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Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence results for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparison against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.<\/jats:p>","DOI":"10.1137\/22m1469833","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T10:41:22Z","timestamp":1686307282000},"page":"A1199-A1213","source":"Crossref","is-referenced-by-count":11,"title":["Efficient Algebraic Two-Level Schwarz Preconditioner for Sparse Matrices"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9355-4042","authenticated-orcid":true,"given":"Hussam","family":"Al Daas","sequence":"first","affiliation":[{"name":"STFC, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Jolivet","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9, CNRS, LIP6, 75252 Paris Cedex 05, France."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0476-2259","authenticated-orcid":true,"given":"Tyrone","family":"Rees","sequence":"additional","affiliation":[{"name":"STFC, Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"M. 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