Previous |  Up |  Next


Rothe's method; finite elements.; Euler’s backward formula; linear parabolic or hyperbolic equations; convergence
Existence and finite element approximation of a hyperbolic-parabolic problem is studied. The original two-dimensional domain is approximated by a polygonal one (external approximations). The time discretization is obtained using Euler’s backward formula (Rothe’s method). Under certain smoothing assumptions on the data (see (2.6), (2.7)) the existence and uniqueness of the solution and the convergence of Rothe’s functions in the space $C(\overline{I},V)$ is proved.
[1] Demerdash, N.A., Gillot, D.H.: A new approach for determination of eddy current and flux penetration in nonlinear ferromagnetic materials. IEEE Trans. MAG-10 (1974), 682–685.
[2] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie, Berlin, 1974. MR 0636412
[3] Hannalla, A.Y., Macdonald, D.C.: Numerical analysis of transient field problems in electrical machines. Proc. IEE 123 (1976), 893–898.
[4] Kačur, J.: Method of Rothe in Evolution Equations. Teubner-Texte zur Mathematik, Band 80, Leipzig, 1985. MR 0834176
[5] Lütke-Daldrup, B.: Numerische Lösung zweidimensionaler nichtlinearer instantionärer Feld- und Wirbelstromprobleme. Archiv für Elektrotechnik 68 (1985), 223–228. DOI 10.1007/BF01575911
[6] Melkes, F., Zlámal M.: Numerical solution of nonlinear quasistationary magnetic fields. Internat. J. Numer. Methods Engineering 19 (1983), 1053–1062. DOI 10.1002/nme.1620190709
[7] Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10 (1973), 229–240. DOI 10.1137/0710022 | MR 0395263
[8] Zlámal, M.: Finite element solution of quasistationary nonlinear magnetic field. RAIRO Anal. Numér. 16 (1982), 161–191. MR 0661454
[9] Ženíšek A.: Finite element variational crimes in parabolic-elliptic problems. Numer. Math. 55 (1989), 343–376. DOI 10.1007/BF01390058 | MR 0993476
[10] Ženíšek, A.: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations. Academic Press, London, 1990. MR 1086876
Partner of
EuDML logo