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<jats:p>We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting.\nThe weak formulation allows for the direct approximation of solutions in the Lebesgue <jats:inline-formula id=\"j_cmam-2018-0199_ineq_9999_w2aab3b7e1880b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mi>p<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0199_eq_0298.png\"\/>\n                        <jats:tex-math>{L^{p}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-space, <jats:inline-formula id=\"j_cmam-2018-0199_ineq_9998_w2aab3b7e1880b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mi mathvariant=\"normal\">\u221e<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0199_eq_0243.png\"\/>\n                        <jats:tex-math>{1&lt;p&lt;\\infty}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nThe greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed.\nWe first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients.\nThe main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in <jats:inline-formula id=\"j_cmam-2018-0199_ineq_9997_w2aab3b7e1880b1b6b1aab1c14b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mi>p<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0199_eq_0298.png\"\/>\n                        <jats:tex-math>{L^{p}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.<\/jats:p>","DOI":"10.1515\/cmam-2018-0199","type":"journal-article","created":{"date-parts":[[2019,7,2]],"date-time":"2019-07-02T09:14:53Z","timestamp":1562058893000},"page":"557-579","source":"Crossref","is-referenced-by-count":19,"title":["The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4430-5167","authenticated-orcid":false,"given":"Ignacio","family":"Muga","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Valpara\u00edso , Casilla 4059 , Valpara\u00edso , Chile"}]},{"given":"Matthew J.\u2009W.","family":"Tyler","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , University of Nottingham , University Park , Nottingham , NG72RD , United Kingdom"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6830-8031","authenticated-orcid":false,"given":"Kristoffer G.","family":"van der Zee","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , University of Nottingham , University Park , Nottingham , NG72RD , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2019,7,2]]},"reference":[{"key":"2023033110340748254_j_cmam-2018-0199_ref_001_w2aab3b7e1880b1b6b1ab2b2b1Aa","unstructured":"P.  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