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A novel extrapolation cascadic multigrid (EXCMG) method is developed to solve large linear systems encountered in geophysical EM modeling. The original curl-curl equation with arbitrary anisotropic conductivity is regularized by including the gradient of a scaled divergence correction term, and a linear edge element method is used to discretize the equation. We develop a novel approach to address the issue of edge unknowns in 3D edge element discretizations on nonuniform rectilinear grids. Inspired by the original EXCMG method for nodal elements, we introduce a new prolongation operator that treats edge unknowns as defined on the midpoints of edges. This operator aims to provide an accurate approximation of the finite-element solution on the refined grid. Using a good initial guess significantly reduces the number of iterations required by the preconditioned biconjugate gradient stabilized solver, which is used as a smoother for the EXCMG algorithm. Numerical experiments are carried out to validate the accuracy and efficiency of our EXCMG method, including models from magnetotellurics and controlled-source EM modeling. The testing results indicate that EXCMG is more efficient than traditional Krylov-subspace iterative solvers, the algebraic multigrid solver, and those depending on the auxiliary-space Maxwell solver, especially for large-scale problems where the number of unknowns exceeds 10 million. The EXCMG method can also be applied to solve other large-scale forward modeling problems encountered in geophysics.<\/jats:p>","DOI":"10.1190\/geo2023-0702.1","type":"journal-article","created":{"date-parts":[[2024,7,27]],"date-time":"2024-07-27T09:53:05Z","timestamp":1722073985000},"page":"E241-E253","update-policy":"https:\/\/doi.org\/10.1190\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["An efficient cascadic multigrid method with regularization technique for 3D electromagnetic finite-element anisotropic modeling"],"prefix":"10.1190","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5768-7972","authenticated-orcid":false,"given":"Kejia","family":"Pan","sequence":"first","affiliation":[{"name":"Central South University 1 , HNP-LAMA, School of Mathematics and Statistics, Changsha, China. kejiapan@csu.edu.cn"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8137-5696","authenticated-orcid":false,"given":"Jinxuan","family":"Wang","sequence":"additional","affiliation":[{"name":"Central South University 1 , HNP-LAMA, School of Mathematics and Statistics, Changsha, China. wjx171210@csu.edu.cn (corresponding author)"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-7021-7985","authenticated-orcid":false,"given":"Zhengguang","family":"Liu","sequence":"additional","affiliation":[{"name":"Central South University 1 , HNP-LAMA, School of Mathematics and Statistics, Changsha, China. zhengguang-liu@outlook.com"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-8168-5572","authenticated-orcid":false,"given":"Ziyi","family":"Ou","sequence":"additional","affiliation":[{"name":"Central South University 1 , HNP-LAMA, School of Mathematics and Statistics, Changsha, China. 222111064@csu.edu.cn"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0363-4653","authenticated-orcid":false,"given":"Rongwen","family":"Guo","sequence":"additional","affiliation":[{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. rongwenguo@csu.edu.cn"},{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. rongwenguo@csu.edu.cn"},{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. rongwenguo@csu.edu.cn"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9125-0053","authenticated-orcid":false,"given":"Zhengyong","family":"Ren","sequence":"additional","affiliation":[{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. renzhengyong@csu.edu.cn"},{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. renzhengyong@csu.edu.cn"},{"name":"Central South University 2 , School of Geosciences and Info-Physics, Changsha, China; , Laboratory of Non-ferrous Resources and Geological Hazard Detection, Changsha, China and , Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, China. renzhengyong@csu.edu.cn"}]}],"member":"186","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"2025121212372427100_r1","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.camwa.2020.06.009","article-title":"MFEM: A modular finite element methods library","volume":"81","author":"Anderson","year":"2021","journal-title":"Computers & Mathematics with Applications"},{"key":"2025121212372427100_r2","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1093\/gji\/ggx149","article-title":"A gauged finite-element potential formulation for accurate inductive and galvanic modelling of 3-D electromagnetic problems","volume":"210","author":"Ansari","year":"2017","journal-title":"Geophysical Journal 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