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The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, <jats:italic>p<\/jats:italic>, equal to one), the condition \u201c<jats:inline-formula><jats:alternatives><jats:tex-math>$$h^2 k^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>h<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:msup>\n                      <mml:mi>k<\/mml:mi>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> sufficiently small\" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of <jats:italic>k<\/jats:italic>) for scattering of a plane wave by a nontrapping obstacle and\/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for <jats:inline-formula><jats:alternatives><jats:tex-math>$$p\\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.<\/jats:p>","DOI":"10.1007\/s00211-021-01253-0","type":"journal-article","created":{"date-parts":[[2021,11,27]],"date-time":"2021-11-27T11:02:42Z","timestamp":1638010962000},"page":"137-178","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A sharp relative-error bound for the Helmholtz h-FEM at high frequency"],"prefix":"10.1007","volume":"150","author":[{"given":"D.","family":"Lafontaine","sequence":"first","affiliation":[]},{"given":"E. 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