Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot’s FEM equations

Generalized Jacobi (GJ) diagonal preconditioner coupled with symmetric quasi-minimal residual (SQMR) method has been demonstrated to be efficient for solving the 2 × 2 block linear system of equations arising from discretized Biot’s consolidation equations. However, one may further improve the performance by employing a more sophisticated non-diagonal preconditioner. This paper proposes to employ a block constrained preconditioner Pc that uses the same 2 × 2 block matrix but its (1, 1) block is replaced by a diagonal approximation. Numerical results on a series of 3-D footing problems show that the SQMR method preconditioned by Pc is about 55% more efficient time-wise than the counterpart preconditioned by GJ when the problem size increases to about 180,000 degrees of freedom. Over the range of problem sizes studied, the Pc-preconditioned SQMR method incurs about 20% more memory than the GJ-preconditioned counterpart. The paper also addresses crucial computational and storage issues in constructing and storing Pc efficiently to achieve superior performance over GJ on the commonly available PC platforms.