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
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DOI：
10.1007/s00211-021-01256-x
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发表时间：
2021-05
影响因子：
2.1
通讯作者：
S. Chandler-Wilde;E. Spence
中科院分区：
文献类型：
--
作者：
S. Chandler-Wilde;E. Spence
众所周知，当有界Lipschitz域的边界为有界域时，经典的双层位势算子在自然迹空间上具有本质范数，且具有特定的范数。这意味着，对于位势理论中的内外Dirichlet和Neumann问题的标准第二类边界积分方程组，Galerkin方法对于任意有限维渐近稠密的子空间序列都是收敛的。长期悬而未决的问题是，本质范数是否对所有的Lipschitzin 2-d也作为算子；或者，对于所有的2-d和3-d的Lipschitzin，或者至少对于3-d中较小的Lipschitz多面体类，较弱的条件是否认为算子是强制算子的紧扰动--这是Galerkin方法对于每个渐近稠密的子空间序列收敛的充要条件。我们消极地解决了这些悬而未决的问题。我们给出了Lipschitz常数等于1的二维Lipschitz域和三维Lipschitz域的例子，以及Lipschitz常数为二的算符不是强制加紧的例子。我们还给出了本质范数为且不是强制算子对任何实数或复数的紧扰动的Lipschitz多面体的例子。然后，通过Hilbert空间中Galerkin方法的一个新结果，我们探讨了这些结果对Galerkin边界元方法在集合中的收敛的影响。最后，我们负地解决了配置法收敛理论中一个相关的公开问题，证明了对于我们的多面体例子，不存在与标准上确模等价的加权范数，对它的本质范数是Don.
It is well known that, with a particular choice of norm, the classical double-layer potential operatorDhas essential normas an operator on the natural trace spacewheneveris 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 infor any sequence of finite-dimensional subspacesthat is asymptotically dense in. Long-standing open questions are whether the essential norm is alsoforDas an operator onfor all Lipschitzin 2-d; or whether, for all Lipschitzin 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operatorsare compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspacesthat is asymptotically dense in. 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 ofDis, and examples with Lipschitz constant two for which the operatorsare not coercive plus compact. We also give, for every, examples of Lipschitz polyhedra for which the essential norm isand for whichis not a compact perturbation of a coercive operator for any real or complexwith. 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 thesetting. 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, equivalent to the standard supremum norm, for which the essential norm ofDonis.