{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:49:06Z","timestamp":1776836946453,"version":"3.51.2"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"343","license":[{"start":{"date-parts":[[2024,5,4]],"date-time":"2024-05-04T00:00:00Z","timestamp":1714780800000},"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                    A singularly perturbed convection-diffusion problem posed on the unit square in\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R squared\">\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:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than\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 Superscript minus left-parenthesis k plus 1 slash 2 right-parenthesis 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>N<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/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:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(N^{-(k+1\/2)})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    accurate in the energy norm induced by the bilinear form of the weak formulation, where\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                    mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish\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 Superscript minus left-parenthesis k plus 1 right-parenthesis 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>N<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(N^{-(k+1)})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This supercloseness property implies a new\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N Superscript minus left-parenthesis k plus 1 right-parenthesis\">\n                        <mml:semantics>\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:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">N^{-(k+1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    bound for the\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                    error between the LDG solution on each type of mesh and the true solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.\n                  <\/p>","DOI":"10.1090\/mcom\/3844","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T09:38:48Z","timestamp":1678873128000},"page":"2065-2095","source":"Crossref","is-referenced-by-count":20,"title":["Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem"],"prefix":"10.1090","volume":"92","author":[{"given":"Yao","family":"Cheng","sequence":"first","affiliation":[]},{"given":"Shan","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Stynes","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,5,4]]},"reference":[{"issue":"238","key":"1","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0025-5718-01-01317-5","article-title":"Optimal a priori error estimates for the \u210e\ud835\udc5d-version of the local discontinuous Galerkin method for convection-diffusion problems","volume":"71","author":"Castillo, Paul","year":"2002","journal-title":"Math. 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