{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:00:57Z","timestamp":1776841257307,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction\u2013diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis upper N right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(N)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N Superscript negative 2 Baseline ln squared upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>ln<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N^{-2}\\ln ^2 N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter\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                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/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 N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Numerical results are presented that verify this rate of convergence.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00753-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1085-1109","source":"Crossref","is-referenced-by-count":15,"title":["A uniformly convergent method for a singularly perturbed semilinear reaction\u2013diffusion problem with multiple solutions"],"prefix":"10.1090","volume":"65","author":[{"given":"Guangfu","family":"Sun","sequence":"first","affiliation":[]},{"given":"Martin","family":"Stynes","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1114-3","volume-title":"Nonlinear singular perturbation phenomena: theory and applications","volume":"56","author":"Chang, K. W.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/038796066X"},{"key":"2","unstructured":"C.M. D\u2019Annunzio, Numerical analysis of a singular perturbation problem with multiple solutions, Ph.D. Dissertation, University of Maryland at College Park, 1986 (unpublished)."},{"key":"3","isbn-type":"print","volume-title":"Uniform numerical methods for problems with initial and boundary layers","author":"Doolan, E. P.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/090678302X"},{"key":"4","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/BF00247733","article-title":"Semilinear elliptic boundary value problems with small parameters","volume":"52","author":"Fife, Paul C.","year":"1973","journal-title":"Arch. Rational Mech. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"8","first-page":"163","article-title":"On numerical solution of a singularly perturbed boundary value problem. II","volume":"17","author":"Herceg, Dragoslav","year":"1987","journal-title":"Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0352-0900","issn-type":"print"},{"key":"9","first-page":"151","article-title":"Stability and monotonicity properties of stiff quasilinear boundary problems","volume":"12","author":"Lorenz, Jens","year":"1982","journal-title":"Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0352-0900","issn-type":"print"},{"issue":"7","key":"10","first-page":"1","article-title":"On a three-point difference scheme for a singular perturbation problem without a first derivative term. I, II","author":"Niijima, Koichi","year":"1980","journal-title":"Mem. Numer. Math."},{"key":"11","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0977-5","volume-title":"Singular perturbation methods for ordinary differential equations","volume":"89","author":"O\u2019Malley, Robert E., Jr.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/038797556X"},{"issue":"176","key":"12","doi-asserted-by":"publisher","first-page":"555","DOI":"10.2307\/2008172","article-title":"A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem","volume":"47","author":"O\u2019Riordan, Eugene","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","volume-title":"Iterative solution of nonlinear equations in several variables","author":"Ortega, J. M.","year":"1970"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1016\/0377-0427(90)90196-7","article-title":"Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline-functions","volume":"29","author":"Roos, H.-G.","year":"1990","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"5","key":"15","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1515\/rnam.1988.3.5.393","article-title":"Grid approximation of singularly perturbed parabolic equations with internal layers","volume":"3","author":"Shishkin, G. I.","year":"1988","journal-title":"Soviet J. Numer. Anal. Math. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0169-2895","issn-type":"print"},{"key":"16","isbn-type":"print","volume-title":"Singular-perturbation theory","author":"Smith, Donald R.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0521300428"},{"key":"17","first-page":"187","article-title":"On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh","volume":"13","author":"Vulanovi\u0107, Relja","year":"1983","journal-title":"Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0352-0900","issn-type":"print"},{"key":"18","unstructured":"R. Vulanovi\u0107, Exponential fitting and special meshes for solving singularly perturbed problems, IV Conference on Applied Mathematics (B. Vrdoljak, ed.) 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