Alternative variational formulations for first order partial differential systems
Author:
Nima Geffen
Journal:
Quart. Appl. Math. 41 (1983), 245-252
MSC:
Primary 49B22; Secondary 35Q20, 49H05
DOI:
https://doi.org/10.1090/qam/719508
MathSciNet review:
719508
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Abstract: Two simple alternative variational principles are derived for a first order differential system with appropriate initial and boundary conditions. The problem is assumed to be well posed, and may be nonlinear, nonhomogeneous and of any type (i.e. elliptic, hyperbolic or mixed). Primitive variables are used, which allows for non-smooth solutions. Redundancy in the system is considered, and applications to fluidynamics and electrodynamics fields given.
L. D. Landau and E. M. Lifschitz, Mechanics and electrodynamics, a shorter of theoretical physics, Vol. 1, Pergamon Press, 1st ed., 1972
- Herbert Goldstein, Classical mechanics, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1980. Addison-Wesley Series in Physics. MR 575343
R. P. Feymann, Lectures on physics, California Institute of Technology Notes, 1965, The principle of least action, pp. 19-1–19-14
R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, 1965
N. Geffen, Finite elements for fluidynamics, Final Scientific Report, Grant No. AFOSR-77-3345, 15–18, July 1979
Milton E. Rose, A numerical scheme to solve $div u = 0$, $curl u = \zeta$, ICASE Report No. 82-8, April 7, 1982
O. Buneman, Ideal gas dynamics in Hamiltonian form with benefit for numerical schemes, Phys. Fluids, to be published; also a presentation at the 7th International Meeting on Numerical Methods in Fluid Dynamics, Stanford University, June 1980
- Nima Geffen, A variational formulation for constrained quasilinear vector systems, Quart. Appl. Math. 35 (1977/78), no. 3, 375–381. MR 459268, DOI https://doi.org/10.1090/S0033-569X-1977-0459268-3
- Nima Geffen, Variational formulations for nonlinear wave propagation and unsteady transonic flow, Z. Angew. Math. Phys. 28 (1977), no. 6, 1037–1043 (English, with German summary). MR 468600, DOI https://doi.org/10.1007/BF01601671
H. A. Meng and R. H. Gallagher, A critical assessment of the simplified hybrid displacement method, Internat. J. Numerical Methods in Engineering, 11, 145–167 (1977), Further comments on the simplified hybrid displacement method, a discussion on the paper above, Ibid., 12, 1457–1484 (1978)
E. Haugeneder and H. A. Meng, Admissible and inadmissible simplifications of variational methods in finite element analysis, presented at the IUTAM Symposium on Variational Methods in Mechanics of Solids, Northwestern Univ., Evanston, Ill., Sept. 11–13, 1978
- E. Haugeneder and H. A. Mang, On an improper modification of a variational principle for finite element plate analysis, Z. Angew. Math. Mech. 59 (1979), no. 11, 637–640. MR 563449, DOI https://doi.org/10.1002/zamm.19790591106
L. D. Landau and E. M. Lifschitz, Mechanics and electrodynamics, a shorter of theoretical physics, Vol. 1, Pergamon Press, 1st ed., 1972
H. Goldstein, Classical mechanics, Addison-Wesley, Second Edition, 1980
R. P. Feymann, Lectures on physics, California Institute of Technology Notes, 1965, The principle of least action, pp. 19-1–19-14
R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, 1965
N. Geffen, Finite elements for fluidynamics, Final Scientific Report, Grant No. AFOSR-77-3345, 15–18, July 1979
Milton E. Rose, A numerical scheme to solve $div u = 0$, $curl u = \zeta$, ICASE Report No. 82-8, April 7, 1982
O. Buneman, Ideal gas dynamics in Hamiltonian form with benefit for numerical schemes, Phys. Fluids, to be published; also a presentation at the 7th International Meeting on Numerical Methods in Fluid Dynamics, Stanford University, June 1980
Nima Geffen, A variational formulation for constrained quasilinear vector systems, Quart. Appl. Math. 375–381, October 1977
---, Variational formulations for nonlinear wave propagation and unsteady transonic flow, ZAMP, 28, 1038–1043, 1977
H. A. Meng and R. H. Gallagher, A critical assessment of the simplified hybrid displacement method, Internat. J. Numerical Methods in Engineering, 11, 145–167 (1977), Further comments on the simplified hybrid displacement method, a discussion on the paper above, Ibid., 12, 1457–1484 (1978)
E. Haugeneder and H. A. Meng, Admissible and inadmissible simplifications of variational methods in finite element analysis, presented at the IUTAM Symposium on Variational Methods in Mechanics of Solids, Northwestern Univ., Evanston, Ill., Sept. 11–13, 1978
---, On an improper modification of a variational principle for finite element plate analysis, ZAMM 59, 637–540 (1979)
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Article copyright:
© Copyright 1983
American Mathematical Society
