationary cylinder computed with Re = 200, see (Ito, Lai, & Li, 2009).


Recent a remarkable progress on augmented IIM is the convergence proof and for variable coefficients. It has been shown in (Li, Ji, & Chen, 2016) that not only the computed solution is second order accurate globally in the strongest maximum norm, but also the computed gradient from each side of the interface. The method is designed for closed smooth interfaces not for open-ended interface such as cracks. For closed interfaces but with corners, see the left plot in Figure 3. The convergence of the augmented method for elliptic interface problems has been shown both in one and two dimensions under appropriate regularity assumptions and a piecewise constant ß(x). For a variable coefficient ß(x), the conclusions are still true if h is small enough, that is, in the asymptotic sense.

New applications of the augmented IIM include multi-physics simulations with different governing equations on different regions such as the Stokes or Navier-Stokes equations coupled with the Darcy’s law in (Li, 2015; Li, Lai, Peng, & Zhang, 2018), determine the crank tips for a Mumford-Shah minimizer of a free boundary value problem in (Li & Mikayelyan, 2016); the 3D compressible bubbles in a incompressible ?uid (Li, Xiao, Cai, Zhao, & Luo, 2015); and an efficient preconditioner for the Schur complement matrix resulted from AIIM in (Angot & Li, 2017; Xia, Li, & Ye, 2015).  


Angot, P., & Li, Z. L. (2017). An augmented IIM & preconditioning technique for jump embedded boundary conditions. International Journal of Numerical Analysis and Modeling, 14(4-5), 712-729.

Chen, G., Li, Z. L., & Lin, P. (2008). A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow. Advances in Computational Mathematics, 29(2), 113-133.

Deng, S. Z., Ito, K., & Li, Z. L. (2003). Three-dimensional elliptic solvers for interface problems and applications. Journal of Computational Physics, 184(1), 215-243.

Hunter, J. K., Li, Z. L., & Zhao, H. K. (2002). Reactive autophobic spreading of drops. Journal of Computational Physics, 183(2), 335-366.

Ito, K., Kunisch, K., & Li, Z. L. (2001). Level-set function approach to an inverse interface problem. Inverse Problems, 17(5), 1225-1242.

Ito, K., Lai, M. C., & Li, Z. (2009). A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains. Journal of Computational Physics, 228(7), 2616-2628.

Jin, S. & Wang, X. (2002), Robust numerical simulation of porosity evolution in chemical vapor in?ltration, Journal of Computational Physics, 179, 557–577.

Li, Z. L. (2016). An augmented Cartesian grid method for Stokes-Darcy fluid-structure interactions. International Journal for Numerical Methods in Engineering, 106(7), 556-575.  

Li, Z. L. (2015). On convergence of the immersed boundary method for elliptic interface problems, Math. Comp. 84 (293), 1169–1188.

Z. Li, K. Ito, and M-C. Lai, An augmented approach for Stokes equations with a discontinuous viscosity and singular forces, Computers and Fluids, 36 (2007), 622–635.

Li, Z. L., Ji, H. F., & Chen, X. H. (2017). Accurate solution and gradient computation for elliptic interface problems with variable coefficients. SIAM Journal on Numerical Analysis, 55(2), 570-597.

Li, Z. L., Lai, M. C., Peng, X. F., & Zhang, Z. Y. (2018). A least squares augmented immersed interface method for solving Navier-Stokes and Darcy coupling equations. Computers & Fluids, 167, 384-399.

Li, Z. L. & A. Mayo (1993), ADI methods for heat equations with discontinuities along an arbitrary interface, AMS Proc. Symp. Appl. Math. W. Gautschi, editor, 48, 311–315.

Li, Z. L., & Mikayelyan, H. (2016). Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer. Interfaces and Free Boundaries, 18(1), 75-90.

Li, Z., Qiao, Z., & Tang, T. (2018). Numerical solution of differential equations: Introduction to finite difference and finite element methods. New York: Cambridge University Press.

Li, Z. L., & Qin, F. F. (2017). An augmented method for 4th order PDEs with discontinuous coefficients. Journal of Scientific Computing, 73(2-3), 968-979.

Li, Z. L., & Wang, C. (2003), A fast ?nite difference method for solving Navier-Stokes equations on irregular domains, J. of Commu. in Math. Sci., 1, 180–196.

Li, Z. L., Xiao, L., Cai, Q., Zhao, H. K., & Luo, R. (2015). A semi-implicit augmented IIM for Navier-Stokes equations with open, traction, or free boundary conditions. Journal of Computational Physics, 297, 182-193.

Li, Z. L., Zhao, H. K., & Gao, H. J. (1999). A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid. Journal of Computational Physics, 152(1), 281-304.

Mori, Y. (2008) Convergence proof of the velocity ?eld for a Stokes ?ow immersed boundary method, Comm. Pure Appl. Math. 61, 1213-1263.

Peskin, C.  & McQueen, D. M. (1995), A general method for the computer simulation of biological systems interacting with ?uids, Symposia of the Society for Experimental Biology 49, 265.

Xia, J. L., Li, Z. L., & Ye, X. (2015). Effective matrix-free preconditioning for the augmented immersed interface method. Journal of Computational Physics, 303, 295-312. 

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits
Editor in Chief
Zhilin Li (Ph.D. (Applied Mathematics))
Professor of Mathematics, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 USA


Prof. Dr. Zhilin Li is associated with Department of Mathematics, North Carolina State University, US since 1997. He was graduated from Nanking Normal University (Mathematics B.S.) in 1982. Whereas, he has completed his Master in Mathematics from University of Washington Applied in 1991. The University of Washington awarded him PhD in Applied Mathematics in 1994. He has published 127 quality manuscript since October 31, 2016. Moreover, he received the Research Grants Current and past from NSF, NIH, NSF/NIGMS, ARO, AFOSR, Oak Ridge, DOE/ARO etc. He is collaborated and affiliated with Juan Alvares, UAH, Spain; Philippe Angot, Aix-Marceille University, France; J. Chen & H. Ji, NNU, China; K Ito, SR Lubkin, NCSU; M-C Lai, Taiwan; R. Luo, UCI; J. Xia, Purdue; Hayk Mikayelyan, Nottingham. Moreover, he has advised and supervised more than 15 graduated students to complete their research projects.

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Abbreviation: Adv Calc Anal
Current Volume: 2 (2017)
Next volume: December, 2018 (Volume 3)
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