Modern investigation of the real-analytic continuability of solutions to boundary problems involves elements of complex and microlocal analysis, as well as the theory of pseudodifferential operators. Apart from its purely mathematical interest, this investigation can lead to significant improvement of numerical methods used in, e.g., acoustic and electromagnetic scattering. In this talk, I shall take as the starting point the desire to improve one such numerical method, namely the so-called Method of Auxiliary Sources (MAS). The latter is a promising numerical scheme, with the potential of replacing the traditional boundary layer formulations in the numerical solution of scattering problems. To address the convergence issues inherent to the MAS, I shall introduce a relevant general real-analytic continuation problem and describe how it can be reformulated in terms of an analytic Cauchy problem in the complex setting. Then, as part of the analysis of this Cauchy problem, I shall propose a new class of pseudodifferential operators as a microanalytical tool well-suited for estimating the extent of real-analytic continuation of solutions to Helmholtz boundary problems in R^n.
|Publication status||Published - 2008|
|Event||Seminar at the Department of Mathematical Sciences, Aalborg University - |
Duration: 1 Jan 2008 → …
|Conference||Seminar at the Department of Mathematical Sciences, Aalborg University|
|Period||01/01/2008 → …|