{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T09:08:59Z","timestamp":1774688939794,"version":"3.50.1"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"delta\">\n  <mml:semantics>\n    <mml:mi>\u03b4<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\delta<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript r Baseline left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>H<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>r<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">H^{r}(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to r less-than-or-equal-to 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>0<\/mml:mn>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>r<\/mml:mi>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">0\\le r \\le 2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, \u201chigher than optimal order\u201d error estimates can be obtained with respect to the mesh parameter <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>. Several examples of semilinear elliptic partial differential equations featuring different situations of existence\/nonexistence, uniqueness\/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.<\/p>","DOI":"10.1090\/s0025-5718-96-00743-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:28Z","timestamp":1027721668000},"page":"943-982","source":"Crossref","is-referenced-by-count":29,"title":["Boundary element monotone iteration scheme for semilinear elliptic partial differential equations"],"prefix":"10.1090","volume":"65","author":[{"given":"Yuanhua","family":"Deng","sequence":"first","affiliation":[]},{"given":"Goong","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Wei-Ming","family":"Ni","sequence":"additional","affiliation":[]},{"given":"Jianxin","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions, with formulas, graphs and mathematical tables","year":"1966"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/0022-0396(76)90126-1","article-title":"Supersolutions, monotone iterations, and stability","volume":"21","author":"Amann, Herbert","year":"1976","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1016\/0022-1236(73)90051-7","article-title":"Dual variational methods in critical point theory and applications","volume":"14","author":"Ambrosetti, Antonio","year":"1973","journal-title":"J. Functional Analysis"},{"key":"4","first-page":"1","article-title":"Survey lectures on the mathematical foundations of the finite element method","author":"Babu\u0161ka, Ivo","year":"1972"},{"key":"5","unstructured":"C.A. Brebbia and S. Walker, Boundary Element Techniques in Engineering, Newnes-Butterworths, London, 1980."},{"key":"6","series-title":"Computational Mathematics and Applications","isbn-type":"print","volume-title":"Boundary element methods","author":"Chen, Goong","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/012170940X"},{"key":"7","unstructured":"G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Vol. II: WKB and Wave Methods, Visualization and Experimentation CRC Press, Boca Raton, Florida, 1993."},{"key":"8","unstructured":"Y. Deng, Boundary element methods for nonlinear partial differential equations,, Ph.D. dissertation, Math. Dept., Texas A&M Univ., College Station, Texas, August 1994."},{"key":"9","first-page":"55","article-title":"Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anistropic inhomogeneous elasticity","author":"Fichera, Gaetano","year":"1961"},{"issue":"4","key":"10","first-page":"299","article-title":"The Aubin-Nitsche lemma for integral equations","volume":"3","author":"Hsiao, G. C.","year":"1981","journal-title":"J. Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0163-5549","issn-type":"print"},{"key":"11","volume-title":"Quelques m\\'{e}thodes de r\\'{e}solution des probl\\`emes aux limites non lin\\'{e}aires","author":"Lions, J.-L.","year":"1969"},{"key":"12","unstructured":"W.M. Ni, Some aspects of semilinear elliptic equations, Lecture Notes published by Institute of Mathematics, National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China, May, 1987."},{"key":"13","volume-title":"Maximum principles in differential equations","author":"Protter, Murray H.","year":"1967"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/BF01396763","article-title":"On the convergence of the Galerkin method for nonsmooth solutions of integral equations","volume":"54","author":"Ruotsalainen, K.","year":"1988","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"15","first-page":"385","article-title":"On the boundary element method for a nonlinear boundary value problem","author":"Ruotsalainen, K.","year":"1987"},{"key":"16","unstructured":"M. Sakakihara, An iterative boundary integral equation method for mildly nonlinear elliptic partial differential equations, Boundary Elements VII (C.A. Brebbia and G. Maier, ed.), vol. . II, Springer-Verlag, Berlin-Heidelberg, 1985, pp. 13.49-13.58."},{"key":"17","series-title":"Lecture Notes in Mathematics, Vol. 309","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0060079","volume-title":"Topics in stability and bifurcation theory","author":"Sattinger, David H.","year":"1973"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00743-0\/S0025-5718-96-00743-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00743-0\/S0025-5718-96-00743-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T23:05:32Z","timestamp":1635894332000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00743-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":17,"journal-issue":{"issue":"215","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["S0025-5718-96-00743-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00743-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996]]}}}