{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:09:33Z","timestamp":1776802173977,"version":"3.51.2"},"reference-count":36,"publisher":"American Mathematical Society (AMS)","issue":"304","license":[{"start":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T00:00:00Z","timestamp":1490659200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00c3\u00adfico y Tecnol\u00c3\u00b3gico","doi-asserted-by":"publisher","award":["11140699"],"award-info":[{"award-number":["11140699"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00c3\u00adfico e Tecnol\u00c3\u00b3gico","doi-asserted-by":"publisher","award":["11140699"],"award-info":[{"award-number":["11140699"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002322","name":"Coordena\u00c3\u00a7\u00c3\u00a3o de Aperfei\u00c3\u00a7oamento de Pessoal de N\u00c3\u00advel Superior","doi-asserted-by":"publisher","award":["11140699"],"award-info":[{"award-number":["11140699"]}],"id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second-order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norms are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h plus epsilon Superscript delta Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b5\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h + \\varepsilon ^\\delta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h squared plus h epsilon Superscript delta Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b5\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h^2 + h\\,\\varepsilon ^\\delta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , respectively, and for the dual variable it is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h plus epsilon Superscript delta Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b5\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h + \\varepsilon ^\\delta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis d i v semicolon dot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>div<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\operatorname {div};\\cdot )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than delta less-than-or-equal-to 1 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03b4\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0&gt;\\delta \\leq 1\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions.\n                  <\/p>","DOI":"10.1090\/mcom\/3108","type":"journal-article","created":{"date-parts":[[2015,10,7]],"date-time":"2015-10-07T13:57:56Z","timestamp":1444226276000},"page":"525-548","source":"Crossref","is-referenced-by-count":18,"title":["On the robustness of multiscale hybrid-mixed methods"],"prefix":"10.1090","volume":"86","author":[{"given":"Diego","family":"Paredes","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fr\u00e9d\u00e9ric","family":"Valentin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henrique","family":"Versieux","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,3,28]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1051\/cocv:1999110","article-title":"Boundary layer tails in periodic homogenization","volume":"4","author":"Allaire, Gr\u00e9goire","year":"1999","journal-title":"ESAIM Control Optim. 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