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The solution to the problem is approximated by the Linearized Backward Euler finite element () method which is dissipative and the Linearized Crank\u2013Nicolson finite element () one which is conservative. Letting\n \n \n $$\\tau $$<\/jats:tex-math>\n \n \u03c4<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the time-step and\n h<\/jats:italic>\n be the width of the finite element partition of the space domain, we provide an optimal order\n \n \n $$O(\\tau +h^2)$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \u03c4<\/mml:mi>\n +<\/mml:mo>\n \n h<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n error estimate in the\n \n \n $$L^2$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n norm for both methods, and an\n \n \n $$O(\\tau ^{\\alpha }+h)$$<\/jats:tex-math>\n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4<\/mml:mi>\n \u03b1<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n h<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n error estimate in the\n \n \n $$H^1$$<\/jats:tex-math>\n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n norm, where\n \n \n $$\\alpha =\\frac{3}{4}$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n =<\/mml:mo>\n \n 3<\/mml:mn>\n 4<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in the () method and\n \n \n $$\\alpha =\\frac{1}{2}$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n =<\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in the () one. For\n \n \n $$d=1$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , no CFL conditions are imposed, while for\n \n \n $$d=2$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or 3, a mesh condition of the form\n \n \n $$\\big (h^{\\frac{2-d}{2}}\\,|\\ln (h)|^{\\frac{d-1}{d}} \\,\\tau ^{\\alpha }+h^{2-\\frac{d}{2}}\\big )=O(1)$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n <\/mml:mrow>\n \n h<\/mml:mi>\n \n \n 2<\/mml:mn>\n -<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:msup>\n \n \n \n |<\/mml:mo>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n h<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:mfrac>\n <\/mml:msup>\n \n \n \u03c4<\/mml:mi>\n \u03b1<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n -<\/mml:mo>\n \n d<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:msup>\n \n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n O<\/mml:mi>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is required. Finally, with results from numerical experiments, we investigate the performance of the methods proposed and analyzed.\n <\/jats:p>","DOI":"10.1007\/s10915-026-03328-2","type":"journal-article","created":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T01:04:12Z","timestamp":1779757452000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schr\u00f6dinger Equation with a Schamel-Type Nonlinearity"],"prefix":"10.1007","volume":"108","author":[{"given":"Panagiotis","family":"Paraschis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0234-0175","authenticated-orcid":false,"given":"Georgios E.","family":"Zouraris","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,26]]},"reference":[{"key":"3328_CR1","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/BF01385769","volume":"59","author":"G Akrivis","year":"1991","unstructured":"Akrivis, G., Dougalis, V.A., Karakashian, O.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schr\u00f6dinger equation. 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