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Here, u is the solution of the original variational problem, ${\\cal T}$ is a certain continuous solution operator, and S is the finite dimensional test and trial space. The abstract analysis can be applied to both finite and boundary element solutions of high\u2010frequency Helmholtz problems. We apply the analysis to investigate the properties of the Brakhage\u2013Werner boundary integral formulation of the Helmholtz problem, discretized by a standard Galerkin boundary element method. In the case of scattering by the unit sphere, we derive the explicit dependence of the error and of the stability condition on the wave number k. We show that $hk \\lesssim 1$ is a sufficient condition for stability and a quasi\u2010optimal error estimate. Further, we show that the constant of quasioptimality is independent of k, which is an improvement over previously available results. Thus, the boundary element method does not suffer from the pollution effect.<\/jats:p>","DOI":"10.1137\/060654177","type":"journal-article","created":{"date-parts":[[2007,1,8]],"date-time":"2007-01-08T18:00:23Z","timestamp":1168279223000},"page":"37-53","source":"Crossref","is-referenced-by-count":27,"title":["A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems"],"prefix":"10.1137","volume":"45","author":[{"given":"L.","family":"Banjai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Sauter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,1,8]]},"reference":[{"key":"R1","unstructured":"M. Abramowitz and I. A. Stegun, eds.\n                      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables\n                      , Dover Publications, New York, 1992."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19980315)41:5<875::AID-NME313>3.0.CO;2-9"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994269186"},{"key":"R4","unstructured":"L. Banjai and W. Hackbusch,\n                      ${\\cal H}$\u2010 and ${\\cal H}^2$\u2010Matrices for Low and High Frequency Helmholtz Equation\n                      , Technical Report 17\/2005, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, 2005."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01220037"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , 2nd ed., Texts in Appl. Math. 15, Springer\u2010Verlag, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2003.1338"},{"key":"R8","unstructured":"A. Buffa and S. Sauter,\n                      On the acoustic single layer potential: Stabilization and Fourier Analysis\n                      , SIAM J. Sci. Comput., to appear."},{"key":"R9","unstructured":"G. Chen and J. Zhou,\n                      Boundary Element Methods\n                      , Comput. Math. Appl., Academic Press, London, 1992."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7091"},{"key":"R11","unstructured":"V. Dom\u00ednguez, I. G. Graham, and V. P. 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