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Comput. <jats:bold>28<\/jats:bold>(128), 937\u2013958 1974), Wahlbin (1991, \u00a79), Demlow et al.(Math. Comput. <jats:bold>80<\/jats:bold>(273), 1\u20139 2011), showing that these bounds hold with constants independent of <jats:italic>k<\/jats:italic>, provided one works in Sobolev norms weighted with <jats:italic>k<\/jats:italic> in the natural way. We prove two main results: (i) a bound on the local <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> error by the best approximation error plus the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> error, both on a slightly larger set, and (ii) the bound in (i) but now with the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the <jats:italic>k<\/jats:italic>-explicit analogue of the main result of Demlow et al. (Math. Comput. <jats:bold>80<\/jats:bold>(273), 1\u20139 2011). The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of <jats:inline-formula><jats:alternatives><jats:tex-math>$$k^{-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and is the <jats:italic>k<\/jats:italic>-explicit analogue of the results of Nitsche and Schatz (Math. Comput. <jats:bold>28<\/jats:bold>(128), 937\u2013958 1974), Wahlbin (1991, \u00a79). Since our Sobolev spaces are weighted with <jats:italic>k<\/jats:italic> in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lesssim k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>\u2272<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.<\/jats:p>","DOI":"10.1007\/s10444-024-10193-w","type":"journal-article","created":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T04:30:39Z","timestamp":1731904239000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0836-3848","authenticated-orcid":false,"given":"M.","family":"Averseng","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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