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We consider GMRES applied to discretisations of the high-frequency Helmholtzequation with strong trapping; recall that in this situation the problem isexponentially ill-conditioned through an increasing sequence of frequencies.Under certain assumptions about the distribution of the eigenvalues, we proveupper bounds on how the number of GMRES iterations grows with the frequency.Our main focus is on boundary-integral-equation formulations of the exteriorDirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, weinvestigate numerically the sharpness (in terms of dependence on frequency) ofboth our bounds and various quantities entering our bounds. This paper istherefore the first comprehensive study of the frequency-dependence of thenumber of GMRES iterations for Helmholtz boundary-integral equations undertrapping.

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?