INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. 2011; 35:1656–1681 Published online 6 November 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.971 Three-dimensional modeling of problems in poro-elasticity via a mixed least-squares method using linear tetrahedral elements Maria Tchonkova 1, ∗, † , John Peters 2 and Stein Sture 3 1 9009 Great Hills Trail, Suite 224, Austin, TX 78759, U.S.A. 2 US Army Engineer Research and Development Center, Vicksburg, MS 39190, U.S.A. 3 Department of Civil, Environmental and Architectural Engineering, Campus Box 428, University of Colorado at Boulder, Boulder, CO 80309, U.S.A. SUMMARY In a previous publication we developed a new mixed least-squares method for poro-elasticity. The approx- imate solution was obtained via a minimization of a least-squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involved four independent types of variables: displacements, stresses, pore pressures and velocities. All of them were approximated by linear continuous triangles. Encouraged by the computational results, obtained from the two-dimensional implementation of the method, we extended our formulation to three dimensions. In this paper we present numerical examples for the performance of continuous linear tetrahedra within the context of the mixed least-squares method. The initial results suggest that the method works well in the nearly and entirely incompressible limits for elasticity. For poro- elasticity, the obtained pore pressures are stable without exhibiting the oscillations, which are observed when the standard Galerkin formulation is used. Copyright  2010 John Wiley & Sons, Ltd. Received 18 November 2009; Revised 17 June 2010; Accepted 18 June 2010 KEY WORDS: poro-elsticity; Darcy flow; mixed finite elements 1. INTRODUCTION The mixed least-squares finite element method, presented in this paper, was originally developed for solving problems in linear elasticity [1, 2]. It involves separate approximations for displacements and stresses and results in a positive-definite coefficient matrix. The method was tested on classical two-dimensional problems in solid mechanics, with well-known exact analytical solutions. Two types of mixed finite elements were introduced in the initial formulation: piece-wise constant displacements-bilinear stresses and bilinear displacements-bilinear stresses. Both these elements exhibited the same rates of convergence for both: the primary function (displacements) and its derivative (stresses) in the compressible and the entirely incompressible limit. In a subsequent publication the mixed least-squares method was compared with the classical and some other least- squares formulations for elasticity [3]. Three major differences between the mixed least-squares method and the standard mixed finite element formulations were pointed out: first, it does not require compatibility between the approximation spaces for the primary variable and its derivative; second, it is a minimization, not a saddle point problem; and third, the resulting coefficient matrix ∗ Correspondence to: Maria Tchonkova, 9009 Great Hills Trail, Suite 224, Austin, TX 78759, U.S.A. † E-mail: maria.tchonkova@att.net Copyright  2010 John Wiley & Sons, Ltd.