Projects per year
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 
Bibliographical note
Funding Information:This paper is dedicated, on the occasion of his 85th birthday, to Wolfgang Wendland (Stuttgart) who has had a leading role in our PDE and BIE community for many years. In particular, we thank Wolfgang for his insightful survey paper [95 ], which prompted the current work, and for many enjoyable and illuminating discussions of secondkind integral equations on nonsmooth domains, dating back to the Joint IMA/SIAM Conference on the State of the Art in Numerical Analysis in 1986. The authors thank Johannes Elschner (WIAS, Berlin), Raffael Hagger and KarlMikael Perfekt (both University of Reading), and Eugene Shargorodsky (King’s College London) for a number of very useful discussions. EAS was supported by UK Engineering and Physical Sciences Research Council grant EP/R005591/1.
Fingerprint
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
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council