{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:24:35Z","timestamp":1787336675145,"version":"3.56.0"},"reference-count":21,"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>In part I the authors reported on the design of a robust and versatile gradient-weighted moving finite element (GWMFE) code in one dimension and on its application to a variety of PDEs and PDE systems. This companion paper does 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. The many potential pitfalls in the design of GWMFE codes and the special features of the implicit one-dimensional (1D) and 2D codes which contribute to their robustness and efficiency are discussed at length in part I; this paper concentrates on issues unique to the 2D case. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of systems. A catalog of inner products which occur in GWMFE is given, with particular attention paid to those involving second-order operators. After presenting an example of the 2D phenomenon of grid collapse and discussing the need for long-time regularization, the paper reports on the application of the 2D code to several nontrivial problems---nonlinear arsenic diffusion in the manufacture of semiconductors, the drift-diffusion equations for semiconductor device simulation, the Buckley--Leverett black oil equations for reservoir simulation, and the motion of surfaces by mean curvature.<\/jats:p>","DOI":"10.1137\/s1064827594269561","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"766-798","source":"Crossref","is-referenced-by-count":55,"title":["Design and Application of a Gradient-Weighted Moving Finite Element Code II: in Two Dimensions"],"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","first-page":"766","volume":"19","author":"Carlson Neil","year":"1998","journal-title":"SIAM J. Sci. Comput."},{"key":"R2","unstructured":"K. Miller,\n                      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":"R3","unstructured":"Neil Carlson, Keith Miller, Gradient weighted moving finite elements in two dimensions, ICASE\/NASA LaRC Ser., Springer, New York, 1988, 151\u201316489i:65094"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994260884"},{"key":"R5","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":"R6","unstructured":"L. H\u00f6rmander,\n                      Linear Partial Differential Operators\n                      , Springer, 1969."},{"key":"R7","unstructured":"Keith Miller, Alternate modes to control the nodes in the moving finite element method , SIAM, Philadelphia, PA, 1983, 165\u201318286j:65133"},{"key":"R8","unstructured":"N. N. Carlson and K. Miller,\n                      Design and application of a gradient\u2010weighted moving finite code, Part II, in 2\u2010D\n                      , Technical Report 237, Center for Applied Mathematics, Purdue University, West Lafayette, IN, May 1994."},{"key":"R8","unstructured":"This report is an earlier unabridged version of the present paper."},{"key":"R9","unstructured":"T. J. R. Hughes,\n                      The Finite Element Method\n                      , Prentice\u2010Hall, Inc., 1987."},{"key":"R10","unstructured":"C. P. Please and P. K. Sweby,\n                      A transformation to assist numerical solution of diffusion equations\n                      , Numer. Anal. Report 5\/86, Department of Mathematics, University of Reading, Reading, UK, 1986."},{"key":"R11","unstructured":"P. A. Markowich,\n                      Semiconductor Equations\n                      , Springer\u2010Verlag, 1990."},{"key":"R12","unstructured":"D. W. Peaceman,\n                      Fundamentals of Numerical Resevoir Simulation\n                      , Elsevier North\u2010Holland, Amsterdam, 1977."},{"key":"R13","unstructured":"K. A. Brakke,\n                      The Motion of a Surface by its Mean Curvature\n                      , Princeton University Press, Princeton, NJ, 1978."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1214446559"},{"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","unstructured":"P. Marcellini and K. Miller,\n                      Regularization for the equation of prescribed mean curvature and for motion by mean curvature\n                      , in Curvature Flows and Related Topics, A. Damlamian, ed., Gakkotosho, Tokyo, 1995."},{"key":"R17","unstructured":"Proceedings of the Conference at Levico, Italy, June 1994."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385643"},{"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."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827594269561","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\/S1064827594269561"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["10.1137\/S1064827594269561"],"URL":"https:\/\/doi.org\/10.1137\/s1064827594269561","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}