×

zbMATH — the first resource for mathematics

Mixed finite element approximation of the vector potential. (English) Zbl 0596.76073
We consider a mixed finite element approximation of the three-dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddle- point formulation of the essential boundary conditions.
In particular, this avoids the wellknown Babuška paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/ the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.

MSC:
76H05 Transonic flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Adams, R.A.: Sobolev spaces. New York: Academic Press 1975 · Zbl 0314.46030
[2] Aubin, J.P.: Behaviour of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.21, 599-637 (1967) · Zbl 0276.65052
[3] Babu?ka, I.: The theory of small changes in the domain of existence in the theory of partial differential equations and its applications. In: Differential Equations and their Applications. New York: Academic Press 1963 · Zbl 0156.10301
[4] Babu?ka, I.: The finite element method with Lagrange multipliers. Numer. Math.20, 179-192 (1973) · Zbl 0258.65108 · doi:10.1007/BF01436561
[5] Bendali, A.: Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary element method. Report, Ecole Polytechnique, 1982
[6] Bendali, A., Dominguez, J.M., Gallic, S.: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl.107, 537-560 (1985) · Zbl 0591.35053 · doi:10.1016/0022-247X(85)90330-0
[7] Bernardi, C.: M?thodes d’?l?ments finis mixtes pour les ?quations de Navier-Stokes. Th?se de 3me cycle, Universit? Paris VI, 1979
[8] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO, Anal. Numer.8, 129-151 (1974) · Zbl 0338.90047
[9] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978 · Zbl 0383.65058
[10] Dominguez, J.M.: Formulation on potentiel vecteur du syst?me de Stokes dans un domaine de ?3. Rapport 88015, Universit? Paris VI, 1983
[11] ElDabaghi, F., Pironneau, O.: Vecteur de courant et fluides parfaits en a?rodynamique num?rique tridimensionelle. Numer. Math.48, 561-589 (1986) · Zbl 0625.76009 · doi:10.1007/BF01389451
[12] Gobert, J.: Sur une in?galit? de co?rcivit?. J. Math. Anal. Appl.36, 518-528 (1971) · Zbl 0221.35016 · doi:10.1016/0022-247X(71)90035-7
[13] Nitsche, J.A.: Ein Kriterium f?r die Quasi-Optimalit?t des Ritzschen Verfahrens. Numer. Math.11, 346-348 (1968) · Zbl 0175.45801 · doi:10.1007/BF02166687
[14] Verf?rth, R.: Finite element approximation of stationary Navier-Stokes equations with slip boundary condition. Habilitationsschrift, Report Nr. 75, Univ. Bochum, 1986 · Zbl 0611.76030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.