{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:28:15Z","timestamp":1776842895793,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Superscript 4 left-parenthesis d minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>d<\/mml:mi>\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/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\">p^{4(d-1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that \u201cregardless of the choice of basis, the condition numbers grow like\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Superscript 4 d\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">p^{4d}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    or faster\". Numerical results are also presented which verify that our theoretical bounds are correct.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00983-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1423-1450","source":"Crossref","is-referenced-by-count":20,"title":["Bounds for eigenvalues and condition numbers in the \ud835\udc5d-version of the finite element method"],"prefix":"10.1090","volume":"67","author":[{"given":"Ning","family":"Hu","sequence":"first","affiliation":[]},{"given":"Xian-Zhong","family":"Guo","sequence":"additional","affiliation":[]},{"given":"I.","family":"Katz","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1137\/0718033","article-title":"The \ud835\udc5d-version of the finite element method","volume":"18","author":"Babu\u0161ka, I.","year":"1981","journal-title":"SIAM J. 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