Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

Abstract

We consider operator preconditioning ${\mathscr{B}}^{-1}\mathcal {A}$, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators $\mathcal {A}, {\mathscr{B}}: {H_{0}^{1}}({\Omega }) \to H^{-1}({\Omega })$ are the standard integral/functional representations of the partial differential operators −∇⋅ (k(x)∇u) and −∇⋅ (g(x)∇u), respectively, and the scalar coefficient functions k(x) and g(x) are assumed to be continuous throughout the closure of the solution domain. The function g(x) is also assumed to be uniformly positive. When the discretized problem, with the preconditioned operator ${\mathscr{B}}_{n}^{-1}\mathcal {A}_{n}$, is solved with Krylov subspace methods, the convergence behavior depends on the distribution of the eigenvalues. Therefore, it is crucial to understand how the eigenvalues of ${\mathscr{B}}_{n}^{-1}\mathcal {A}_{n}$ are related to the spectrum of ${\mathscr{B}}^{-1}\mathcal {A}$. Following the path started in the two recent papers published in SIAM J. Numer. Anal. [57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211], the first part of this paper addresses the open question concerning the distribution of the eigenvalues of ${\mathscr{B}}_{n}^{-1}\mathcal {A}_{n}$ formulated at the end of the second paper. The approximation of the spectrum studied in the present paper differs from the eigenvalue problem studied in the classical PDE literature which addresses individual eigenvalues of compact (solution) operators.

In the second part of this paper, we generalize some of our results to general bounded and self-adjoint operators ${\mathcal A, B}: V\to V^{\#}$, where V^# denotes the dual of V. More specifically, provided that ${\mathscr{B}}$ is coercive and that the standard Galerkin discretization approximation properties hold, we prove that the whole spectrum of ${\mathscr{B}}^{-1}\mathcal {A}:V\to V$ is approximated to an arbitrary accuracy by the eigenvalues of its finite dimensional discretization ${\mathscr{B}}_{n}^{-1}\mathcal {A}_{n}$.

Appendix: : Approximations of the spectrum of self-adjoint operators

Using [25, Chapter 3] and [24, Chapter 8], we first recall several results concerning the convergence of linear self-adjoint operators defined on infinite dimensional Hilbert spaces. By the Hellinger-Toeplitz Theorem (see [30, Theorem 5.7.2, p. 260]), any linear self-adjoint operator $\mathcal {Z}: V \to V$ on a Hilbert space V is closed and, according to the Banach closed-graph theorem, bounded (continuous).

Consider a bounded linear operator $\mathcal {G}: V \to V$ (not necessarily self-adjoint, therefore we for the moment change the notation) and a sequence of its bounded linear approximations $\{\mathcal {G}_{n} \}, \mathcal {G}_{n}: V \to V$, that can converge to $\mathcal {G}$ in different ways:

• pointwise (strongly), i.e., $\mathcal {G}_{n} \stackrel {p}{\rightarrow } \mathcal {G}$

  $$ \text{iff for all} x \in V, \quad \lim_{n \rightarrow \infty} \| \mathcal{G} x - \mathcal{G}_{n} x\| = 0 ;$$

• uniformly (in norm), iff $\lim _{n \rightarrow \infty } \| \mathcal {G} - \mathcal {G}_{n} \| = 0 ;$

• stably, i.e. $\mathcal {G}_{n} \stackrel {s}{\rightarrow } \mathcal {G}$ iff

  • $\mathcal {G}_{n} \stackrel {p}{\rightarrow } \mathcal {G}$, and

  • the inverse operators $\{ \mathcal {G}_{n}^{-1} \}$ are uniformly bounded, i.e., for some C >0, $\|\mathcal {G}_{n}^{-1}\| \leq C$ for all n.

Clearly, uniform convergence implies pointwise convergence, but the converse implication does not hold. Since the class of compact operators is closed with respect to uniform convergence, the uniform convergence concept can not be used to investigate the convergence of compact to non-compact operators, such as to bounded continuously invertible operators defined on infinite dimensional Hilbert spaces.

The spectral theory for bounded linear operators is based on the concept of the operator resolvent

$$ \mathcal{R}(\mu) := (\mu \mathcal{I} - \mathcal{G})^{-1}$$

and on the resolvent set

$$ \rho(\mathcal{G}) := \left\{ \mu \in \mathbb{C}; \mu \mathcal{I} - \mathcal{G} \text{has a bounded inverse} \right\}. $$

(42)

It is interesting to notice that, for any $\mu \in \rho (\mathcal {G})$, a sequence of shifted operators $\{\mu \mathcal {I} - \mathcal {G}_{n}\}$ converge to $\mu \mathcal {I} - \mathcal {G}$ stably if, and only if, $\{\mathcal {G}_{n}\}$ and the resolvents $\{\mathcal {R}_{n}(\mu )\}, \mathcal {R}_{n}(\mu ):= (\mu \mathcal {I} - \mathcal {G}_{n})^{-1}$, converge to $\mathcal {G}$ and $\mathcal {R}(\mu )$ pointwise, respectively, i.e.,¹⁴

$$ \mu \mathcal{I} - \mathcal{G}_{n} \stackrel{s}{\rightarrow} \mu \mathcal{I} - \mathcal{G} \quad \text{iff} \quad \mathcal{G}_{n} \stackrel{p}{\rightarrow} \mathcal{G} \text{and} \mathcal{R}_{n}(\mu) \stackrel{p}{\rightarrow} \mathcal{R}(\mu). $$

