Seminar in Numerical Analysis: Pierre-Henri Tournier (UPMC - University Pierre and Marie Curie)
This work deals with preconditioning the time-harmonic Maxwell equations with absorption, where the preconditioner is constructed using two-level overlapping Additive Schwarz Domain Decomposition, and the PDE is discretised using finite-element methods of fixed, arbitrary order. The theory shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters are chosen appropriately, then classical two-level overlapping Additive Schwarz Domain Decomposition preconditioning performs optimally – in the sense that GMRES converges in a wavenumber-independent number of iterations – for the problem with absorption. This work is an extension of the theory proposed in  for the Helmholtz equation. Numerical experiments illustrate this theoretical result and also (i) explore replacing the PEC boundary conditions on the subdomains by impedance boundary conditions, and (ii) show that the preconditioner for the problem with absorption is also an effective preconditioner for the problem with no absorption. The numerical results include examples arising from applications: a problem with absorption arising from medical imaging shows the robustness of the preconditioner against heterogeneity, and a scattering problem by the COBRA cavity shows good scalability of the preconditioner with up to 3000 processors. Finally, additional numerical results for the elastic wave equation are presented for benchmarks in seismic inversion.
 I. G. Graham, E. A. Spence, and E. Vainikko. Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Mathematics of Computation, 86(307):2089–2127, 2017.
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