{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:00:58Z","timestamp":1776841258025,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"261","license":[{"start":{"date-parts":[[2008,5,14]],"date-time":"2008-05-14T00:00:00Z","timestamp":1210723200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Consider the problem\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"minus epsilon squared normal upper Delta u plus u equals f\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-\\epsilon ^2\\Delta u+u=f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with homogeneous Neumann boundary condition in a bounded smooth domain in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R Superscript upper N\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The whole range\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than epsilon less-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03f5\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0&gt;\\epsilon \\le 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; the mesh is fixed and independent of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03f5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . A precise analysis of how the error at each point depends on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/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=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03f5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is presented. As an application, first order error estimates in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which are uniform with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03f5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-02015-7","type":"journal-article","created":{"date-parts":[[2007,10,29]],"date-time":"2007-10-29T06:30:33Z","timestamp":1193639433000},"page":"21-39","source":"Crossref","is-referenced-by-count":4,"title":["Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem"],"prefix":"10.1090","volume":"77","author":[{"given":"Dmitriy","family":"Leykekhman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2007,5,14]]},"reference":[{"issue":"7","key":"1","first-page":"1168","article-title":"On the Galerkin finite-element method for elliptic quasilinear singularly perturbed boundary value problems. I","volume":"28","author":"Blatov, I. A.","year":"1992","journal-title":"Differentsial\\cprime nye Uravneniya","ISSN":"https:\/\/id.crossref.org\/issn\/0374-0641","issn-type":"print"},{"issue":"252","key":"2","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1090\/S0025-5718-05-01762-X","article-title":"A parameter robust numerical method for a two dimensional reaction-diffusion problem","volume":"74","author":"Clavero, C.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","first-page":"179","article-title":"Investigation of the Green\u2019s matrix of a homogeneous parabolic boundary value problem","volume":"23","author":"\u00c8\u012ddel\u2032man, S. D.","year":"1970","journal-title":"Trudy Moskov. Mat. Ob\\v{s}\\v{c}.","ISSN":"https:\/\/id.crossref.org\/issn\/0134-8663","issn-type":"print"},{"key":"4","series-title":"Studies in Mathematics and its Applications","isbn-type":"print","volume-title":"Asymptotic analysis of singular perturbations","volume":"9","author":"Eckhaus, Wiktor","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0444853065"},{"key":"5","unstructured":"N. Kopteva, Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, to appear in Math. Comp."},{"key":"6","unstructured":"J.P. Krasovskii, Properties of Green\u2019s function and generalized solutions of elliptic boundary value problems, Soviet Mathematics (Translations of Doklady Academy of Sciences of the USSR) 10\u00b2 (1969), 54\u2013120."},{"issue":"223","key":"7","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1090\/S0025-5718-98-00959-4","article-title":"Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates","volume":"67","author":"Schatz, Alfred H.","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"11-12","key":"8","doi-asserted-by":"publisher","first-page":"1349","DOI":"10.1002\/(sici)1097-0312(199811\/12)51:11\/12<1349::aid-cpa5>3.0.co;2-1","article-title":"Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations","volume":"51","author":"Schatz, A. H.","year":"1998","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"138","key":"9","doi-asserted-by":"publisher","first-page":"414","DOI":"10.2307\/2006424","article-title":"Interior maximum norm estimates for finite element methods","volume":"31","author":"Schatz, A. H.","year":"1977","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"157","key":"10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2007461","article-title":"On the quasi-optimality in \ud835\udc3f_{\u221e} of the \ud835\udc3b\u0307\u00b9-projection into finite element spaces","volume":"38","author":"Schatz, A. H.","year":"1982","journal-title":"Math. 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