{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:53Z","timestamp":1787333753252,"version":"build-2736575974"},"reference-count":51,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["WO671\/11-1"],"award-info":[{"award-number":["WO671\/11-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Matrix-free techniques play an increasingly important role in large-scale simulations. Schur complement techniques and massively parallel multigrid solvers for second-order elliptic partial differential equations significantly benefit from reduced memory traffic and consumption. The matrix-free approach often restricts solver components to purely local operations, for instance, to the most basic schemes like Jacobi- or Gauss\u2013Seidel-Smoothers in multigrid methods. An incomplete LU(0)-decomposition (ILU) cannot be calculated from local information and is therefore not applicable to an on-the-fly computation typically needed for matrix-free calculations. It requires storing and factorizing a sparse matrix, contradicting the low memory requirements in large-scale scenarios. Here, we propose a matrix-free ILU realization. More precisely, we introduce a memory-efficient matrix-free ILU-Smoother component for low-order conforming finite elements on tetrahedral hybrid grids. Hybrid grids consist of an unstructured macro-mesh which is subdivided into structured micro-meshes. The ILU operates on the degrees of freedom assigned to the interior of macro-tetrahedra. This ILU-Smoother can be applied to the efficient matrix-free evaluation of the Steklov\u2013Poincar\u00e9 operator from domain-decomposition methods, as well as for the finite element tearing and interconnecting dual-primal and balancing domain decomposition by constraints methods. After introducing and formally defining our smoother, we investigate its performance on refined macro-tetrahedra. On the macro-tetrahedra the ILU-Smoother is implemented via surrogate matrix polynomials, which we combine with a fast on-the-fly evaluation scheme, resulting in an efficient matrix-free algorithm. We obtain the polynomial coefficients by solving a least-squares problem on a small part of the factorized ILU matrices to remain memory efficient. The convergence rates of this smoother in relation to the polynomial order are thoroughly studied.<\/jats:p>","DOI":"10.1137\/22m1529415","type":"journal-article","created":{"date-parts":[[2023,12,4]],"date-time":"2023-12-04T04:01:22Z","timestamp":1701662482000},"page":"C304-C329","source":"Crossref","is-referenced-by-count":1,"title":["A Matrix-Free ILU Realization Based on Surrogates"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3757-2136","authenticated-orcid":true,"given":"Daniel","family":"Drzisga","sequence":"first","affiliation":[{"name":"Lehrstuhl f\u00fcr Numerische Mathematik, Fakult\u00e4t f\u00fcr Mathematik (M2), Technische, Universit\u00e4t M\u00fcnchen, 85748 Garching bei M\u00fcnchen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Wagner","sequence":"additional","affiliation":[{"name":"Corresponding author. Lehrstuhl f\u00fcr Numerische Mathematik, Fakult\u00e4t f\u00fcr Mathematik (M2), Technische, Universit\u00e4t M\u00fcnchen, 85748 Garching bei M\u00fcnchen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Barbara","family":"Wohlmuth","sequence":"additional","affiliation":[{"name":"Lehrstuhl f\u00fcr Numerische Mathematik, Fakult\u00e4t f\u00fcr Mathematik (M2), Technische, Universit\u00e4t M\u00fcnchen, 85748 Garching bei M\u00fcnchen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,12,4]]},"reference":[{"key":"ref1","unstructured":"ILU Implementation - Source Code, https:\/\/doi.org\/10.5281\/zenodo.7199022."},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/16M1079506"},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"H. Anzt , \nT. Ribizel , \nG. Flegar , \nE. Chow , and \nJ. 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