Commun. Comput. Phys., 14 (2013), pp. 1322-1346. An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures Jian Du 1*, Robert D. Guy 2, Aaron L. Fogelson 3, Grady B. Wright 4, James P. Keener 31 Department of Mathematical Science, Florida Institute of Technology, Melbourne, FL 32901, USA. 2 Department of Mathematics, University of California Davis, Davis, CA 95616, USA. 3 Department of Mathematics and Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA. 4 Department of Mathematics, Boise State University, Boise, ID 83725, USA. Received 18 May 2012; Accepted (in revised version) 21 March 2013 Available online 5 July 2013 doi:10.4208/cicp.180512.210313a Abstract Many problems in biology involve gels which are mixtures composed of a polymer network permeated by a fluid solvent (water). The two-fluid model is a widely used approach to described gel mechanics, in which both network and solvent coexist at each point of space and their relative abundance is described by their volume fractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel and regions of pure solvent, resulting in a degenerate network momentum equation where the network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction of one phase goes to zero in part of the computational domain. A small and constant network volume fraction is temporarily added throughout the domain in setting up the discrete linear equations and the same set of equation is solved everywhere. These equations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficient and produces an accurate solution of these equations for the full range of relevant regularization parameter values. AMS subject classifications: 76T99, 76D07, 76D27, 65F22, 76M20, 65M12 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Gel swelling, preconditioner, two-fluid model. *Corresponding author. Email: jdu@fit.edu (J. Du), guy@math.ucdavis.edu (R. D. Guy), fogelson@math.utah.edu (A. L. Fogelson), wright@math.boisestate.edu (G. B. Wright), keener@math.utah.edu (J. P. Keener)