Commun. Comput. Phys., 15 (2014), pp. 451-469. A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant Jian-Jun Xu 1*, Yunqing Huang 1, Ming-Chih Lai 2, Zhilin Li 31 School of Mathematical and Computational Sciences, Xiangtan University, Hunan 410005, China; Hunan Key Lab for Computation and Simulation in Science and Engineering, Hunan 410005, China. 2 Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan. 3 Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA; School of Mathematical Sciences, Nanjing Normal University, Nanjing, China. Received 24 October 2012; Accepted (in revised version) 31 May 2013 Available online 10 September 2013 doi:10.4208/cicp.241012.310513a Abstract In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation. AMS subject classifications: 76D07, 65M06 Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: 3D immersed interface method, level-set method, insoluble surfactant, Stokes interfacial flow, drop deformation. *Corresponding author. Email: jjxu21@xtu.edu.cn (J.-J. Xu), huangyq@xtu.edu.cn (Y. Q. Huang), mclai@math.nctu.edu.tw (M.-C. Lai), zhilin@math.ncsu.edu (Z. Li)