We consider an implicit fractional step procedure for the time discretization of the non-stationary Stokes equations in smoothly bounded domains of ℝ³. We prove optimal convergence properties uniformly in time in a scale of Sobolev spaces, under a certain regularity of the solution. We develop a representation for the solution of the discretized equations in the form of potentials and the uniquely determined solution of some system of boundary integral equations. For the numerical computation of the potentials and the solution of the boundary integral equations a boundary element method of collocation type is carried out.
@article{bwmeta1.element.bwnjournal-article-bcpv29z1p135bwm,
author = {Varnhorn, Werner},
title = {Finite differences and boundary element methods for non-stationary viscous incompressible flow},
journal = {Banach Center Publications},
volume = {29},
year = {1994},
pages = {135-154},
zbl = {0801.76063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv29z1p135bwm}
}
Varnhorn, Werner. Finite differences and boundary element methods for non-stationary viscous incompressible flow. Banach Center Publications, Tome 29 (1994) pp. 135-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv29z1p135bwm/
[000] [
[001] [1] W. Borchers, Über das Anfangsrandwertproblem der instationären Stokes Gleichung, Z. Angew. Math. Mech. 65 (1985), T329-T330. | Zbl 0614.76030
[002] [2] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Sem. Mat. Univ. Padova 31 (1964), 308-340. | Zbl 0116.18002
[003] [3] M. Costabel, Principles of boundary element methods, preprint 998, Technische Hochschule Darmstadt, 1986.
[004] [4] P. Deuring, W. von Wahl and P. Weidemaier, Das lineare Stokes-System im ℝ³ (1. Vorlesungen über das Innenraumproblem), Bayreuth. Math. Schr. 27 (1988), 1-252. | Zbl 0667.35059
[005] [5] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal. 16 (1964), 269-315. | Zbl 0126.42301
[006] [6] F. K. Hebeker, Efficient boundary element methods for three-dimensional exterior viscous flow, Numer. Methods Partial Differential Equations 2 (1986), 273-297. | Zbl 0645.76035
[007] [7] G. C. Hsiao, P. Kopp and W. L. Wendland, A Galerkin collocation method for some integral equations of the first kind, Computing 25 (1980), 89-130. | Zbl 0419.65088
[008] [8] G. C. Hsiao, P. Kopp and W. L. Wendland, Some applications of a Galerkin-collocation method for boundary integral equations of the first kind, Math. Methods Appl. Sci. 6 (1984), 280-325. | Zbl 0546.65091
[009] [9] J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte Math. 80, Teubner, Leipzig 1985. | Zbl 0582.65084
[010] [10] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York 1969. | Zbl 0184.52603
[011] [11] P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (1956), 267-293. | Zbl 0072.08903
[012] [12] R. Leis, Vorlesungen über partielle Differentialgleichungen zweiter Ordnung, Bibliographisches Institut, Mannheim 1967.
[013] [13] M. Mc Cracken, The resolvent problem for the Stokes equations on halfspace in , SIAM J. Math. Anal. 12 (1981), 201-228.
[014] [14] F. K. G. Odquist, Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten, Math. Z. 32 (1930), 329-375. | Zbl 56.0713.04
[015] [15] W. I. Smirnow, Lehrgang der höheren Mathematik 4, Deutscher Verlag der Wissenschaften, Berlin 1979.
[016] [16] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam 1977.
[017] [17] W. Varnhorn, Efficient quadrature for a boundary element method to compute threedimensional Stokes flow, Internat. J. Numer. Methods Fluids 9 (1989), 185-191. | Zbl 0658.76034