{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T19:10:21Z","timestamp":1735931421745,"version":"3.32.0"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T00:00:00Z","timestamp":1715817600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100008902","name":"Los Alamos National Laboratory","doi-asserted-by":"publisher","award":["89233218CNA000001"],"award-info":[{"award-number":["89233218CNA000001"]}],"id":[{"id":"10.13039\/100008902","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A Schr\u00f6dinger equation for the system\u2019s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials\u2019 properties.\nRecent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method.\nIn particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schr\u00f6dinger equation offers a very effective way of solving the three-dimensional Schr\u00f6dinger eigenvalue problem.\nWe investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the \u210e- and \ud835\udc5d-refinement in the <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0115_ineq_0001.png\"\/>\n                        <jats:tex-math>L^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and energy norm for both the eigenvalues and the eigenfunctions.\nFinally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schr\u00f6dinger equation with the harmonic potential, for which the exact solution is known.<\/jats:p>","DOI":"10.1515\/cmam-2023-0115","type":"journal-article","created":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T12:51:04Z","timestamp":1715777464000},"page":"39-60","source":"Crossref","is-referenced-by-count":0,"title":["The Partition of Unity Finite Element Method for the Schr\u00f6dinger Equation"],"prefix":"10.1515","volume":"25","author":[{"given":"Daniele","family":"Boffi","sequence":"first","affiliation":[{"name":"Computer, Electrical and Mathematical Science and Engineering Division , King Abdullah University of Science and Technology , Thuwal , Saudi Arabia ; and Dipartimento di Matematica \u201cF. Casorati\u201d, Universit\u00e0 di Pavia, Pavia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ondrej","family":"Certik","sequence":"additional","affiliation":[{"name":"CCS-3 , 5112 Los Alamos National Laboratory , Los Alamos , New Mexico , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Gardini","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica \u201cF. Casorati\u201d , 19001 Universit\u00e0 di Pavia , Pavia , Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gianmarco","family":"Manzini","sequence":"additional","affiliation":[{"name":"T-5 , 5112 Los Alamos National Laboratory , Los Alamos , New Mexico , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,5,16]]},"reference":[{"key":"2025010318340356234_j_cmam-2023-0115_ref_001","unstructured":"S. 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