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Abstract
It is well known that, with a particular choice of norm, the classical doublelayer 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 secondkind 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 finitedimensional 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 2d; or whether, for all Lipschitz Γ in 2d and 3d, or at least for the smaller class of Lipschitz polyhedra in 3d, the weaker condition holds that the operators ±12I+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 2d and 3d 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 ±12I+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.
Original language  English 

Pages (fromto)  299–371 
Number of pages  73 
Journal  Numerische Mathematik 
Volume  150 
Issue number  2 
Early online date  24 Dec 2021 
DOIs  
Publication status  Published  28 Feb 2022 
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Dive into the research topics of 'Coercivity, essential norms, and the Galerkin method for secondkind integral equations on polyhedral and Lipschitz domains'. Together they form a unique fingerprint.Projects
 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council