Semi-implicit and Newton-like finite element methods are developed for the stationary
compressible Euler equations. The Galerkin discretization of the inviscid
fluxes is potentially oscillatory and unstable. To suppress numerical oscillations,
the spatial discretization is performed by a high-resolution finite element scheme
based on algebraic flux correction. A multidimensional limiter of TVD type is
employed. An important goal is the efficient computation of stationary solutions
in a wide range of Mach numbers, which is a challenging task due to oscillatory
correction factors associated with TVD-type flux limiters. A semi-implicit scheme
is derived by a time-lagged linearization of the nonlinear residual, and a Newtonlike
method is obtained in the limit of infinite CFL numbers. Special emphasis is
laid on the numerical treatment of weakly imposed characteristic boundary conditions.
It is shown that the proposed approach offers unconditional stability as
well as higher accuracy and better convergence behavior than in the case of algorithms
in which the boundary conditions are implemented in a strong sense.
Keywords: compressible Euler equations, steady-state solutions, finite elements,
high-resolution schemes, algebraic flux correction, boundary conditions