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A key core of our approach is employing a harmonic map to handle the general physical domains. This technique ensures broad geometric applicability, making the method highly effective for both complex star-shaped and nonstar-shaped domains. Moreover, this method is rigorously proved, with optimal convergence results established under\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$H^1$$<\/jats:tex-math>\n                        <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>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -norm, which is independent of the domain boundary\u2019s smoothness. The effectiveness and generality of the scheme are validated through some numerical examples on a wide variety of complex geometries.\n                  <\/jats:p>","DOI":"10.1007\/s10915-026-03241-8","type":"journal-article","created":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T01:18:07Z","timestamp":1774919887000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Spectral Method with Harmonic Map for Elliptic PDEs on General Two-Dimensional Domains"],"prefix":"10.1007","volume":"107","author":[{"given":"Shan","family":"Shi","sequence":"first","affiliation":[]},{"given":"Xiaoyun","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Fanhai","family":"Zeng","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0009-7334-772X","authenticated-orcid":false,"given":"Hui","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,31]]},"reference":[{"key":"3241_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109733","volume":"421","author":"W Bao","year":"2020","unstructured":"Bao, W., Chen, L., Jiang, X., Ma, Y.: A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schr\u00f6dinger operator. 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