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**An efficient diagonal preconditioner for finite element solution of Biot’s consolidation equations.**
*(English)*
Zbl 1076.74558

Summary: Finite element simulations of very large-scale soil-structure interaction problems (e.g. excavations, tunnelling, pile-rafts, etc.) typically involve the solution of a very large, ill-conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large-scale Biot’s finite element equations using the symmetric quasi-minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element-by-element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three-dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix.

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

76S05 | Flows in porous media; filtration; seepage |

74L10 | Soil and rock mechanics |

### Keywords:

Biot’s consolidation; preconditioner; generalized Jacobi; Schur complement; quasi-minimal residual method; element-by-element (EBE) iteration### Software:

KELLEY
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\textit{K. K. Phoon} et al., Int. J. Numer. Methods Eng. 55, No. 4, 377--400 (2002; Zbl 1076.74558)

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