{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:01:56Z","timestamp":1776769316894,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study the uniform approximation of boundary layer functions\n \n \n \n \n exp<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n x<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\exp (-x\/d)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n x<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n x\\in (0,1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n d<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n d\\in (0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , by the\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n h<\/mml:mi>\n p<\/mml:mi>\n <\/mml:mrow>\n hp<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n versions of the finite element method. For the\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version (with fixed mesh), we prove super-exponential convergence in the range\n \n \n \n \n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n ><\/mml:mo>\n e<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 2<\/mml:mn>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n p + 1\/2 > e\/(2d)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also establish, for this version, an overall convergence rate of\n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n (<\/mml:mo>\n \n p<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:msqrt>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathcal O}(p^{-1}\\sqrt {\\ln p})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the energy norm error which is\n uniform<\/italic>\n in\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and show that this rate is sharp (up to the\n \n \n \n \n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:msqrt>\n \\sqrt {\\ln p}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n term) when\n robust<\/italic>\n estimates uniform in\n \n \n \n \n d<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n d\\in (0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are considered. For the\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version with variable mesh (i.e., the\n \n \n \n \n h<\/mml:mi>\n p<\/mml:mi>\n <\/mml:mrow>\n hp<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version), we show that exponential convergence, uniform in\n \n \n \n \n d<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n d\\in (0,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , is achieved by taking the first element at the boundary layer to be of size\n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n (<\/mml:mo>\n p<\/mml:mi>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathcal O}(pd)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is as small as, e.g.,\n \n \n \n \n 10<\/mml:mn>\n \n \n \u2212\n \n <\/mml:mo>\n 8<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n 10^{-8}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . They also illustrate the superiority of the\n \n \n \n \n h<\/mml:mi>\n p<\/mml:mi>\n <\/mml:mrow>\n hp<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n approach over other methods, including a low-order\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n version with optimal \u201cexponential\" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00781-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1403-1429","source":"Crossref","is-referenced-by-count":116,"title":["The \ud835\udc5d and \u210e\ud835\udc5d versions of the finite element method for problems with boundary layers"],"prefix":"10.1090","volume":"65","author":[{"given":"Christoph","family":"Schwab","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manil","family":"Suri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"D. N. Arnold and R. S. Falk. 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Boundary layer resolving methods for singularly perturbed problems, submitted to I.M.A. J. Numer. Anal."},{"key":"9","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1017\/S0305004100035945","article-title":"Error bounds for the Liouville-Green (or \ud835\udc4a\ud835\udc3e\ud835\udc35) approximation","volume":"57","author":"Olver, F. W. J.","year":"1961","journal-title":"Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-1981","issn-type":"print"},{"key":"10","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0122947606"},{"key":"11","doi-asserted-by":"crossref","unstructured":"H. Hakula, Y. Leino, and J. Pitk\u00e4ranta. Scale resolution, layers and high-order numerical modeling of shells, to appear in Comp. Meth. Appl. Mech. Eng., 1996.","DOI":"10.1016\/0045-7825(95)00939-6"},{"key":"12","unstructured":"H. Kraus. 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