{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T09:37:49Z","timestamp":1791452269811,"version":"4.3.3"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We analyze the error of a finite element domain embedding method for elliptic equations on a domain $\\omega$ with curved boundary. The domain is embedded in a rectangle R on which uniform mesh and linear continuous elements are employed. The numerical scheme is based on an extension of the differential equation from $\\omega$ to R by regularization with a small parameter $\\epsilon$ (for Neumann and Robin problems), or penalty with a large parameter $\\epsilon^{-1}$ (for the Dirichlet problem), or a mixture of these (for a mixed boundary value problem). For Neumann and Robin problems, we prove that when $\\epsilon\\le h$ (the mesh size), the error in the $H^1(\\omega)$ norm is of the optimal order $\\mathcal{O}(h)$. For the Dirichlet problem, when $\\epsilon\\le h^{1\/2}$, the error is $\\mathcal{O}(h^{1\/2})$ that is not optimal. If the mesh is adjusted around $\\partial\\omega$ to fit it, then the optimal convergence rate $\\mathcal{O}(h)$ holds for the Dirichlet problem if $\\omega$ is convex and $\\epsilon\\le h$. If $\\omega$ is not convex, then the convergence rate can only be improved to $\\mathcal{O}(h^{2\/3})$ by such mesh adjustment, with the parameter being $\\epsilon=h^{2\/3}$. In this latter case, a parameter smaller than $h^{2\/3}$ thwarts the convergence rate, which is verified by a numerical result.<\/jats:p>","DOI":"10.1137\/060671681","type":"journal-article","created":{"date-parts":[[2008,8,1]],"date-time":"2008-08-01T18:08:34Z","timestamp":1217614114000},"page":"2843-2866","source":"Crossref","is-referenced-by-count":15,"title":["Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids"],"prefix":"10.1137","volume":"46","author":[{"given":"Sheng","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,8,1]]},"reference":[{"key":"R1","unstructured":"R.A. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2005.09.046"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0726074"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1973-0351118-5"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"N.S. Bakhvalov and A.V. Knyazev,\n                      Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients\n                      , in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 180, D. Keyes and J. Xu, eds., AMS, Providence, RI, 1994.","DOI":"10.1090\/conm\/180\/01968"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503426074"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0727068"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0727055"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"F. Brezzi, J.L. Lions, and O. Pironneau,\n                      The Chimera method for a model problem\n                      , in Numerical Mathematics and Advanced Applications -ENUMATH 2001, F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, eds., Springer, New York, 2003, pp. 817\u2013826.","DOI":"10.1007\/978-88-470-2089-4_74"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"P.G. Ciarlet,\n                      Finite Element Method for Elliptic Equations\n                      , North-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R11","unstructured":"S. Del Pino and O. Pironneau,\n                      A fictitious domain based general PDE solver\n                      , in Numerical Methods for Scientific Computing: Variational Problems and Applications, Y. Kuznetsov, P. Neittanmaki, and O. Pironneau, eds., Internat. Center Numer. Methods Eng., Barcelona, 2003."},{"key":"R12","unstructured":"K.O. Friedrichs and H.B. Keller,\n                      A finite difference scheme for generalized Neumann problems\n                      , in Numerical Solutions of Partial Differential Equations, J.H. Bramble, ed., Academic Press, New York and London, 1966."},{"key":"R13","unstructured":"D. Gilbarg and N.S. Trudinger,\n                      Elliptic Partial Differential Equations of Second Order\n                      , Springer, Berlin, 1983."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02576766"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.0118"},{"key":"R16","unstructured":"P. Grisvard,\n                      Singularities in Boundary Value Problems\n                      , Masson, Springer, New York, 1992."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01378-3"},{"key":"R18","unstructured":"Yu. A. Kuznetsov,\n                      Domain decomposition and fictitious domain methods with distributed Lagrange multipliers\n                      , in Domain Decomposition Methods in Science and Engineering, N. Debit, M. Garbey, R. Hoppe, D. Keyes, Y. Kuznetsov, and J. Periaux, eds., Internat. Center Numer. Methods Eng., Barcelona, 2002."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0076-z"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903433819"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"E.M. Stein,\n                      Singular Integrals and Differentiability Properties of Functions\n                      , Princeton University Press, Princeton, NJ, 1970.","DOI":"10.1515\/9781400883882"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2005.10.004"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.06.025"},{"key":"R24","unstructured":"S. Zhang,\n                      Sharp Convergence Rate of Domain Embedding Methods for Various Boundary Conditions\n                      , http:\/\/www.math.wayne.edu\/~sheng\/embedding.pdf (July 14, 2008)."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060671681","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:22Z","timestamp":1787325262000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060671681"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060671681"],"URL":"https:\/\/doi.org\/10.1137\/060671681","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}