Analysis of the Finite Element Variational Crimes in the Numerical Approximation of Transonic Flow

The paper presents a detailed theory of the finite element approximations of two-dimensional transonic potential flow. We consider the boundary value problem for the full potential equation in a general bounded domain fi with mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed as usual in practice: the domain Q is approximated by a polygonal domain, conforming piecewise linear triangular elements are used, and the integrals are evaluated by numerical quadratures. Using a new version of entropy compactification of transonic flow and the theory of finite element variational crimes for nonlinear elliptic problems, we prove the convergence of approximate solutions to the exact physical solution of the continuous problem, provided its existence can be shown. Introduction The investigation of transonic flow represents a very interesting part of fluid dynamics, both from physical and mathematical points of view. The interest resides in specific phenomena in high-speed gas flow and in the character of equations describing transonic flow. Although transonic flow problems play an extremely important role in the design of high-speed airplanes, turbomachines, and compressors, the fundamental general questions concerning the existence and uniqueness of solutions are still open. Some results in this direction were obtained, e.g., by DiPerna [11], Morawetz [29], and Feistauer, Mandel, and Ñecas [15, 16, 17, 18, 32]. The publications [15, 16, 32] emphasize the importance of the second law of thermodynamics represented as an entropy condition; in [11, 17, 18, 29] the viscosity method is studied. In contrast to the lack of theoretical results there exists a series of methods for the simulation of various types of transonic flow. Here we shall deal with the numerical solution of the transonic flow model based on the full potential equation. Most numerical methods for the solution of transonic potential flow use finite differences, upwinding in the density and line relaxation, and often apply Received by the editor October 25, 1990 and, in revised form, March 19, 1992. 1991 Mathematics Subject Classification. Primary 65N30; Secondary 76M10, 76H05.