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Math."],"published-print":{"date-parts":[[2022,4]]},"abstract":"Abstract<\/jats:title>For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $${\\mathbf {A}}, {\\mathbf {b}},\\gamma $$<\/jats:tex-math>\n \n A<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n \u03b3<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$L^\\infty $$<\/jats:tex-math>\n \n L<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and symmetric and uniformly positive definite coefficient matrix $${\\mathbf {A}}$$<\/jats:tex-math>\n A<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in $$H({{\\,\\mathrm{div}\\,}})\\times L^2$$<\/jats:tex-math>\n \n H<\/mml:mi>\n \n (<\/mml:mo>\n \n \n div<\/mml:mi>\n \n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \u00d7<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as well as in in $$L^2\\times L^2$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \u00d7<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to $$L^\\infty $$<\/jats:tex-math>\n \n L<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not<\/jats:italic> apply immediately to the mixed formulation in $$H({{\\,\\mathrm{div}\\,}})\\times L^2$$<\/jats:tex-math>\n \n H<\/mml:mi>\n \n (<\/mml:mo>\n \n \n div<\/mml:mi>\n \n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \u00d7<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. But it allows the uniform approximation of some $$L^2$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contributions and can be combined with a recent $$L^2$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.<\/jats:p>","DOI":"10.1007\/s00211-022-01282-3","type":"journal-article","created":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T13:17:20Z","timestamp":1648819040000},"page":"975-992","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs"],"prefix":"10.1007","volume":"150","author":[{"given":"C.","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Neela","family":"Nataraj","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amiya K.","family":"Pani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,4,1]]},"reference":[{"issue":"211","key":"1282_CR1","first-page":"943","volume":"64","author":"T Arbogast","year":"1995","unstructured":"Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. 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