{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T17:50:23Z","timestamp":1779126623242,"version":"3.51.4"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"285","license":[{"start":{"date-parts":[[2014,6,5]],"date-time":"2014-06-05T00:00:00Z","timestamp":1401926400000},"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                    Assume that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">V_h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a space of piecewise polynomials of a degree less than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">r\\geq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on a family of quasi-uniform triangulation 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                    . There exists the well-known upper bound of the approximation error by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">V_h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>V<\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">V_h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the upper-bound error estimate is also sharp.\n                  <\/p>\n                  <p>This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.<\/p>","DOI":"10.1090\/s0025-5718-2013-02724-x","type":"journal-article","created":{"date-parts":[[2013,6,5]],"date-time":"2013-06-05T18:42:18Z","timestamp":1370457738000},"page":"1-13","source":"Crossref","is-referenced-by-count":36,"title":["Lower bounds of the discretization error for piecewise polynomials"],"prefix":"10.1090","volume":"83","author":[{"given":"Qun","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hehu","family":"Xie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinchao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,6,5]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"211","key":"2","doi-asserted-by":"publisher","first-page":"943","DOI":"10.2307\/2153478","article-title":"On the implementation of mixed methods as nonconforming methods for second-order elliptic problems","volume":"64","author":"Arbogast, Todd","year":"1995","journal-title":"Math. 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