Computers & Geosciences 31 (2005) 1028–1041 An adaptive multigrid approach for the simulation of contaminant transport in the 3D subsurface Ming-Hsu Li a,Ã , Hwai-Ping Cheng b , Gour-Tsyh Yeh c a Institute of Hydrological Sciences, National Central University , #300 Jungda Rd., JungLi, Taiwan 320, ROC b Engineer Research & Development Center, Coastal & Hydraulics Laboratory, US Army Corps of Engineers, Vicksburg, MS 39180, USA c Department of Civil and Environmental Engineering, University of Central Florida, Orlando, FL 32816, USA Received 26 November 2004; received in revised form 14 March 2005; accepted 14 March 2005 Abstract This paper presents an adaptive multigrid approach, combining adaptive local grid refinement and multigrid methods, in conjunction with the Lagrangian–Eulerian finite element method to simulate contaminant transport in the 3D subsurface. Adaptive local grid refinement can improve solution accuracy without sacrificing computational efficiency because computer efforts are focused on the rough regions (i.e., requiring high spatial resolution) of the problem domain. To implement adaptive grids, a backward/forward particle tracking technique is applied in the Lagrangian step, and the interpolation errors of the Lagrangian concentrations are compared with prescribed error tolerances to determine rough regions. A modular setting of the grid generation is then used to generate locally zooming grids and to prepare information for applying multigrid methods. The Lagrangian concentrations of the newly generated nodes at the finest grid level are also evaluated by performing a backward tracking. Multigrid strategies which can effectively eliminate the smooth component error through coarse grid correction are finally applied in the Eulerian step to solve the matrix equations for further saving of computer time. Example problems are used to demonstrate the success of this integrated approach. r 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive local grid refinement; Multigrid method; Finite element method; Contaminant transport; Subsurface 1. Introduction Advection–dispersion equations describe the main mechanisms governing contaminant transport in the subsurface. Along with the development of computer technologies, numerical models have become the most effective tool to provide quantitative solutions for analyzing and predicting the migration of contaminants in the subsurface. However, a variety of numerical errors (e.g., numerical dispersion and oscillation) may be introduced due to the mixed hyperbolic and parabolic nature of advection–dispersion equations. To overcome these difficulties and to solve these equations accurately, several numerical techniques have been developed in conjunction with the mixed Lagrangian–Eulerian ap- proach in the past decade, such as LEZOOM (Yeh, 1990), ELLAM (Celia et al., 1990), EPCOF (Yeh et al., 1992), and LEZOOMPC (Cheng et al., 1998a). No matter what spatial discretization techniques are employed, one eventually must compute the solution of ARTICLE IN PRESS www.elsevier.com/locate/cageo 0098-3004/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2005.03.010 Ã Corresponding author. Tel.: +886 3 4227151 ext. 65691; fax: +886 3 4222964. E-mail addresses: mli@cc.ncu.edu.tw (M.-H. Li), Hwai-Ping.Cheng@erdc.usace.army.mil (H.-P. Cheng), gyeh@mail.ucf.edu (G.-T. Yeh).