Commun. Comput. Phys., 15 (2014), pp. 1045-1067. Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver Wenqiang Feng 1, Xiaoming He 1*, Yanping Lin 2, Xu Zhang 31 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA. 2 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong; Department of Mathematical and Statistics Science, University of Alberta, Edmonton, AB, T6G 2G1, Canada. 3 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA. Received 15 March 2013; Accepted (in revised version) 17 October 2013 Available online 21 January 2014 doi:10.4208/cicp.150313.171013s Abstract This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L^2 and semi-H^1 norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location. AMS subject classifications: 65F10, 65N30, 35J15, 35K20 Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Interface problems, immersed finite elements, algebraic multigrid method. *Corresponding author. Email: fw253@mst.edu (W. Feng), hex@mst.edu (X. He), yanlin@ualberta.ca (Y. Lin), xuz@vt.edu (X. Zhang)