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Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques, is particularly challenging due to the presence of sharp boundary layers and shocks causing jumps and discontinuities in the solution, and numerical issues such as loss of the maximum principle in the discretization. These complications cause instabilities, admitting large oscillations in the numerical solution when using traditional methods. Drawing connections to the simplex-averaged finite-element method [S. Wu and J. Xu, SIAM J.\u00a0Numer.\u00a0Anal., 58 (2020), pp.\u00a0884\u2013906], this paper develops a mimetic finite-difference (MFD) discretization using exponentially averaged coefficients to overcome instability of the numerical solution as the diffusion coefficient approaches zero. The finite-element framework allows for transparent analysis of the MFD, such as proving well-posedness and deriving error estimates. Numerical tests are presented confirming the stability of the method and verifying the error estimates.<\/jats:p>","DOI":"10.1137\/22m1514337","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T05:01:50Z","timestamp":1701234110000},"page":"A2973-A3000","source":"Crossref","is-referenced-by-count":11,"title":["A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6603-8840","authenticated-orcid":true,"given":"James H.","family":"Adler","sequence":"first","affiliation":[{"name":"Department of Mathematics, Tufts University, Medford, MA 02155 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Casey","family":"Cavanaugh","sequence":"additional","affiliation":[{"name":"Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7533-0416","authenticated-orcid":true,"given":"Xiaozhe","family":"Hu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tufts University, Medford, MA 02155 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andy","family":"Huang","sequence":"additional","affiliation":[{"name":"Radiation and Electrical Science, Sandia National Laboratories, Albuquerque, NM 87123 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nathaniel","family":"Trask","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,11,29]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/20M1382568"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"D. 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