Tractability of the Helmholtz equation with non-homogeneous Neumann boundary conditions: the relation to the \(L_{2}\)-approximation
type: Item
instance of: scholarly article
title: Tractability of the Helmholtz equation with non-homogeneous Neumann boundary conditions: the relation to the \(L_{2}\)-approximation
zbMATH Open document ID: 1207.65022
author: A. G. Werschulz
published in: Journal of Complexity
publication date: 2010-01-12T00:00:00Z
review text: The Helmholtz equation \(-\Delta u + qu = f\) is considered in the \(d\)-dimensional unit cube. The tractability of the problem corresponds to the minimal number of function evaluations (\(q\) and \(f\)), which an arbitrary algorithm requires to approximate the exact solution satisfying an error criterion characterised by a norm of a Sobolev space and some \(\varepsilon > 0\). \textit{A. G. Werschulz} and \textit{H. Wozniakowski} [Math. Comput. 76, No.~258, 745--776 (2007; Zbl 1135.65006)] analysed the tractability of the Helmholtz equation in case of homogeneous Dirichlet and homogeneous Neumann boundary conditions. Now the author investigates the case of inhomogeneous Neumann boundary conditions \(\partial_{\nu} u = g\). Thereby, the input functions \(q,f,g\) are restricted to weighted reproducing kernel Hilbert spaces. In case of product weights and finite-order weights, it follows that the problem is polynomially tractable (complexity grows polynomially in \(\varepsilon^{-1}\) and \(d\)) assuming appropriate bounds on the weights. Corresponding estimates for the exponents of this tractability are proved for an absolute as well as a normalised error criterion. In this analysis, parts of the problem can be reduced to a common \(L_{2}\)-approximation. Although homogeneous Dirichlet and Neumann problems are strongly polynomially tractable (complexity is independent of \(d\)) under certain assumptions, the author indicates that the bounds on the complexity are not independent of the dimension \(d\) in case of inhomogeneous Neumann problems. The tractability of inhomogeneous Dirichlet problems shall be investigated in a further paper.
reviewed by: Roland Pulch
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 5656616
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: non-homogeneous Neumann boundary conditions
zbMATH Keywords: tractability
zbMATH Keywords: complexity
zbMATH Keywords: error criterion
zbMATH Keywords: weighted reproducing kernel Hilbert space
zbMATH Keywords: Sobolev space
zbMATH Keywords: minimal number of function evaluations
MaRDI profile type: Publication
full work available at URL: j.jco.2009.09.001
OpenAlex ID: W2056494632
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DOI: 10.1016/J.JCO.2009.09.001
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description: scientific article; zbMATH DE number 5656616