RMCP: Relaxed Mixed Constraint Preconditioners for saddle point linear systems arising in geomechanics

Abstract

A major computational issue in the Finite Element (FE) integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system. Because of ill-conditioning, the iterative solution, which is recommended in large size 3D settings, requires the computation of a suitable preconditioner to guarantee convergence. In this paper the coupled system is solved by a Krylov subspace method preconditioned by a Relaxed Mixed Constraint Preconditioner (RMCP) which is a generalization based on a parameter ω of the Mixed Constraint Preconditioner (MCP) developed in [7]. Choice of optimal ω is driven by the spectral distribution of suitable symmetric positive definite (SPD) matrices. Numerical tests performed on realistic 3D problems reveal that RMCP accelerates Krylov subspace solvers by a factor up to three with respect to MCP.

Eigenvalue distribution of relaxed mixed constraint preconditioner for saddle point problems

“…ω A in detail and promote some corresponding presented results in [36,37,44]. The conclusions are drawn in Section 4.…”

“…Recently, drawing on the previous works: [3436], Bergamaschi and Martínez [37] discussed a family of relaxed mixed constraint preconditioner (RMCP) as follows:…”

“…where ω is a real acceleration parameter, PA is a suitable approximation of the (1,1) block A and PS is a suitable approximation of the Schur complement matrix S = BP −1 A B T + C. A detailed spectral analysis of RMCP was presented in [37]. In this paper, we focus on the relaxed mixed constraint preconditioner (RMCP) for symmetric saddle point problems (1.1).…”

An inexact relaxed DPSS preconditioner for saddle point problem

“…Therefore, many researchers devote themselves to the preconditioned iterative methods (1.1) (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). And kinds of preconditioners for saddle point matrix are studied, such as symmetric indefinite preconditioners [28,29], inexact constraint preconditioners [29][30][31][32][33][34] and primal-based penalty preconditioners [35]. In [36], Pan et al employed the positive-definite and skew-Hermitian splitting (PSS) technique proposed in [37] to derive a deteriorated PSS (DPSS) preconditioner for nonsymmetric saddle point problem, where A is positive definite but nonsymmetric.…”

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

“…Recently, the eigenvalue distribution of the nonsymmetric saddle point matrix has been deeply studied. On one hand, eigenvalue bounds for can be used to analyze the spectral properties of preconditioners such as symmetric indefinite preconditioners, inexact constraint preconditioners and primal‐based penalty preconditioners for ; see, for example, . On the other hand, eigenvalue estimates for can give theoretical basis for the CG method for in a nonstandard inner product; see, for example, .…”