Abstract
A convergence analysis for a piecewise quadratic finite element method to solve vector biharmonic problems inR 3 in primal variables is presented. The application to the equations of the vector potential for the flow of incompressible viscous fluids is focused. For simplicity, only the particular case of stationary stokesian flows is treated in detail, but it is showed that the method applies as well to more complex flows of such fluids.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams Sobolev Space. Academic Press, New York, 1975.
O. Alcino Mendes, Resolução numérica do problema de Stokes em formulação potencial vector por um método de elementos finitos parametrado. Doctoral Dissertation, Pontifícia Universidade Católica do Rio de Janeiro, Brazil, 1988, in press.
P.G. Ciarlet, The finite Element Method for Elliptic Problems. North-Hollland, Amsterdam, 1978.
B. Dupire, Problemas variacionais, sua aproximação e formulações mistas. Doctoral Dissertation, Pontificia Universidade Católica do Rio de Janeiro, 1985.
T. Fujino, An analysis of hydrodynamic and plate structure problems by the finite element method. Recent Advances in Matrix Methods in Structural Analysis and Design (eds. R.H. Gallagher, Y. Yamada, T.J. Oden,), The University of Alabama Press, 1971, 20–24.
S.M. Gallic, Système de Stokes stationnaire en dimension 3; Formulation en ψ et formulation enu, p dans le cas axisymétrique. Thèse de Doctorat de 3e Cycle, Université Pierre et Marie Curie, Paris, 1982.
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, New York, 1985.
V.A. Kondratchev, Boundary value problems for elliptic equations with conical angular points. Trans. Moscow Math. Soc.16 (1967), 227–313.
P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem. RAIRO Série Rouge, Analyse Numérique R-1, 1975, 9–53.
L.S.D. Morley, On the constant moment plate bending element. J. Strain Anal.,6 (1971), 20–24.
A. Muzzio and L. Quartapelle, Decoupled solution of vector Poisson equations with boundary condition compling. Technical Report of the Istituto di Fisica del Politecnico di Milano, Milan, Italy, July 1987.
L. Quartapelle and V. Ruas, Uncoupled finite element solution of biharmonic problems for vector potentials. Proceedings of the International Conference on Finite Element Methods in Flow Problems, Huntsville, Ala., April 1989, 1104–1111.
V. Ruas Santos, Finite element solution of three-dimensional viscous flow problems using nonstandard degrees of freedom. Japan J. Appl. Maths.,2 (1985), 415–431.
V. Ruas, A quadratic finite element for solving biharmonic problems inR n. Numer. Math.,52 (1988), 33–43.
V. Ruas, Sur la formulation variationnelle des équations d’écoulements visqueux en fonction du potentiel vecteur. CRAS de Paris, Sér. II,306 (1988), 677–680.
R. Témam, Navier-Stokes Equations. North-Holland, Amsterdam, 1977.
Author information
Authors and Affiliations
About this article
Cite this article
Ruas, V. Approximation of the vector potential for viscous incompressible flow via the constant stress finite element. Japan J. Appl. Math. 7, 345–362 (1990). https://doi.org/10.1007/BF03167848
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167848