{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:24:58Z","timestamp":1787336698769,"version":"3.56.0"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:p>This paper reports on the design of a robust and versatile gradient-weighted moving finite element (GWMFE) code in one dimension and its application to a variety of difficult PDEs and PDE systems. A companion paper, part II, will do the same for the two-dimensional (2D) case. These moving node methods are especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of PDE systems. There are many possible pitfalls in the design of GWMFE codes; section 5 discusses special features of the implicit one-dimensional (1D) and 2D codes which contribute greatly to their robustness and efficiency. Section 6 uses a few simple examples to illustrate the workings of the method, some difficulties, and reasons for the standard choices of the internodal viscosity regularization coefficient. Section 7 reports numerical trials on several more difficult PDE systems. Section 8 discusses the failure of the method on certain steady-state convection problems. Section 9 describes a simple nonlinear \"Krylov subspace\" accelerator for Newton's method, a routine which greatly decreases the number of Jacobian evaluations required for our stiff ODE solver.<\/jats:p>","DOI":"10.1137\/s106482759426955x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"728-765","source":"Crossref","is-referenced-by-count":107,"title":["Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension"],"prefix":"10.1137","volume":"19","author":[{"given":"Neil N.","family":"Carlson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Keith","family":"Miller","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0718070"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0718071"},{"key":"R3","first-page":"57","volume":"67","author":"Alexander Roger","year":"1979","journal-title":"Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(81)90207-2"},{"key":"R5","unstructured":"M. J. Djomehri and K. Miller,\n                      A moving finite element code for general systems of PDE\u2019s in 2\u2010D\n                      , Center for Pure Appl. Math. Report PAM\u201057, University of California, Berkeley, October 1981."},{"key":"R6","unstructured":"M. J. Djomehri,\n                      Moving Finite Element Solution of Systems of Partial Differential Equations in One Dimension\n                      , PhD thesis, Department of Mathematics, University of California, Berkeley, CA, December 1982."},{"key":"R6","unstructured":"Center for Pure Appl. Math. Report PAM\u2010125, January 1983, unpublished."},{"key":"R7","unstructured":"M. J. Djomehri, S. Doss, R. Gelinas, and K. Miller,\n                      Applications of the moving finite element method for systems in 2\u2010D\n                      , Center for Pure Appl. Math. Report PAM\u2010427, University of California, Berkeley, September 1988."},{"key":"R8","unstructured":"K. Miller,\n                      Alternate modes to control the nodes in the moving finite element method\n                      , in Adaptive Computational Methods for Partial Differential Equations, SIAM, 1983, pp. 165\u2013182."},{"key":"R9","unstructured":"Recent results on finite element methods with moving nodes\n                      , in Accuracy Estimates and Adaptive Refinement in Finite Element Calculations, I. Babuska, O. C. Zienkiewicz, and et al., eds., Wiley and Sons, 1986, pp. 325\u2013338."},{"key":"R10","unstructured":"N. N. Carlson and K. Miller,\n                      Gradient weighted moving finite elements in two dimensions\n                      , in Finite Elements Theory and Application, D. L. Dwoyer, M. Y. Hussaini, and R. G. Voight, eds., Springer Verlag, 1988, pp. 151\u2013164."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0729006"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994260884"},{"key":"R13","unstructured":"N. N. Carlson and K. Miller,\n                      Design and application of a gradient\u2010weighted moving finite code, Part I, in 1\u2010D\n                      , Technical Report 236, Center for Applied Mathematics, Purdue University, May 1994."},{"key":"R13","unstructured":"This report is an earlier unabridged version of the present paper."},{"key":"R14","unstructured":"Design and application of a gradient\u2010weighted moving finite code, Part II, in 2\u2010D\n                      . Submitted to SIAM J. Sci. Comput., May 1994."},{"key":"R15","unstructured":"A. Kuprat,\n                      Creation and Annihilation of Nodes for the Moving Finite Element Method\n                      , PhD thesis, Department of Mathematics, University of California, Berkeley, CA, May 1992."},{"key":"R16","unstructured":"S. Nazari,\n                      Rotational Surfaces in Euclidean and Hyperbolic Spaces, Mean Curvature Motion, and the Moving Finite Element Method\n                      , PhD thesis, Department of Mathematics, University of California, Berkeley, CA, May 1993."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(90)90148-T"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90413-S"},{"key":"R19","unstructured":"P. A. Zegeling,\n                      Moving\u2010Grid Methods for Time\u2010Dependent Partial Differential Equations\n                      , PhD thesis, University of Amsterdam and Centre for Mathematics and Computer Science (CWI), Amsterdam, 1992."},{"key":"R19","unstructured":"Several of the chapters appear separately as [17] and [18]."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(89)90204-0"},{"key":"R21","unstructured":"I. Lie and J. O. Langseth,\n                      A moving finite element method with solution\u2010dependent norm, and the ODE\u2010aspects of the MFE equations\n                      , preprint, NRDE, Div. for Electronics, P. O. Box 25, N\u20102007 Kjellev, Norway, September 1988."},{"key":"R22","unstructured":"A. J. Wathen,\n                      Moving Finite Elements and Oil Reservoir Modeling\n                      , PhD thesis, Department of Mathematics, University of Reading, Reading, UK, 1984."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/5.2.161"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/0723051"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/7.4.449"},{"key":"R26","unstructured":"M. J. Baines,\n                      Locally adaptive moving finite elements\n                      , in Numerical Methods for Fluid Dynamics II, K. W. Morton and M. J. Baines, eds., Oxford University Press, 1986, pp. 1\u201314."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(86)90004-8"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90016-2"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90017-4"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/0728070"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/12.4.545"},{"key":"R32","unstructured":"M. J. Baines,\n                      Moving Finite Elements\n                      , Oxford University Press, 1994."},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620211110"},{"key":"R34","unstructured":"P. A. Markowich,\n                      Semiconductor Equations\n                      , Springer\u2010Verlag, 1990."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(78)90023-2"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/0904040"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/0911026"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/0723039"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S106482759426955X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:26:19Z","timestamp":1787333179000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S106482759426955X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["10.1137\/S106482759426955X"],"URL":"https:\/\/doi.org\/10.1137\/s106482759426955x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}