{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:29:23Z","timestamp":1776796163776,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"283","license":[{"start":{"date-parts":[[2013,10,30]],"date-time":"2013-10-30T00:00:00Z","timestamp":1383091200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper develops and analyzes two local discontinuous Galerkin (LDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are stable for all positive wave number\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and all positive mesh 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                    . Energy norm 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                    -norm error estimates are derived for both LDG methods in all mesh parameter regimes including pre-asymptotic regime (i.e.,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k squared h greater-than-or-equivalent-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>\n                              \u2273\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k^2 h\\gtrsim 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). To analyze the proposed LDG methods, they are recast and treated as (nonconforming) mixed finite element methods. The crux of the analysis is to show that the sesquilinear form associated with each LDG method satisfies a coercivity property in all mesh parameter regimes. These coercivity properties then easily infer the desired discrete stability estimates for the solutions of the proposed LDG methods. In return, the discrete stabilities not only guarantee the well-posedness of the LDG methods but also play a crucial role in the error analysis. Numerical experiments are also presented in the paper to validate the theoretical results and to compare the performance of the proposed two LDG methods.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02652-4","type":"journal-article","created":{"date-parts":[[2012,10,30]],"date-time":"2012-10-30T16:01:16Z","timestamp":1351612876000},"page":"1269-1296","source":"Crossref","is-referenced-by-count":33,"title":["Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number"],"prefix":"10.1090","volume":"82","author":[{"given":"Xiaobing","family":"Feng","sequence":"first","affiliation":[]},{"given":"Yulong","family":"Xing","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2012,10,30]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","article-title":"Unified analysis of discontinuous Galerkin methods for elliptic problems","volume":"39","author":"Arnold, Douglas N.","year":"2001","journal-title":"SIAM J. 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