{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:43:19Z","timestamp":1776796999621,"version":"3.51.2"},"reference-count":68,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,10,30]],"date-time":"2014-10-30T00:00:00Z","timestamp":1414627200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods.<\/p>","DOI":"10.1090\/s0025-5718-2013-02760-3","type":"journal-article","created":{"date-parts":[[2013,10,30]],"date-time":"2013-10-30T10:21:29Z","timestamp":1383128489000},"page":"1083-1120","source":"Crossref","is-referenced-by-count":21,"title":["Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients"],"prefix":"10.1090","volume":"83","author":[{"given":"Blanca","family":"Ayuso de Dios","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Holst","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunrong","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ludmil","family":"Zikatanov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,10,30]]},"reference":[{"key":"1","series-title":"Van Nostrand Mathematical Studies, No. 2","volume-title":"Lectures on elliptic boundary value problems","author":"Agmon, Shmuel","year":"1965"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1051\/m2an:2007006","article-title":"Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case","volume":"41","author":"Antonietti, Paola F.","year":"2007","journal-title":"M2AN Math. 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