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

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, us...

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Bibliographic Details
Published in:Computational geosciences Vol. 14; no. 2; pp. 289 - 299
Main Authors: Wilson, John David, Naff, Richard L.
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.03.2010
Springer Nature B.V
Subjects:
Algorithms
Earth and Environmental Science
Earth Sciences
Finite element analysis
Fluid mechanics
Geotechnical Engineering & Applied Earth Sciences
Hydraulics
Hydrogeology
Mathematical Modeling and Industrial Mathematics
Original Paper
Soil Science & Conservation
Conjugate gradient
Mixed finite element method
Multigrid
Subsurface flow
Distorted grids
Nested iteration
Lowest-order Raviart–Thomas
ISSN:1420-0597, 1573-1499
Online Access:Get full text
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Summary: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.
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ISSN:1420-0597
1573-1499
DOI:10.1007/s10596-009-9152-z