A finite element approximation of three dimensional motion of a Bingham fluid

— In this paper, we approximate solutions of an initiai-boundary value problem associated with the motion of a Bingham fluid in a three dimensional domain. The method o f approximation consists of the backward Euler scheme in the time variable and conforming piecewise hnear finite éléments in the space variables augmented by the penalty method. The convergence of this scheme is proved under a mild assumption on the data. Error estimâtes are also obtained when the data satisfy restrictive assumptions Résumé. — Dans cet article nous approximons les solutions d'un problème aux conditions aux limites et valeurs initiales, associé au mouvement d'un fluide de Bingham dans un domaine tridimensionnel. La méthode de discrétisation se compose d'un schéma d'Euler en temps et d'éléments finis conformes linéaires par morceaux en espace avec pénalisation La convergence de ce schéma est démontrée moyennant une hypothèse faible sur les données Des estimations d'erreur sont aussi obtenues lorsque les données satisfont des hypothèses restrictives supplémentaires 0. INTRODUCTION The purpose of this paper is to discuss a certain finite element method to approximate solutions of an initiai-boundary value problem associated with the motion of a Bingham fluid in a three dimensional domain. According to Duvaut and Lions [4], [5], the initiai-boundary value problem is formulated as / " M > u j + a(u, w -u) + b(u,u, w) + + / (*>)-ƒ(«) s* ( / , w u ) in ( 0 , r ) , (0.1) (*) Received m October 1987. This research was supportée! by AFOSR under contract AFOSR-86-0085 and by NSF-grant DMS-8521848. (*) Department of Mathematics Virginia Polytechmc Institute and State University, Blacksburg, VA 24061-4097. MAN Modélisation mathématique et Analyse numérique 0399-0516/89/02/293/41/$ 6.10 Mathematical Modellmg and Numencal Analysis © AFCET Gauthier-Vülars 294 J. u. KIM for each test function w such that V. w — 0 in n and w = 0 on an, V . M = 0 in ü x ( O J ) , (0.2) M = o on an x [o, T], (0.3) u(x,O) = uo(x) in n . (0.4) Here, n is a bounded convex domain in R with smooth boundary 6n, u(x, t) dénotes the velocity of the fluid and f(x, t) stands for external force. The density, the yield limit and the viscosity are assumed to be positive constants. In particular, the density is taken to be one. We employ the notation : 3 f a(u, w) = V 2 \x \ Dl}(u) Dl}(w) dx, JJL = viscosity " v ' 2\dx, dx, / ' J(u) = 2 g \ Dn(u) m dx , gf = yield limit J £1 1 3