Stationary Variational Principle Research Articles

Existence and uniqueness of solutions to a stationary finite element model of the biharmonic equation

This paper describes the finite element approximations based on a stationary variational principle of the biharmonic equation. Independent approximations for the solution and its second derivatives are used in the element: the normal derivatives of the solution and its second derivatives are approximated independently on the element boundary. Thus, the present model is a ‘mixed-hybrid’ model. Existence and uniqueness of solutions to the exact weak (or variational) problem are established and the associated finite element approximations are described. Existence and uniqueness of the finite element solutions are proved and a priori error estimates are given.

Approximate solutions of the Liouville equation. II. Stationary variational principles

Stationary variational functionals for the Laplace transform of the Liouville distribution are constructed. The value of the functional is the autocorrelation function that one wishes to compute. It is shown that the functionals may be transformed to a renormalized form. Trial functions not involving the potential explicitly give rise to time-dependent autocorrelation functions determined only by equilibrium spatial correlation functions. Another class of functionals is constructed by independently varying the parity symmetric and antisymmetric parts of the distribution function. Trial functions need only be assumed for one of these—the optimum value of the other one is given exactly. This procedure is used to improve the simplest known theories for velocity and density autocorrelation functions.

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.

On the convergence of a mixed finite-element method for plate bending problems

In this paper we justify a finite element method for biharmonic boundary value problems. The method is based on a stationary variational principle (the Reissner principle), and was introduced by Hellan, Herrmann and Visser. We prove error estimates and the existence of a finite element solution.

Variational Derivation of Discrete Ordinate-Like Approximations

The derivation of discrete ordinate and discrete ordinate-like approximations from variational principles for the one-speed transport equation is explored here. Standard discrete ordinate approximations are derived from a first-order stationary variational principle. The derivation yields a prescription for ordinates to be used given a selection of weights. Resultant quadrature schemes are compared numerically with those in common use. These new schemes derived using the weights of SN quadratures do not show significant variations in performance from the parent SN schemes. In the second portion of the paper, a new “modified” discrete ordinate approximation, MDN, is found by applying the same techniques as in the derivation of the standard approximation, this time, however, using an extremum second-order variational principle. The new approximation is compared through several numerical examples with standard discrete ordinate, simplified PN, and standard PN approximations. The MDN results do show a mitigation of the ray effects associated with standard discrete ordinate calculations (DN), but for gross region-wise absorption rates its accuracy for low orders is more like that of simplified PN rather than of PN or DN approximations. It is concluded that a low-order MDN approximation should not be a candidate to replace diffusion theory. The approximation may, however, have some application as a calculational standard.