Analysis of the finite element variational crimes in the numerical approximation of transonic flow

Abstract

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 Ω \Omega with mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed as usual in practice: the domain Ω \Omega 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.