Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

08/14/2019
by   Martin Halla, et al.
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We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal. Optim. 17 (1996)]) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT 55(1) (2015)], is immediately applicable to analyze approximations of eigenvalue problems related to [A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Num. Math. 138(4) (2018)] and is already applied in [G. Unger, preprint (2017)].

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1. Weak T-coercivity and T-compatibility

Let be a Hilbert space with scalar body and scalar product and associated norm . Let be the space of bounded linear operators from to with operator norm for . For we denote its adjoint operator by , i.e.  for all . For a closed subspace let be the space of bounded linear operators from to with norm for and denote the orthogonal projection from to . Henceforth we assume that is a sequence of closed subspaces of such that converges point-wise to the identity, i.e.  for each .

Definition 1.1.

Let and be bijective. The operator is called

  1. coercive, if ,

  2. weakly coercive, if there exists a compact operator such that is coercive,

  3. -coercive if is coercive,

  4. weakly -coercive if is weakly coercive.

Due to the Lemma of Lax-Milgram every coercive operator is invertible. Every weakly -coercive operator is Fredholm with index zero. For a (weakly) coercive operator it is true that the Galerkin approximations inherit the (weak) coercivity, while for (weakly) -coercive operators it is in general wrong.

We note that if is weakly coercive, then is so too. Vice-versa, if is weakly coercive, then so is . Hence we could alternatively define to be (weakly) right -coercive, if is (weakly) coercive. However, we stick to the former variant because it is more convenient.

For an operator or or a sum of such we define the “discrete norm”

(3)
Definition 1.2.

Consider and . We say that converges to in discrete norm, if

(4)

We define in the following what we mean by -compatible approximations of weakly -coercive operators.

Definition 1.3.

Let be weakly -coercive. Then we call the sequence of Galerkin approximations -compatible, if is a sequence of index zero Fredholm operators and there exists a sequence of index zero Fredholm operators such that converges to in discrete norm: .

Definition 1.4.

A sequence is said to be compact, if for every subsequence exists in turn a converging subsubsequence.

Definition 1.5.

A sequence is called regular, if for every bounded sequence the compactness of already implies the compactness of .

Next we briefly elaborate on the notion of regularity for readers who are totally unfamiliar with this concept. Regularity of Galerkin approximations is a meaningful generalization of stability and well suited for the approximation analysis of eigenvalue problems. Consider for example bijective and its Galerkin approximation . In this case regularity of implies stability: Assume that is not stable. Thus there exists with for each such that . If is regular, there exists a subsequence and such that . It follows . Since is bijective, it follows which is a contradiction to .

On the other hand, consider a holomorphic Fredholm operator function with non-empty resolvent set and sequences of eigenvalues with normalized eigenelements of the Galerkin approximation (i.e. ) such that (see Section 2 for definitions and details). If is regular for each , then is indeed an eigenvalue of (i.e. there occurs no spectral pollution): Due to the continuity of with respect to , implies . If is regular, there exists a subsequence and such that . It follows and , i.e.  is an eigenvalue of with normalized eigenelement .

Our next goal is to prove in Theorem 1.8 that -compatible Galerkin approximations of weakly -coercive operators are regular. In preparation we formulate the next two lemmata.

Lemma 1.6.

Let and be a sequence of operators with and . Then there exist a constant and an index such that

(5)

for all . If is bijective and is Fredholm with index zero for each , then there exist a constant and an index such that is also bijective for all and

(6)
Proof.

Let . With the triangle inequality we deduce

and hence

Since the right hand side of the previous inequality is bounded. Similar, with the inverse triangle inequality we deduce

and hence

It hold and . Thus let be such that and for all . It follows

for all . For the last claim let be such that for all . Again with the inverse triangle inequality and

it follows

for all . We deduce that is injective. Since is Fredholm with index zero its bijectivity follows. The norm estimate holds due to . ∎

Lemma 1.7.

Let be weakly -coercive and be compact such that is coercive. Let be a -compatible Galerkin approximation of . Then there exist and , such that is invertible and

(7)

for all .

Proof.

Let be large enough such that is bijective (see Lemma 1.6). We compute

with coercivity constant

Since is uniformly bounded from above and below (see Lemma 1.6) and converges to in discrete norm by assumption, it follows the existence of and such that

for all . Hence is injective. Since is Fredholm with index zero and is compact, is Fredholm with index zero too. Thus is bijective. The norm estimate follows now from

for any bijective . ∎

Theorem 1.8.

Let be weakly -coercive and

be a -compatible Galerkin approximation. Then is regular.

Proof.