(43)

Indeed, using the resolvent identity

$$ \mathcal{R}_{n}(\mu) - \mathcal{R}(\mu) = \mathcal{R}_{n}(\mu) (\mathcal{G}_{n} - \mathcal{G}) \mathcal{R}(\mu), $$

the right implication follows immediately from the definition of stable convergence. Conversely, from the pointwise convergence of $\mathcal {R}_{n}(\mu )$ and the uniform boundedness principle (Banach–Steinhaus theorem) we conclude that $\{\mathcal {R}_{n}(\mu )\}=\{ (\mu \mathcal {I} - \mathcal {G}_{n})^{-1} \}$ is uniformly bounded and the result follows.

We will now present a proof of Theorem 4.1; cf. also [24, chapter VIII, § 1.2, Theorem 1.14, p. 431] and [25, chapter 5, section 4, Theorem 5.12, p. 239-240].

Theorem A.1 (Approximation of the spectrum of self-adjoint operators)

Let $\mathcal {Z}$ be a linear self-adjoint operator on a Hilbert space V and let $\{\mathcal {Z}_{n}\}$ be a sequence of linear self-adjoint operators on V converging to $\mathcal {Z}$ pointwise (strongly). Then, for any point $\lambda \in \text {sp} (\mathcal {Z})$ in the spectrum of $\mathcal {Z}$, and for any of its neighbourhoods, the intersection of the spectrum $\text {sp}(\mathcal {Z}_{j})$ with this neighbourhood is, for j sufficiently large, nonempty.

Proof

Consider any point $\lambda \in \text {sp}(\mathcal {Z}) \subset \mathbb {R}$ in the spectrum of $\mathcal {Z}$. Then, for any ε >0, the point μ := λ + ιε, where ι is the complex unit, belongs to the resolvent set $\rho (\mathcal {Z})$ (because $\mathcal {Z}$ is self-adjoint). For self-adjoint operators, the norm of the resolvent at any point in the resolvent set is equal to the inverse of the distance of the given point to the spectrum (see, e.g., [25, Proposition 2.32]). Therefore, for all n,

$$ \begin{array}{@{}rcl@{}} \| \mathcal{R}(\mu) \| &:=&\|(\mu \mathcal{I} - \mathcal{Z})^{-1}\| = \frac{1}{\text{dist}(\mu, \text{sp}(\mathcal{Z}))}= \frac{1}{\varepsilon} , \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} \| \mathcal{R}_{n}(\mu) \| &:=&\|(\mu \mathcal{I} - \mathcal{Z}_{n})^{-1}\| = \frac{1}{\text{dist}(\mu, \text{sp}(\mathcal{Z}_{n}))}\leq \frac{1}{\varepsilon} . \end{array} $$

(45)

The inequality in (45) follows from the assumption that $\{\mathcal {Z}_{n}\}$ is a sequence of self-adjoint operators, i.e., $\text {sp}(\mathcal {Z}_{n}) \subset \mathbb {R}$. Note that inequality (45) provides the uniform boundedness of $\{ \mathcal {R}_{n}(\mu ) \}$, which, together with the pointwise convergence of $\{\mathcal {Z}_{n}\}$, yields that

$$ \mu \mathcal{I} - \mathcal{Z}_{n} \stackrel{s}{\rightarrow} \mu \mathcal{I} - \mathcal{Z}. $$

Using (43), we thus have the pointwise convergence of $\{\mathcal {R}_{n}(\mu ) \}$, i.e., for any x ∈ V,∥x∥ =1,

$$ \| \mathcal{R}(\mu) x \| = \lim_{n \rightarrow \infty} \| \mathcal{R}_{n}(\mu) x \|. $$

Consider a fixed x ∈ V,||x∥ =1, such that

$$ \frac{1}{2 \varepsilon} \leq \| \mathcal{R}(\mu) x \| \leq \frac{1}{\varepsilon}. $$

(The existence of such a x ∈ V follows from (44).) Then, from the pointwise convergence, there must exist n^∗ such that for all n ≥ n^∗

$$ \| \mathcal{R}_{n}(\mu)\| \geq \| \mathcal{R}_{n}(\mu) x \| \geq \frac{1}{3 \varepsilon}. $$

Recall that $\| \mathcal {R}_{n}(\mu ) \|=\mathrm {[dist}(\mu , \text {sp}(\mathcal {Z}_{n}))]^{-1}$. Therefore, there exists a point $\lambda _{n} \in \text {sp}(\mathcal {Z}_{n})$ such that

$$ |\lambda_{n} - \mu| \leq 3 \varepsilon \quad \text{and, consequently,} \quad |\lambda_{n} - \lambda| \leq 4 \varepsilon, $$

which finishes the proof. □