{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:45:55Z","timestamp":1776829555061,"version":"3.51.2"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,11,19]],"date-time":"2002-11-19T00:00:00Z","timestamp":1037664000000},"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                    The multigrid\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V\">\n                        <mml:semantics>\n                          <mml:mi>V<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">V<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle algorithm using the Richardson relaxation scheme as the smoother is studied in this paper. For second-order elliptic boundary value problems, the contraction number of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V\">\n                        <mml:semantics>\n                          <mml:mi>V<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">V<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps, without assuming full elliptic regularity. As a consequence, the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper V\">\n                        <mml:semantics>\n                          <mml:mi>V<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">V<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle convergence result of Braess and Hackbusch is generalized to problems without full elliptic regularity.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01361-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"507-525","source":"Crossref","is-referenced-by-count":28,"title":["Convergence of the multigrid \ud835\udc49-cycle algorithm for second-order boundary value problems without full elliptic regularity"],"prefix":"10.1090","volume":"71","author":[{"given":"Susanne","family":"Brenner","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,19]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1137\/0722038","article-title":"Sharp estimates for multigrid rates of convergence with general smoothing and acceleration","volume":"22","author":"Bank, Randolph E.","year":"1985","journal-title":"SIAM J. 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