Solving time-harmonic scattering problems based on the condition: Theory
Please always quote using this URN: urn:nbn:de:0297-zib-6279
- The pole condition is a general concept for the theoretical analysis and the numerical solution of a variety of wave propagation problems. It says that the Laplace transform of the physical solution in radial direction has no poles in the lower complex half-plane. In the present paper we show that for the Helmholtz equation with a radially symmetric potential the pole condition is equivalent to Sommerfeld's radiation condition. Moreover, a new representation formula based on the pole condition is derived and used to prove existence, uniqueness and asymptotic properties of solutions. This lays the foundations of a promising new algorithm to solve time-harmonic scattering problems numerically and provides a new approach for analyzing existing algorithms such as the Perfectly Matched Layer (PML) method and the Bayliss-Gunzburger-Turkel (BGT) algorithm.
Author: | Thorsten Hohage, Frank Schmidt, Lin Zschiedrich |
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Document Type: | ZIB-Report |
Tag: | Laplace transform; Sommerfeld radiation condition; transparent boundary conditions |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Cxx Representations of solutions / 35C10 Series solutions |
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Cxx Representations of solutions / 35C15 Integral representations of solutions | |
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Cxx Representations of solutions / 35C20 Asymptotic expansions | |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N99 None of the above, but in this section | |
Date of first Publication: | 2001/01/09 |
Series (Serial Number): | ZIB-Report (01-01) |
ZIB-Reportnumber: | 01-01 |
Published in: | Appeared under the title "Solving time-harmonic scattering problems based on the condition I: Theory in: SIAM Journal on Mathematical Analysis 35 (2003) 183-210 |