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Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 126)

Abstract

Optimized Schwarz methods use better transmission conditions than the classical Dirichlet conditions that were used by Schwarz. These transmission conditions are optimized for the physical problem that needs to be solved to lead to fast convergence. The optimization is typically performed in the geometrically simplified setting of two unbounded subdomains using Fourier transforms. Recent studies for both homogeneous and heterogeneous domain decomposition methods indicate that the geometry of the physical domain has actually an influence on this optimization process. We study here this influence for an advection diffusion equation in a bounded domain using separation of variables. We provide theoretical results for the min-max problems characterizing the optimized transmission conditions. Our numerical experiments show significant improvements of the new transmission conditions which take the geometry into account, especially for strong tangential advection.

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Notes

  1. 1.

    In the unbounded domain analysis, one minimizes over all frequencies \(k:=\frac {\pi l}{L}\in [k_{\min },k_{\max }]\), with \(k_{\min }:=\frac {\pi }{L}\) and \(k_{\max }=\frac {\pi }{h}\), where \(h=\frac {L}{N+1}\) is the mesh size and N the number of mesh points on the interface Γ. From (6), we see that \(k_{\min }\) corresponds to l = 1, and for l = N, \(\frac {\pi N}{L}\approx \frac {\pi }{h}=k_{\max }\), like in e.g. [3].

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Correspondence to Tommaso Vanzan .

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Gander, M.J., Vanzan, T. (2019). Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_87

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