An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation

• Published in: International journal for numerical methods in fluids Vol. 55; no. 2; pp. 163 - 175
• Main Authors: Gander, M. J., Halpern, L., Magoulès, F.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 20.09.2007; Wiley
• Subjects: Acoustics; domain decomposition; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Helmholtz equation; Linear acoustics; Mathematics; Numerical Analysis; Physics; Schwarz; transmission conditions; transmission conditions; Helmholtz equation; Acoustics; Asymptotic approximation; Overlap; Experimental study; Schwarz; Domain decomposition; Modeling
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.1433

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ωγ) for γ⩾1, then the optimized asymptotic convergence factor is 1 – O(ω(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.