Tractability of the Helmholtz equation with non-homogeneous Neumann boundary conditions: the relation to the \(L_{2}\)-approximation
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.
non-homogeneous Neumann boundary conditions
weighted reproducing kernel Hilbert space
scientific article; zbMATH DE number 5656616
scientific article