Multigrid preconditioned conjugate-gradient solver for mixed finite-element method |
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| Authors: | John David Wilson Richard L Naff |
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| Institution: | 1.U.S. Geological Survey,Denver Federal Center,Denver,USA |
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| Abstract: | 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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