{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:15:00Z","timestamp":1787328900314,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schr\u00f6dinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in $L^2$ and $H^1$ are proved for the associated semidiscrete and fully discrete Crank\u2013Nicolson\u2013Galerkin approximations. The problem involving the Schr\u00f6dinger equation is motivated by considering the standard \u201cparabolic\u201d (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long-range sound propagation in the sea, in the specific case of a domain with a rigid bottom of variable topography. This model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations. In the (real) parabolic case, numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type.<\/jats:p>","DOI":"10.1137\/070710858","type":"journal-article","created":{"date-parts":[[2009,8,6]],"date-time":"2009-08-06T18:03:14Z","timestamp":1249581794000},"page":"2752-2781","source":"Crossref","is-referenced-by-count":9,"title":["Galerkin Methods for Parabolic and Schr\u00f6dinger Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics"],"prefix":"10.1137","volume":"47","author":[{"given":"D. C.","family":"Antonopoulou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V. A.","family":"Dougalis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. E.","family":"Zouraris","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,8,6]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670130503"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1121\/1.399089"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1991250606431"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999367460"},{"key":"R5","unstructured":"D. C. Antonopoulou,\n                      Theory and Numerical Analysis of Parabolic Approximations\n                      , Ph.D. thesis (in Greek), University of Athens, Athens, Greece, 2006."},{"key":"R6","unstructured":"D. C. Antonopoulou, V. A. Dougalis, and G. E. Zouraris,\n                      Galerkin methods for parabolic and Schr\u00f6dinger equations with dynamical boundary conditions and applications to underwater acoustics\n                      , arXiv:0904.3900."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0148005"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"C. Bandle and W. Reichel,\n                      A linear parabolic problem with non-dissipative dynamical boundary conditions\n                      , in Recent Advances on Elliptic and Parabolic Issues, Proceedings of the 2004 Swiss-Japanese Seminar, M. Chipot and H. Ninomiya, eds., World Scientific, River Edge, NJ, 2006, pp. 46\u201379.","DOI":"10.1142\/9789812774170_0003"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.4171\/RLM\/453"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0707006"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Texts Appl. Math. 15, Springer-Verlag, New York, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R12","unstructured":"J. Crank,\n                      The Mathematics of Diffusion\n                      , 2nd ed., Clarendon Press, Oxford, 1975."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"V. A. Dougalis, N. A. Kampanis, F. Sturm, and G. E. Zouraris,\n                      Numerical solution of the parabolic equation in range-dependent waveguides\n                      , in Effective Computational Methods for Wave Propagation, N. A. Kampanis et al., eds., Chapman and Hall\/CRC, Boca Raton, FL, 2008, pp. 175\u2013207.","DOI":"10.1201\/9781420010879.ch6"},{"key":"R14","unstructured":"V. A. Dougalis and G. E. Zouraris,\n                      Finite difference methods for the parabolic equation with interface conditions\n                      , to appear."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1080\/03605309308820976"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1121\/1.399448"},{"key":"R17","unstructured":"D. Lee and G. Botseas,\n                      IFD: An Implicit Finite-Difference Computer Model for Solving the Parabolic Equation\n                      , NUSC Report No. 6659, Naval Underwater Systems Center, New London, CT, 1982."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1121\/1.386918"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(87)90206-9"},{"key":"R20","unstructured":"J. L. Lions and E. Mag\u00e9nes,\n                      Probl\u00e8mes aux Limites Non Homog\u00e8nes et Applications, Vol\n                      . 2, Travaux et Recherches Math\u00e9matiques 17, Dunod, Paris, 1968."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1121\/1.400525"},{"key":"R22","unstructured":"F. Sturm,\n                      Mod\u00e9lisation math\u00e9matique et num\u00e9rique d'un probl\u00e8me de propagation en acoustique sous-marine: Prise en compte d'un environnement variable tridimensionnel\n                      , Th\u00e8se de Docteur en Sciences, Universit\u00e9 de Toulon et du Var, La Garde, France, 1997."},{"key":"R23","doi-asserted-by":"crossref","unstructured":"F. D. Tappert,\n                      The parabolic approximation method\n                      , in Wave Propagation and Underwater Acoustics, J. B. Keller and J. S. Papadakis, eds., Lecture Notes in Phys. 70, Springer-Verlag, Berlin, 1977, pp. 224\u2013287.","DOI":"10.1007\/3-540-08527-0_5"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"V. Thom\u00e9e,\n                      Galerkin finite element methods for parabolic problems\n                      , Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 1997.","DOI":"10.1007\/978-3-662-03359-3"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1080\/03605300801970960"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0710076"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070710858","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:07:09Z","timestamp":1787324829000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070710858"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":26,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/070710858"],"URL":"https:\/\/doi.org\/10.1137\/070710858","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}