Mixed finite elements and Newton‐type linearizations for the solution of Richards' equation

Abstract

SUMMARYWe present the development of a two-dimensional Mixed-Hybrid Finite Element (MHFE) model for the solution of the non-linear equation of variably saturated ow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest-order Raviart-Thomas (RT0) elements and is 'exactly' mass conserving. Hybridization is used to overcome the ill-conditioning of the mixed system. The scheme is globally ÿrst-order in space. Nevertheless, numerical results employing non-uniform meshes show second-order accuracy of the pressure head and normal uxes on speciÿc grid points. The non-linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE-Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques.

A study on iterative methods for solving Richards’ equation

“…The Newton method, which is quadratically convergent was very successfully applied to Richards' equation in e.g. [7,8,20,23,27]. The drawback of Newton's method is that it is only locally convergent and involves the computation of derivatives.…”

Mixed finite elements for solving 2-D diffusion-type equations

“…In addition, for steady state conditions, the system (15) is symmetric but not positive definite, which makes it difficult to handle with powerful iterative solvers based on conjugate gradients [Golub and Van Loan, 1989;Brezzi and Fortin, 1991;Bergamaschi and Putti, 1999].…”

“…[8] In the field of water resources, applicability and worth of MFEs were shown for a wide range of problems, including steady state [Arbogast et al, 1997;Chavent and Roberts, 1991;Durlofsky, 1994;James and Graham, 1999] and transient [Chavent and Roberts, 1991;Ackerer et al, 1999;Younes et al, 1999a] single-phase flow, flow in unsaturated media [Farthing et al, 2003], multiphase flow [Chen and Ewing, 1997;Dawson et al, 1998;Bergamaschi and Putti, 1999;Huber and Helmig, 1999;Nayagum et al, 2004;Hoteit and Firoozabadi, 2008], flow with heat transfer [Chounet et al, 1999;Holstad, 2001], multiphase flow in fractured media Firoozabadi, 2005, 2008], and numerical upscaling [Durlofsky, 1998;Ma et al, 2006]. The MFE method was also successfully used to obtain a locally mass conservative multiscale approach.…”

Locally conservative, stabilized finite element methods for variably saturated flow

“…This complication is avoided by finite element formulations that include an explicit velocity representation, such as mixed FEMs and local discontinuous Galerkin (LDG) methods [24,25]. While they have proven successful for many subsurface flow problems including Richards' equation [12,14,26], mixed FEMs are not without drawbacks. The resulting linear systems are saddle point problems without hybridization [24], and the number of unknowns is greater than a nodal CG approximation of the same order on the same mesh [25].…”