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A finite-difference (FD) discretization based on \n \n $$P_1$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-element in the z<\/jats:italic>-direction and a finite-element (FE) discretization based on \n \n $$P_1^{NC}$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n \n NC<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-nonconforming element in the (x<\/jats:italic>,\u00a0y<\/jats:italic>)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of \n \n $$P_1^{NC}$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n \n NC<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-nonconforming finite element methods in the 2D elliptic equation and the error estimation of the \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>-norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the \n \n $$P_1$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> element in the z<\/jats:italic>-direction and the FE method based on the Crouzeix-Raviart element, the \n \n $$P_1$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> linear element, the Park-Sheen element, and the \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> bilinear element, respectively, in the (x<\/jats:italic>,\u00a0y<\/jats:italic>)-plane.<\/jats:p>","DOI":"10.1007\/s10444-025-10219-x","type":"journal-article","created":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T07:46:05Z","timestamp":1737704765000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A difference finite element method based on nonconforming finite element methods for 3D elliptic problems"],"prefix":"10.1007","volume":"51","author":[{"given":"Jianjian","family":"Song","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6857-7883","authenticated-orcid":false,"given":"Dongwoo","family":"Sheen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yinnian","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,24]]},"reference":[{"issue":"1\u20132","key":"10219_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0045-7825(96)01107-3","volume":"142","author":"M Ainsworth","year":"1997","unstructured":"Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. 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