Computational Geosciences

, Volume 14, Issue 2, pp 289–299

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

Open AccessOriginal paper

DOI: 10.1007/s10596-009-9152-z

Cite this article as:
Wilson, J.D. & Naff, R.L. Comput Geosci (2010) 14: 289. doi:10.1007/s10596-009-9152-z


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 methodLowest-order Raviart–ThomasSubsurface flowMultigridDistorted gridsConjugate gradientNested iteration
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© The Author(s) 2009

Authors and Affiliations

  1. 1.U.S. Geological SurveyDenver Federal CenterDenverUSA