Without loss of generality let be a bounded sequence, and be such that . Let be compact such that is coercive. Let and . Since is compact and is bounded, there exist a subsequence and such that . It follows

Due to Lemma 1.7 there exist and , such that for all operator is invertible and . For we compute

The first term on the right hand side of the latter inequality converges to zero, as previously discussed. The second and third term converge to zero, because converges point-wise to the identity. Hence

2. Holomorphic eigenvalue problems

We refer the reader to [13] and [20, Appendix] for theory on holomorphic (Fredholm) operator functions. Let be an open, connected and non-empty subset of . Let be an operator function. An operator function is called holomorphic, if it is complex differentiable. An operator function is called Fredholm, if is Fredholm for each . We denote the resolvent set and spectrum of an operator function as

(8)

For an operator function we denote by the operator function defined by for each and by the operator function defined by for each . Note that for a holomorphic operator function the operator function defined by is holomorphic as well. For a holomorphic operator function denote by the derivative of . It is well known (see e.g. [12, Theorem 8.2]) that for a holomorphic Fredholm operator function such that is bijective for at least one , the spectrum is discrete, has no accumulation points in and every is an eigenvalue. That is, there exists such that . In this case we call an eigenelement. An ordered collection of elements in is called a Jordan chain at if is an eigenelement corresponding to and if

(9)

The elements of a Jordan chain are called generalized eigenelements and the closed linear hull of all generalized eigenelements of at is called the generalized eigenspace for at . For an eigenelement we denote by the maximal length of a Jordan chain at beginning with and

(10)

The maximal length of a Jordan chain is always finite, see e.g. [20, Lemma A.8.3]. Next we generalize Definitions 1.11.3, 1.5 and Theorem 1.8 to operator functions.

Definition 2.1.

Let be operator functions and . is (weakly) (-)coercive, if is (weakly) (-)coercive for each .

Definition 2.2.

Let be weakly -coercive. Then we call the sequence of Galerkin approximations -compatible, if is compatible for each .

Definition 2.3.

Let be an operator function. The sequence of Galerkin approximations is regular, if is regular for each

Theorem 2.4.

Let be weakly -coercive and

be a -compatible Galerkin approximation. Then is regular.

Proof.

Follows from Theorem 2.4. ∎

Next we prepare to apply [18][19].

Lemma 2.5.

Let be a holomorphic Fredholm operator function and let be a sequence of closed subspaces of with orthogonal projections onto , such that converges point-wise to the identity. Then the Galerkin scheme is a discrete approximation scheme in the sense of [18].

Proof.

For a Galerkin scheme it holds with the notation of [18]

Assumptions a1)-a4) of [18] follow all from the point-wise convergence of . ∎

Next we generalize Theorem 4.3.7 of [22].

Lemma 2.6.

Let be open, be a Hilbert space and be the space of bounded linear operators from to . Let be a holomorphic Fredholm operator function with non-empty resolvent set and be a sequence of closed subspaces of with orthogonal projections onto , such that converges point-wise to the identity, i.e.  for all . Let be the Galerkin approximation of defined by for each . Let the assumptions of [18, Theorem 2, Theorem 3] and [19, Theorem 2, Theorem 3] be satisfied. Let be a compact set with rectifiable boundary and . Then there exist and such that for all

(11)

for all and all with .

Proof.

We proceed as in [22]: Theorem 4.3.7 of [22] requires a special form of the operator function . However its proof uses this assumption only to apply Lemma 4.2.1 of [22]. Hence we need to establish the result of [22, Lemma 4.2.1] without the assumption on the special form of . However, the result of [22, Lemma 4.2.1] already follows from [18, Theorem 2 ii)].

Next we apply [18][19] and Lemma 2.6.

Proposition 2.7.

Let be open, connected and non-empty, be a Hilbert space and be the space of bounded linear operators from to itself. Let be a holomorphic Fredholm operator function with non-empty resolvent set . Let be a sequence of closed subspaces of with orthogonal projections onto , such that converges point-wise to the identity, i.e.  for each . Let be the Galerkin approximation of defined by for each . Assume that is Fredholm with index zero for each and . Assume that is a regular approximation of (see Definition 2.3). Then the following results hold.

  1. For every eigenvalue of exists a sequence converging to with being an eigenvalue of for almost all .

  2. Let be a sequence of normalized eigenpairs of , i.e.

    and , so that , then

    1. is an eigenvalue of ,

    2. is a compact sequence and its cluster points are normalized eigenelements of .

  3. For every compact the sequence is stable on , i.e. there exist and such that for all and all .

  4. For every compact with rectifiable boundary exists an index such that

    (12)

    for all , whereby denotes the generalized eigenspace of an operator function at .

Let be a compact set with rectifiable boundary , and

(13)

whereby denotes the complex conjugate of and the adjoint operator function of defined by for each . Then there exist and such that for all

  1. (14)

    for all , whereby denotes the maximal length of a Jordan chain of at the eigenvalue ,

  2. (15)

    whereby is the weighted mean of all the eigenvalues of in

    (16)
  3. (17)

    for all and all with .

Proof.

The first three claims follow with [18, Theorem 2], if we can proof that the required assumptions are satisfied. First of all a Galerkin scheme is a discrete approximation scheme due to Lemma 2.5. The operator function are holomorphic by assumption. It follows that is also holomorphic. and are index zero Fredholm operator functions by assumption. Assumption b1