Original paper

Computational Geosciences

, Volume 14, Issue 2, pp 289-299

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Multigrid preconditioned conjugate-gradient solver for mixed finite-element method

  • John David WilsonAffiliated withU.S. Geological Survey, Denver Federal Center Email author 
  • , Richard L. NaffAffiliated withU.S. Geological Survey, Denver Federal Center


The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor.


Mixed finite element method Lowest-order Raviart–Thomas Subsurface flow Multigrid Distorted grids Conjugate gradient Nested iteration