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Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of<jats:italic>both<\/jats:italic>our bounds<jats:italic>and<\/jats:italic>various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.<\/jats:p>","DOI":"10.1007\/s10444-022-09931-9","type":"journal-article","created":{"date-parts":[[2022,6,4]],"date-time":"2022-06-04T07:02:30Z","timestamp":1654326150000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?"],"prefix":"10.1007","volume":"48","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2522-6837","authenticated-orcid":false,"given":"P.","family":"Marchand","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J.","family":"Galkowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E. 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