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Applications of the finite element method to problems of unsteady fluid flow was presented for a restricted class of problems in potential flow by Visser. 1 This was suggested as a treatment for general transient problems and was elaborated by Zienkiewicz and Cheung. 2) Attempts to formulate a more general class of the problems have recently been presented; for viscous incompressible fluid flow3))7); for general compressible Newtonian fluid flow8))10). Applications of the method to a general class of field equations was recently presented from a view point of applied mathematics11). The above papers may be classified into three main procedures from a view point of methodology of the finite element formulation; via the Ritz method1),2) which is used only when a classical extremal principle exists; via a kind of virtual work principle3),4); via the method of weighted residuals, the Galerkin,s method5))11). These recent papers which especially come under the last procedure demonstrate that the formulation of the finite element can be based on the governing differential equations directly or on the global form of the law of conservation of energy indirectly via the method, when an appropriate variational principle does not exist in the classical sense. On the other hand, new methods of formulating variational principles based on the concept of the local potential was originated by Glansdorff and Prigogine12). This technics has been applied in various fields of hydrodynamics by Glansdorff and Prigogine13), Schechter and Himmelblau14), Takagi,5) and others. An important application of the variational principle is as a means of developing methods to obtain approximate solutions. The extremal principles based on the concept of the local potential, of course, can be used as the basis for developing approximation methods. One is the finite-difference method,s and the other is the direct methods. Schechter17) has suggested that all of the direct methods, that is, the Ritz method, solution by partial integration, eigenvalues and eigenfunctions and others which are used for classical extremal principles are applicable to the local potential and pointed out that the direct methods using the local potential involve the Galerkin,s method with the added advantage of treating the boundary conditions directly, though they do not give clear upper or lower bounds of the solution. However, these direct methods have been applied to the local potential for the whole region of interest. Therefore, it seems difficult to apply the methods to the problems which involve complex geometries and boundary conditions. The finite element method is one procedure to overcome such a difficulty. Unfortunately, however, the concept of the local potential has been considered18) to play no role in finite element formulations by reason that the concept falls outside of the range of validity ap-

THE LOCAL POTENTIAL APPROACH TO FINITE ELEMENT METHOD IN UNSTEADY VISCOUS INCOMPRESSIBLE FLUID FLOW