A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

Abstract

Many applications in structural mechanics require the numerical solution of sequences of linear systems typically issued from a finite element discretization of the governing equations on fine meshes. The method of Lagrange multipliers is often used to take into account mechanical constraints. The resulting matrices then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. A popular strategy is then to combine preconditioning and deflation to yield an efficient method. We propose an alternative that is applicable to the general case and not only to matrices with a saddle point structure. In this approach, we consider to update an existing algebraic or application-based preconditioner, using specific available information exploiting the knowledge of an approximate invariant subspace or of matrix-vector products. The resulting preconditioner has the form of a limited memory quasi-Newton matrix and requires a small number of linearly independent vectors. Numerical experiments performed on three large-scale applications in elasticity highlight the relevance of the new approach. We show that the proposed method outperforms the deflation method when considering sequences of linear systems with varying matrices.

Appendix

We give in this section the two theorems cited in Sect. 4.1.

Theorem 1

Let $A \in \mathbb {R}^{N \times N}$ be a nonsingular matrix and assume that $S \in \mathbb {R}^{N \times k}$ is of full column rank k, with $k \le N$ and denote ${\mathscr {S}}={\mathcal {R}}(S)$. The preconditioner H given in Definition 1 is nonsingular if and only if

$$\begin{aligned} \mathbb {R}^{N} = (A {\mathscr {S}})^{\perp } \oplus {\mathscr {S}}. \end{aligned}$$

(22)

Proof

Using relation (7), the preconditioner given in Definition 1 can be written as

$$\begin{aligned} H = P_{(A{\mathscr {S}})^{\perp }} + S(S^TA^TAS)^{-1}S^TA^T. \end{aligned}$$

(23)

Hence, the preconditioned operator HA simply reads

$$\begin{aligned} HA = P_{(A{\mathscr {S}})^{\perp }}A + I_N-Q. \end{aligned}$$

(24)

Since $\mathcal{{R}}(P_{(A{\mathscr {S}})^{\perp }}A) = (A {\mathscr {S}})^{\perp }$, $\mathcal{{N}}(P_{(A{\mathscr {S}})^{\perp }}A) = {\mathscr {S}}$, $\mathcal{{N}}(I_N-Q) = (A^T A {\mathscr {S}})^{\perp }$ and $\mathcal{{R}}(I_N-Q) = {\mathscr {S}}$, necessary and sufficient conditions for HA to be invertible are given by

$$\begin{aligned} \mathbb {R}^{N} = (A {\mathscr {S}})^{\perp } \oplus {\mathscr {S}} \; \text{ and } \; \mathbb {R}^{N} = (A^T A {\mathscr {S}})^{\perp } \oplus {\mathscr {S}}. \end{aligned}$$

We note that the condition $\mathbb {R}^{N} = (A^T A {\mathscr {S}})^{\perp } \oplus {\mathscr {S}}$ holds since $A^TA$ is symmetric positive definite. This completes the proof, since A is supposed to be nonsingular. $\square$

Theorem 2

Let $A \in \mathbb {R}^{N \times N}$ be a nonsingular matrix and H be given by (15) in Definition 1. Assume that the columns of $W \in \mathbb {R}^{N \times k}$ and $W_{\perp } \in \mathbb {R}^{N \times (N-k)}$ form an orthonormal basis for $A{\mathscr {S}}$ and $(A{\mathscr {S}})^{\perp }$, respectively. The spectrum of the preconditioned operator AH is then given by

$$\begin{aligned} \varLambda (AH)=\{1\} \cup \varLambda (W_{\perp }^TAW_{\perp }). \end{aligned}$$

Proof

Using relations (7) and (23) leads to

$$\begin{aligned} AH=AP_{(A{\mathscr {S}})^{\perp }} + I_N-P_{(A{\mathscr {S}})^{\perp }}. \end{aligned}$$

(25)

Since the columns of $[W, W_{\perp }]$ form an orthonormal basis of $\mathbb {R}^{N}$, we have

$\varLambda (AH)=\varLambda ([W,W_{\perp }]^TAH[W,W_{\perp }])$ and

$$\begin{aligned}{}[W, W_{\perp }]^TAH[W,W_{\perp }] = \begin{pmatrix} W^T AH W &{} W^TAHW_{\perp } \\ W_{\perp }^TAHW &{} W_{\perp }^TAHW_{\perp } \end{pmatrix}. \end{aligned}$$

Using relation (25) and basic properties of the orthogonal projection $P_{(A{\mathscr {S}})^{\perp }}$, we finally obtain

$$\begin{aligned}{}[W, W_{\perp }]^TAH[W, W_{\perp }]=\begin{pmatrix} I_k &{} W^TAW_{\perp } \\ 0_{N-k,k} &{} W_{\perp }^TAW_{\perp } \end{pmatrix}, \end{aligned}$$

which completes the proof. $\square$