Abstract
As is well known, variational methods belong to the fundamental principles in mathematics and in mechanics. They are of interest not only from the theoretical but also from the numerical point of view. Indeed, when discretized the variational principles immediately provide numerical schemes for solving the underlying problem numerically. In particular, variational principles in various forms are videly used in order to establish Ritz — Galerkin schemes in finite element spaces.
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References
COURANT, R. and HILBERT, D. : Methoden der mathematischen Physik. Springer-Verlag. Heidelberger Taschenbücher vol. 30 und 31.
TREFFTZ, E.: Ein Gegenstück zum Ritzschen Verfahren. Verhandl. d. 2. Internat. Kongreß f. techn. Mechanik. Zürich 1926, 131–138.
EKELAND, I. and TEMAM, R.: Convex analysis and variational problems. North-Holland Publ. Company, Amsterdam 1976.
SEWELL, M.J.: Dual approximation principles and optimization in continuum mechanics. Phil. Trans. Roy. Soc. (London) A 265 (1969) 319–351.
NOBLE, B., SEWELL, M.J.: On dual extremum principles in Applied Mathematics. J. Inst. Math. Appl. 9 (1972) 123–193.
ROBINSON, P.D.: Complementary variational principles. In: Rall, L.B. (Ed.): Nonlinear functional analysis and applications. Academic Press 1971, pp. 509–576.
ODEN, J.T. and REDDY, J.N.: Variational methods in theoretical mechanics. Springer-Verlag Berlin, Heidelberg, New York 1976.
VELTE, W.: Direkte Methoden der Variationsrechnung. Teubner Studienbücher Mathematik. Teubner Verlag Stuttgart 1976.
NECAS, J.: Les méthodes directes en théorie des équations elliptiques. Masson et Cie., Paris, Academia, Prag 1967.
DUVAUT, G. and LIONS, J.L.: Les inéquations an mécanique et en physique. Dunaud, Paris 1972.
KINDERLEHRER, D., STAMPACCHIA, G.: An introduction into variational inequalities and their applications. Academic Press 1980.
FICHERA, G.: Boundary value problems of elasticity with unilateral constraints. S. 392–424 in: Flügge, S. (Herausg.): Handbuch der Physik vol. VI a/2. Springer-Verlag Berlin, Heidelberg, New York.
LIONS, J.L., STAMPACCHIA, G.: Variational inequalities. Comm. Pure Appl. Math. 20 (1969) 493–519.
HASLINGER, J.: Dual finite element analysis for an inequality of the 2nd order. Aplikace Matematiky 24 (1979), 118–132.
BREZZI, F., HAGER, W.W., RAVIART, P.A.: Error estimates for the finite element solution of variational inequalities, part I. Primal theory. Part II. Mixed methods. Numer.Math. 28 (1977) 431–443. 31 (1978) 1–16.
RAVIART, P.A., THOMAS, J.M.: A mixed finite element method for 2nd oder elliptic problems. In: Mathematical aspects of the FEM (Rom, December 1975). Springer Lecture Notes in Math. vol. 606.
BREZZI, F.: On the existence, uniqueness and approximations of saddle-point problems arising from Lagrangian multipliers. R.A.I.R.O. Anal. Numer. 11 (1977) 209–216.
Additional references
ALLEN, G.: Variational inequalities, complementary problems and duality theorems. J. f. Math. Anal. and Appl. 58 (1977) 1–10.
ARTHURS, A.M.: Complementary variational principles. Oxford Math. Monographs, Oxford 1970.
COTTLE, R.V., GIANESSI, F., LIONS, J.-L.: Variational Inequalities and Complementary Problems. Theory and Applications. (Internat. School of Math., 19.–30. Juni 1978 in Erice Sicily) John Wiley and Sons 1980.
FRAEIJS DE VEUBEKE, B.: Displacement and equilibrium models in the finite element method. S. 145–197 in: O.C. Zienkiewicz, G.S. Holister (Eds.) Stress Analysis, John Wiley 1965.
FRAEIJS DE VEUBEEKE, B.: The duality principles of elastodynamics. Finite elements applications. S. 357–377 in: Lectures in finite element methods in continuum mechanics. Ed. by J.T. Oden, E.R. De Arantes e Oliveira The Univ. of Alabama Press, Huntsville 1973.
GWINNER, J.: Nichtlineare Variationsgleichungen mit Anwendungen. Verlag Haag u. Herchen, Frankfurt/M. 1978
GWINNER, J.: Bibliography on non-differentiable optimization and non-smooth analysis. [Vierhundert Referenzen.] J. of Computational and Applied Math. 7 (1981) 277–285.
PRAGER, W., SYNGE, J.L.: Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5 (1947) 241–259.
TONTI, E.: Variational principles in elastostatics. Mechanica 2 (1967) 201–208.
WERNER, B.: Complementary variational principles and non-conforming Trefftz elements. Internat. Series Numer.Math. (ISNM) vol. 56 (1981) 180–192.
BABUšKA, I.: The finite element method with Lagrangian multipliers. Nutner. Math. 20 (1973) 179–192.
BRAMBLE, J.H.: The Lagrange multiplier method for Dirichlet’s problem. Math, of Computation 155 (1981) 1–12.
BREZZI, F., RAVIART, P.A.: Mixed finite element methods for 4th order elliptic equations. In: J.H. Miller (ed.): Proc. Royal Irish Academy Conference on Numerical Analysis. Topics in Numerical Analysis III, 1976. Academic Press 1977, 33–56.
BUFLER, H.: Variationsgleichungen und finite Elemente. Bayer. Akad. d. Wiss. Math.-Nat. Klasse Bd. 1975, 155–187.
CIARLET, P.G., RAVIART, P.A.: A mixed finite element method for the biharmonic equation. In: C. de Boor (Ed.): Mathematical aspectes of finite elements in partial differential equations. Academic Press 1874, pp. 125–145.
HASLTNGER, J.: Dual finite element analysis for an inequality of the 2nd order. Aplikace Matematiky 24 (1979) 118–132.
HLAVáČEK, I.: Dual finite element analysis for unilateral boundary value problems. Aplikace Matematiky 22 (1977) 14–51.
HLAVáČEK, I.: Convergence of dual finite element approximations for unilateral boundary value problems. Aplikace matematiky 25 (1980) 375–386.
QUARTERONI, A.: On mixed methods for fourth-order problems. Computer methods in applied mechanics and engineering 24 (1980) 13–34.
RANNACHER, R.: On nonconforming and mixed finite element methods for plate bending problems. The linear case. RAIRO Numer. Anal. 13 (1979) 369–387.
SCAPOLA, T. : A mixed finite element method for the biharmonic problem. R.A.I.R.O. Anal. Numér. 14 (1980) 55–79.
SCHOLZ, R.: Approximation von Sattelpunkten mit finiten Elementen. Bonner Math. Schriften 89 (1976) 53–66.
SCHOLZ, R.: A mixed method for 4th order problems using linear finite elements. R.A.I.R.O. Anal. Numer. 12 (1978) 85–90.
WUNDERLICH, W.: Mixed models for plates and shells: Principles — Elements -Examples. S. 215–241 in: S.N. Atluri, R.H. Gallagher, O.C. Zienkiewicz (ed.): Hybrid and finite element methods. John Wiley 1983.
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Velte, W. (1988). Complementary Variational Principles. In: Stein, E., Wendland, W. (eds) Finite Element and Boundary Element Techniques from Mathematical and Engineering Point of View. CISM International Centre for Mechanical Sciences, vol 301. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2826-8_1
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DOI: https://doi.org/10.1007/978-3-7091-2826-8_1
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