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Additional info for Advances in Chemical Physics: Proceedings of the Conference on Instability and Dissipative Strutures in Hydrodynamics, Volume 32
A basis given by the Cartesian product of the +i’s and ti’s discussed in Section IV was used. Thus, at each interior node, we have four degrees of freedom ( P , P,,, P , y , and P,,). At the upper and lower boundaries, P is known and P , ,SO (actually symmetry abouty = O was used to cut the number of unknowns in about half). SOME REMARKS ON VARIATIONAL METHODS 49 Y ~ Fig. 6. The pressure was initially assumed to be identically equal to 1 and the local potential iterative scheme was used. Figure 7 shows a typical converged solution.
Unpublished work; 1968; see also J. L. Robinson, J. Fluid Mech, 30,579 (1967). 8. J. W. Deardorff, J. Afmos. , 21, 419 (1964). J. W. Deardorff, J. Amos. , 22, 419 (1965). J. E. Fromm, Phys. Fluids, 8 1757 (1965). T. D. Foster, J. , 37, 81 (1969). G. Veronis, J. , 26, 49 (1966). F. B. Lipps, and R. C. J. Somerville, Phys. Fluids, 14, 759 (1971). 9. L. N. Howard, J. Fluid Mech, 17, 405 (1963). L. N. Howard, Ann. Rev. Fluid Mech. 4, 473 (1972). See also F. H. Angew. Math. Phys. 20, 1 (1969a); F.
Clearly any piecewise linear approximation to may be written as + The connection with other approximate methods derived from variational principle is now clear-the finite element method can be considered a special case of the Ritz-Galerkin method. , interface problems) effectively. Moreover higher order element methods yield extremely accurate approximate solutions22with relatively few degrees of freedom. SOME REMARKS ON VARIATlONAL METHODS 43 S ELEMENT NUMBER K EXTERNAL NODE NUMBER (b) Flg. 1.
Advances in Chemical Physics: Proceedings of the Conference on Instability and Dissipative Strutures in Hydrodynamics, Volume 32