{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T13:43:25Z","timestamp":1762177405445,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T00:00:00Z","timestamp":1624492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11772165","11961054","11902170"],"award-info":[{"award-number":["11772165","11961054","11902170"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Boundary or interior layer problems of high-dimensional convection\u2013diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection\u2013diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does\/do not contain boundary or interior layer(s).<\/jats:p>","DOI":"10.3390\/sym13071123","type":"journal-article","created":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T23:22:14Z","timestamp":1624576934000},"page":"1123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection\u2013Diffusion Problems with Boundary\/Interior Layers"],"prefix":"10.3390","volume":"13","author":[{"given":"Tianlong","family":"Ma","sequence":"first","affiliation":[{"name":"Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0502-5314","authenticated-orcid":false,"given":"Lin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fujun","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Science, Inner Mongolia University of Science & Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongbin","family":"Ge","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,24]]},"reference":[{"key":"ref_1","unstructured":"Batchelor, G.K. 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