zbMATH — the first resource for mathematics

Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries. (English) Zbl 0543.65074
Author’s summary: Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an integral approximation of the dual problem for second order elliptic equations is defined. A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation can be obtained by solving a system of linear algebraic equations with a positive definite matrix.
Reviewer: D.R.Westbrook

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX Cite
Full Text: EuDML
[1] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0383.65058
[2] P. G. Ciarlet P. A. Raviart: Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. 1 (1972), 217-249. · Zbl 0261.65079
[3] P. Doktor: On the density of smooth functions in certain subspaces of Sobolev spaces. Comment. Math. Univ. Carolin. 14, 4 (1973), 609-622. · Zbl 0268.46036
[4] B. M. Fraeijs de Veubeke M. Hogge: Dual analysis for heat conduction problems by finite elements. Internat. J. Numer. Methods Engrg. 5 (1972), 65-82. · Zbl 0251.65061
[5] V. Girault P. A. Raviart: Finite element approximation of the Navier-Stokes equations. Springer-Verlg, Berlin, Heidelberg, New York, 1979. · Zbl 0413.65081
[6] J. Haslinger I. Hlaváček: Contact between elastic perfectly plastic bodies. Apl. Mat. 27 (1982), 27-45. · Zbl 0495.73094
[7] J. Haslinger I. Hlaváček: Convergence of a finite element method based on the dual variational formulation. Apl. Mat. 21 (1976), 43 - 65.
[8] I. Hlaváček: The density of solenoidal functions and the convergence of a dual finite element method. Apl. Mat. 25 (1980), 39-55.
[9] M. Křížek: Conforming equilibrium finite element methods for some elliptic plane problems. RAIRO Anal. Numer. 17 (1983), 35–65. · Zbl 0541.76003
[10] O. A. Ladyzenskaya: The mathematical theory of viscous incompressible flow. Gordon & Breach, New York, 1969.
[11] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[12] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York, 1981.
[13] P. Neittaanmäki J. Saranen: On finite element approximation of the gradient for solution of Poisson equation. Numer. Math. 37 (1981), 333-337. · Zbl 0463.65073
[14] J. Penman J. R. Fraser: Complementary and dual energy finite element principles in magnetostatics. IEEE Trans. on Magnetics 18 (1982), 319-324.
[15] G. Strang G. J. Fix: An analysis of the finite element method. Prentice Hall, New Jersey, 1973. · Zbl 0278.65116
[16] J. M. Thomas: Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes. Thesis, Université Paris VI, 1977.
[17] M. Zlámal: Curved elements in the finite element method. Čislennyje metody mechaniki splošnoj sredy, SO AN SSSR, 4 (1973), No. 5, 25-49.
[18] M. Zlámal: Curved elements in the finite element method. SIAM J. Numer. Anal. 10 (1973), 229-240. · Zbl 0285.65067
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.