Some examples concerning applicability of the Fredholm-Radon method in potential theory

Král, Josef; Wendland, Wolfgang L.

doi:10.21136/am.1986.104208

Cited by 22 publications

(9 citation statements)

References 3 publications

(14 reference statements)

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“…The final long-standing open question that we address, flagged by Wendland [95, §4.1] (and see [3,49,59,60]), is concerned specifically with the case when is Lipschitz polyhedral, in which case D is well-defined also as a bounded operator on C( ). To explain this conjecture we note the following general relationship, in a Banach space X equipped with a norm • X , between the essential spectral radius r X ,ess (A) of a bounded linear operator A and its essential norm A X ,ess .…”

Section: The Open Questions We Addressmentioning

confidence: 99%

“…on C( ), equivalent to the standard maximum norm, for which the induced essential norm of D is also < 1/2. Motivated by the numerical analysis of collocation methods for (1.5) in the case D * = ±D, Král and Wendland [59] consider, specifically, weighted norms equivalent to the standard maximum norm. Given w ∈ L ∞ ( ) with, for some c − > 0, ) In [59] they construct examples of Lipschitz (and non-Lipschitz) polyhedral and w for which…”

Section: The Open Questions We Addressmentioning

confidence: 99%

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Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Chandler-Wilde

Spence

2021

Numer. Math.

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It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm $$<1/2$$ < 1 / 2 as an operator on the natural trace space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) whenever $$\Gamma $$ Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) for any sequence of finite-dimensional subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . Long-standing open questions are whether the essential norm is also $$<1/2$$ < 1 / 2 for D as an operator on $$L^2(\Gamma )$$ L 2 ( Γ ) for all Lipschitz $$\Gamma $$ Γ in 2-d; or whether, for all Lipschitz $$\Gamma $$ Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $$\pm \frac{1}{2}I+D$$ ± 1 2 I + D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$L^2(\Gamma )$$ L 2 ( Γ ) . We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is $$\ge 1/2$$ ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators $$\pm \frac{1}{2}I +D$$ ± 1 2 I + D are not coercive plus compact. We also give, for every $$C>0$$ C > 0 , examples of Lipschitz polyhedra for which the essential norm is $$\ge C$$ ≥ C and for which $$\lambda I+D$$ λ I + D is not a compact perturbation of a coercive operator for any real or complex $$\lambda $$ λ with $$|\lambda |\le C$$ | λ | ≤ C . We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the $$L^2(\Gamma )$$ L 2 ( Γ ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on $$C(\Gamma )$$ C ( Γ ) , equivalent to the standard supremum norm, for which the essential norm of D on $$C(\Gamma )$$ C ( Γ ) is $$<1/2$$ < 1 / 2 .

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Section: The Open Questions We Addressmentioning

confidence: 99%

Section: The Open Questions We Addressmentioning

confidence: 99%

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Chandler-Wilde

Spence

2021

Numer. Math.

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“…The final long-standing open question that we address, flagged by Wendland [93, §4.1] (and see [57,3,58,48]), is concerned specifically with the case when Γ is Lipschitz polyhedral, in which case D is well-defined also as a bounded operator on C(Γ). To explain this conjecture we note the following general relationship, in a Banach space X equipped with a norm • X , between the essential spectral radius r X,ess (A) of a bounded linear operator A and its essential norm A X,ess .…”

Section: The Open Questions We Addressmentioning

confidence: 99%

“…on C(Γ), equivalent to the standard maximum norm, for which the induced essential norm of D is also < 1/2. Motivated by the numerical analysis of collocation methods for (1.5) in the case D * = ±D, Král and Wendland [57] consider, specifically, weighted norms equivalent to the standard maximum norm. Given w ∈ L ∞ (Γ) with, for some c − > 0, (1.12)…”

Section: The Open Questions We Addressmentioning

confidence: 99%

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Chandler-Wilde¹,

Spence²

2021

Preprint

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It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm < 1/2 as an operator on the natural trace space H 1/2 (Γ) whenever Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H 1/2 (Γ) for any sequence of finite-dimensional subspaces (HN ) ∞ N =1 that is asymptotically dense in H 1/2 (Γ). Longstanding open questions are whether the essential norm is also < 1/2 for D as an operator on L 2 (Γ) for all Lipschitz Γ in 2-d; or whether, for all Lipschitz Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators ± 1 2 I + D are compact perturbations of coercive operators -this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces (HN ) ∞ N =1 that is asymptotically dense in L 2 (Γ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ≥ 1/2, and examples with Lipschitz constant two for which the operators ± 1 2 I + D are not coercive plus compact. We also give, for every C > 0, examples of Lipschitz polyhedra for which the essential norm is ≥ C and for which λI + D is not a compact perturbation of a coercive operator for any real or complex λ with |λ| ≤ C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L 2 (Γ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Γ), equivalent to the standard supremum norm, for which the essential norm of D on C(Γ) is < 1/2.

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“…y 6 B by (cf. (11)). We see that q e Lf° so that, denoting by p* the cr-essential supremum of the family Lf°, we get q ^ p* cr-a.e.…”

Section: Propositionmentioning

confidence: 99%

Essential norms of a potential theoretic boundary integral operator in $L\sp 1$

Král¹,

Medková²

1998

Math. Bohem.

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Some examples concerning applicability of the Fredholm-Radon method in potential theory

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Cited by 22 publications

References 3 publications

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Essential norms of a potential theoretic boundary integral operator in $L\sp 1$

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